View source for Topological insulator

{{Short description|State of matter with insulating bulk but conductive boundary}}{{science review|date=February 2024}}
{{Use American English|date=January 2019}}
[[File:Topological insulator informal phase diagram-en.svg|thumb|An (informal) phase diagram with topological insulators, trivial insulators, and conductors. There is no path from the topological insulators to the trivial insulators that does not cross the conducting phase. The diagram depicts a <math>\mathbb{Z}_2</math>topological invariant, since there are two "islands" of insulators.]]
[[File:Topological insulator band structure.svg|thumb|An idealized [[band structure]] for a 3D time-reversal symmetric topological insulator. The [[Fermi level]] falls within the bulk band gap which is traversed by topologically-protected spin-textured Dirac surface states.<ref name=":0">{{Cite journal|last=Moore|first=Joel E.|date=2010|title=The birth of topological insulators|journal=Nature|language=En|volume=464|issue=7286|pages=194–198|doi=10.1038/nature08916|issn=0028-0836|pmid=20220837|bibcode=2010Natur.464..194M|s2cid=1911343}}</ref><ref name="HM">{{Cite journal|last1=Hasan|first1=M.Z. |last2=Moore|first2=J.E.  |date=2011|title=Three-Dimensional Topological Insulators|journal=Annual Review of Condensed Matter Physics|language=En|volume=2|pages=55–78|doi=10.1146/annurev-conmatphys-062910-140432|bibcode=2011ARCMP...2...55H|arxiv=1011.5462|s2cid=11516573 }}</ref>]]
A '''topological insulator''' is a material whose interior behaves as an [[electrical insulator]] while its surface behaves as an [[Electrical conductivity|electrical conductor]],<ref name="KM">{{Cite journal|last = Kane | first = C. L.  |author2=Mele, E. J.   | year = 2005 | title = Z<sub>2</sub> Topological Order and the Quantum Spin Hall Effect | journal = [[Physical Review Letters]] | volume = 95 | issue = 14 | pages = 146802 | doi = 10.1103/PhysRevLett.95.146802 | pmid = 16241681  | bibcode=2005PhRvL..95n6802K|arxiv = cond-mat/0506581 | s2cid = 1775498 }}</ref> meaning that electrons can only move along the surface of the material.

A topological insulator is an insulator for the same reason a "[[Triviality (mathematics)|trivial]]" (ordinary) insulator is: there exists an [[Band gap|energy gap]] between the [[valence and conduction bands]] of the material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator.<ref>{{Cite journal |last1=Zhu |first1=Zhiyong |last2=Cheng |first2=Yingchun |last3=Schwingenschlögl |first3=Udo |date=2012-06-01 |title=Band inversion mechanism in topological insulators: A guideline for materials design |url=https://link.aps.org/doi/10.1103/PhysRevB.85.235401 |journal=Physical Review B |language=en |volume=85 |issue=23 |pages=235401 |doi=10.1103/PhysRevB.85.235401 |bibcode=2012PhRvB..85w5401Z |hdl=10754/315777 |issn=1098-0121|hdl-access=free }}</ref> The topological insulator cannot be [[Continuous function|continuously]] transformed into a trivial one without untwisting the bands, which closes the band gap and creates a conducting state. Thus, due to the continuity of the underlying field, the [[Surface state|border]] of a topological insulator with a trivial insulator (including [[vacuum]], which is topologically trivial) is forced to support a conducting state.<ref name=":10">{{Cite journal |last1=Qi |first1=Xiao-Liang |last2=Zhang |first2=Shou-Cheng |date=2011-10-14 |title=Topological insulators and superconductors |url=https://link.aps.org/doi/10.1103/RevModPhys.83.1057 |journal=Reviews of Modern Physics |language=en |volume=83 |issue=4 |pages=1057–1110 |doi=10.1103/RevModPhys.83.1057 |arxiv=1008.2026 |bibcode=2011RvMP...83.1057Q |s2cid=118373714 |issn=0034-6861}}</ref>

Since this results from a global property of the topological insulator's [[band structure]], local (symmetry-preserving) perturbations cannot damage this surface state.<ref name=":11">{{Cite journal |last1=Hasan |first1=M. Z. |last2=Kane |first2=C. L. |date=2010-11-08 |title=Colloquium: Topological insulators |url=https://link.aps.org/doi/10.1103/RevModPhys.82.3045 |journal=Reviews of Modern Physics |volume=82 |issue=4 |pages=3045–3067 |doi=10.1103/RevModPhys.82.3045|bibcode=2010RvMP...82.3045H |s2cid=16066223 |arxiv=1002.3895 }}</ref> This is unique to topological insulators: while ordinary insulators can also support conductive surface states, only the surface states of topological insulators have this robustness property.

This leads to a more formal definition of a topological insulator: an insulator which cannot be [[Adiabatic theorem|adiabatically]] transformed into an ordinary insulator without passing through an intermediate conducting state.<ref name=":10" /> In other words, topological insulators and trivial insulators are separate regions in the [[phase diagram]], connected only by conducting phases. In this way, topological insulators provide an example of a [[state of matter]] not described by the [[Landau theory|Landau symmetry-breaking theory]] that defines ordinary states of matter.<ref name=":11" />

The properties of topological insulators and their surface states are highly dependent on both the dimension of the material and its underlying [[Symmetry in quantum mechanics|symmetries]], and can be classified using the so-called [[Periodic table of topological invariants|periodic table of topological insulators]]. Some combinations of dimension and symmetries forbid topological insulators completely.<ref name=":12">{{Cite journal |last=Kitaev |first=Alexei |date=2009-05-14 |title=Periodic table for topological insulators and superconductors |url=https://aip.scitation.org/doi/abs/10.1063/1.3149495 |journal=AIP Conference Proceedings |volume=1134 |issue=1 |pages=22–30 |doi=10.1063/1.3149495 |arxiv=0901.2686 |bibcode=2009AIPC.1134...22K |s2cid=14320124 |issn=0094-243X}}</ref> All topological insulators have at least [[U(1)|U(1) symmetry]] from particle number conservation, and often have [[time-reversal symmetry]] from the absence of a magnetic field. In this way, topological insulators are an example of [[symmetry-protected topological order]].<ref>{{Cite journal |last=Senthil |first=T. |date=2015-03-01 |title=Symmetry-Protected Topological Phases of Quantum Matter |url=https://www.annualreviews.org/doi/10.1146/annurev-conmatphys-031214-014740 |journal=Annual Review of Condensed Matter Physics |language=en |volume=6 |issue=1 |pages=299–324 |doi=10.1146/annurev-conmatphys-031214-014740 |arxiv=1405.4015 |bibcode=2015ARCMP...6..299S |s2cid=12669555 |issn=1947-5454}}</ref> So-called "topological invariants", taking values in <math>\mathbb{Z}_2</math>&nbsp;or <math>\mathbb{Z}</math>, allow classification of insulators as trivial or topological, and can be computed by various methods.<ref name=":12" />

The surface states of topological insulators can have exotic properties. For example, in time-reversal symmetric 3D topological insulators, surface states have their [[Spin quantum number|spin]] locked at a right-angle to their momentum (spin-momentum locking). At a given energy the only other available electronic states have different spin, so [[Umklapp scattering|"U"-turn scattering]] is strongly suppressed and conduction on the surface is highly metallic.

Despite their origin in quantum mechanical systems, analogues of topological insulators can also be found in classical media. There exist [[Photonic topological insulator|photonic]],<ref>{{Cite journal |last1=Khanikaev |first1=Alexander B. |last2=Hossein Mousavi |first2=S. |last3=Tse |first3=Wang-Kong |last4=Kargarian |first4=Mehdi |last5=MacDonald |first5=Allan H. |last6=Shvets |first6=Gennady |date=March 2013 |title=Photonic topological insulators |url=https://www.nature.com/articles/nmat3520 |journal=Nature Materials |language=en |volume=12 |issue=3 |pages=233–239 |doi=10.1038/nmat3520 |pmid=23241532 |arxiv=1204.5700 |bibcode=2013NatMa..12..233K |s2cid=39748656 |issn=1476-4660}}</ref> [[Magnetic topological insulators|magnetic]],<ref>{{Cite journal |last1=Tokura |first1=Yoshinori |last2=Yasuda |first2=Kenji |last3=Tsukazaki |first3=Atsushi |date=February 2019 |title=Magnetic topological insulators |url=https://www.nature.com/articles/s42254-018-0011-5 |journal=Nature Reviews Physics |language=en |volume=1 |issue=2 |pages=126–143 |doi=10.1038/s42254-018-0011-5 |bibcode=2019NatRP...1..126T |s2cid=53694955 |issn=2522-5820}}</ref> and [[Acoustics|acoustic]]<ref>{{Cite journal |last1=He |first1=Cheng |last2=Ni |first2=Xu |last3=Ge |first3=Hao |last4=Sun |first4=Xiao-Chen |last5=Chen |first5=Yan-Bin |last6=Lu |first6=Ming-Hui |last7=Liu |first7=Xiao-Ping |last8=Chen |first8=Yan-Feng |date=December 2016 |title=Acoustic topological insulator and robust one-way sound transport |url=http://www.nature.com/articles/nphys3867 |journal=Nature Physics |language=en |volume=12 |issue=12 |pages=1124–1129 |doi=10.1038/nphys3867 |arxiv=1512.03273 |bibcode=2016NatPh..12.1124H |s2cid=119255437 |issn=1745-2473}}</ref> topological insulators, among others.

==Prediction==
The first models of 3D topological insulators were proposed by B. A. Volkov and O. A. Pankratov in 1985,<ref name=Volkov85>{{Cite journal|last1=Volkov|first1=B. A.|last2=Pankratov|first2=O. A.|date=1985-08-25|title=Two-dimensional massless electrons in an inverted contact |url=http://jetpletters.ru/ps/1420/article_21570.shtml|journal=JETP Letters|language=en|volume=42|issue=4|pages=178–181}}</ref> 
and subsequently by Pankratov, S. V. Pakhomov, and Volkov in 1987.<ref name=Pankratov87>{{Cite journal|last1=Pankratov|first1=O. A.|last2=Pakhomov|first2=S. V.|last3=Volkov|first3=B. A.|date=1987-01-01|title=Supersymmetry in heterojunctions: Band-inverting contact on the basis of Pb1-xSnxTe and Hg1-xCdxTe|url=https://dx.doi.org/10.1016/0038-1098%2887%2990934-3|journal=Solid State Communications|language=en|volume=61|issue=2|pages=93–96|doi=10.1016/0038-1098(87)90934-3|bibcode=1987SSCom..61...93P|issn=0038-1098}}</ref> Gapless 2D Dirac states were shown to exist at the band inversion contact in [[PbTe]]/[[SnTe]]<ref name=Volkov85/> and [[HgTe]]/[[CdTe]]<ref name=Pankratov87/> heterostructures. 
Existence of interface Dirac states in HgTe/CdTe was experimentally verified by [[Laurens W. Molenkamp|Laurens W. Molenkamp's]] group in 2D topological insulators in 2007.<ref name="König07">{{Cite journal|last1=König|first1=Markus|last2=Wiedmann|first2=Steffen|last3=Brüne|first3=Christoph|last4=Roth|first4=Andreas|last5=Buhmann|first5=Hartmut|last6=Molenkamp|first6=Laurens W.|last7=Qi|first7=Xiao-Liang|last8=Zhang|first8=Shou-Cheng|date=2007-11-02|title=Quantum Spin Hall Insulator State in HgTe Quantum Wells|url=https://www.science.org/doi/10.1126/science.1148047|journal=Science|language=en|volume=318|issue=5851|pages=766–770|doi=10.1126/science.1148047|issn=0036-8075|pmid=17885096|arxiv=0710.0582|bibcode=2007Sci...318..766K|s2cid=8836690}}</ref>

Later sets of theoretical models for the 2D topological insulator (also known as the quantum spin Hall insulators) were proposed by [[Charles L. Kane]] and [[Eugene J. Mele]] in 2005,<ref>{{Cite journal|last1=Kane|first1=C. L.|last2=Mele|first2=E. J.|date=2005-11-23|title=Quantum Spin Hall Effect in Graphene|url=https://link.aps.org/doi/10.1103/PhysRevLett.95.226801|journal=Physical Review Letters|volume=95|issue=22|pages=226801|doi=10.1103/PhysRevLett.95.226801|pmid=16384250|arxiv=cond-mat/0411737|bibcode=2005PhRvL..95v6801K|s2cid=6080059}}</ref> and also by [[B. Andrei Bernevig]]  and [[Shoucheng Zhang]] in 2006.<ref>{{Cite journal|last1=Bernevig|first1=B. Andrei|last2=Zhang|first2=Shou-Cheng|date=2006-03-14|title=Quantum Spin Hall Effect|url=https://link.aps.org/doi/10.1103/PhysRevLett.96.106802|journal=Physical Review Letters|volume=96|issue=10|pages=106802|doi=10.1103/PhysRevLett.96.106802|pmid=16605772|arxiv=cond-mat/0504147|bibcode=2006PhRvL..96j6802B|s2cid=2618285}}</ref> The <math>\mathbb{Z}_2</math> topological invariant was constructed and the importance of the time reversal symmetry was clarified in the work by Kane and Mele.<ref>{{Cite journal|last1=Kane|first1=C. L.|last2=Mele|first2=E. J.|date=2005-09-28|title=${Z}_{2}$ Topological Order and the Quantum Spin Hall Effect|url=https://link.aps.org/doi/10.1103/PhysRevLett.95.146802|journal=Physical Review Letters|volume=95|issue=14|pages=146802|doi=10.1103/PhysRevLett.95.146802|pmid=16241681|arxiv=cond-mat/0506581|bibcode=2005PhRvL..95n6802K |s2cid=1775498}}</ref> Subsequently, Bernevig, Taylor L. Hughes and Zhang made a theoretical prediction that 2D topological insulator with one-dimensional (1D) helical edge states would be realized in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride.<ref>{{Cite journal|last1=Bernevig|first1=B. Andrei|last2=Hughes|first2=Taylor L.|last3=Zhang|first3=Shou-Cheng|date=2006-12-15|title=Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells|url=https://www.science.org/doi/10.1126/science.1133734|journal=Science|language=en|volume=314|issue=5806|pages=1757–1761|doi=10.1126/science.1133734|issn=0036-8075|pmid=17170299|arxiv=cond-mat/0611399|bibcode=2006Sci...314.1757B|s2cid=7295726}}</ref> The transport due to 1D helical edge states was indeed observed in the experiments by Molenkamp's group in 2007.<ref name="König07"/>

Although the topological classification and the importance of time-reversal symmetry was pointed in the 2000s, all the necessary ingredients and physics of topological insulators were already understood in the works from the 1980s.

In 2007, it was predicted that 3D topological insulators might be found in binary compounds involving [[bismuth]],<ref>{{Cite journal| doi = 10.1103/PhysRevB.79.195322 | volume = 79 | pages = 195322 | last = Roy | first = Rahul  | title = Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect| journal = Physical Review B | date = 2009-05-21 |arxiv = cond-mat/0607531 | s2cid = 119407081 }}</ref><ref>{{Cite journal| doi = 10.1103/PhysRevLett.98.106803 | volume = 98 | issue =  10| pages = 106803 | author1= Liang Fu|author2= C. L. Kane|author3=E. J. Mele  | title = Topological insulators in three dimensions| journal = Physical Review Letters  | date = 2007-03-07 | pmid = 17358555 |arxiv = cond-mat/0607699| bibcode = 2007PhRvL..98j6803F | s2cid = 6037351 }}</ref><ref>{{Cite journal| doi = 10.1103/PhysRevB.76.045302 | volume = 76 | issue = 4 | pages = 045302 | last = Fu | first = Liang |author2=C. L. Kane  | title = Topological insulators with inversion symmetry | journal = Physical Review B | date = 2007-07-02 |arxiv = cond-mat/0611341 |bibcode = 2007PhRvB..76d5302F | s2cid = 15011491 }}</ref><ref>{{Cite journal| doi = 10.1088/1367-2630/9/9/356 | issn = 1367-2630 | volume = 9 | issue = 9 | pages = 356 | last = Shuichi Murakami | title = Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase | journal = New Journal of Physics | year = 2007 |arxiv = 0710.0930 |bibcode = 2007NJPh....9..356M | s2cid = 13999448 }}</ref> and in particular "strong topological insulators" exist that cannot be reduced to multiple copies of the [[Quantum spin Hall effect|quantum spin Hall state]].<ref>{{Cite journal| last = Kane | first = C. L. |author2=Moore, J. E.  | year = 2011 | title = Topological Insulators | journal = [[Physics World]] | volume =  24 | issue =   2| pages = 32–36 | doi =  10.1088/2058-7058/24/02/36| url = http://physics.gmu.edu/~pnikolic/articles/Topological%20insulators%20(Physics%20World,%20February%202011).pdf | bibcode = 2011PhyW...24b..32K }}</ref>

== Experimental realization ==
2D Topological insulators were first realized in system containing [[HgTe]] quantum wells sandwiched between [[cadmium telluride]] in 2007.

The first 3D topological insulator to be realized experimentally was [[Bismuth antimonide|Bi<sub>1 − x</sub> Sb <sub>x</sub>]].<ref name=":1">{{Cite journal |last1=Fu |first1=Liang |last2=Kane |first2=C. L. |date=2007-07-02 |title=Topological insulators with inversion symmetry |journal=Physical Review B |volume=76 |issue=4 |pages=045302 |arxiv=cond-mat/0611341 |bibcode=2007PhRvB..76d5302F |doi=10.1103/PhysRevB.76.045302 |s2cid=15011491}}</ref><ref name=":2">{{Cite journal|last1=Hasan|first1=M. Zahid|last2=Moore|first2=Joel E.|date=2011|title=Three-Dimensional Topological Insulators|journal=Annual Review of Condensed Matter Physics|language=en|volume=2|issue=1|pages=55–78|doi=10.1146/annurev-conmatphys-062910-140432|issn=1947-5454|arxiv=1011.5462|bibcode=2011ARCMP...2...55H|s2cid=11516573}}</ref><ref>{{Cite journal| doi = 10.1038/nature06843 | last = Hsieh | first = David | author2=Dong Qian |author3=Andrew L. Wray |author4=Yuqi Xia |author5=Yusan Hor |author6=Robert Cava |author7=M. Zahid Hasan |author-link7=M. Zahid Hasan | title = A Topological Dirac insulator in a quantum spin Hall phase | journal = Nature | volume = 452 | issue = 9 | pages = 970–4 | year = 2008 | pmid = 18432240 | bibcode=2008Natur.452..970H|arxiv = 0902.1356 | s2cid = 4402113 }}</ref> Bismuth in its pure state, is a [[semimetal]] with a small electronic band gap. Using [[angle-resolved photoemission spectroscopy]],  and many other measurements, it was observed that Bi<sub>1 − x</sub>Sb<sub>x</sub> alloy exhibits an odd [[surface state]] (SS) crossing between any pair of [[Kramers theorem|Kramers]] points and the bulk features massive Dirac fermions.<ref name=":2" />  Additionally, bulk Bi<sub>1 − x</sub>Sb<sub>x</sub> has been predicted to have 3D [[Dirac particle]]s.<ref>{{Cite journal|last=Buot|first=F. A.|date=1973-09-01|title=Weyl Transform and the Magnetic Susceptibility of a Relativistic Dirac Electron Gas|journal=Physical Review A|volume=8|issue=3|pages=1570–81|doi=10.1103/PhysRevA.8.1570|bibcode=1973PhRvA...8.1570B}}</ref>  This prediction is of particular interest due to the observation of charge [[Fractional quantum Hall effect|quantum Hall fractionalization]] in 2D graphene<ref>{{Cite journal|last1=Kane|first1=C. L.|last2=Mele|first2=E. J.|date=2005-11-23|title=Quantum Spin Hall Effect in Graphene|journal=Physical Review Letters|volume=95|issue=22|pages=226801|doi=10.1103/PhysRevLett.95.226801|pmid=16384250|arxiv=cond-mat/0411737|bibcode=2005PhRvL..95v6801K|s2cid=6080059}}</ref> and pure bismuth.<ref>{{Cite journal|last1=Behnia|first1=Kamran|last2=Balicas|first2=Luis|last3=Kopelevich|first3=Yakov|date=2007-09-21|title=Signatures of Electron Fractionalization in Ultraquantum Bismuth|journal=Science|language=en|volume=317|issue=5845|pages=1729–31|doi=10.1126/science.1146509|issn=0036-8075|pmid=17702909|arxiv=0802.1993|bibcode=2007Sci...317.1729B|s2cid=15306515}}</ref>

Shortly thereafter symmetry-protected surface states were also observed in pure [[antimony]], [[bismuth selenide]], [[bismuth telluride]] and [[antimony telluride]] using [[angle-resolved photoemission spectroscopy]] (ARPES).<ref>{{Cite journal| doi = 10.1103/RevModPhys.82.3045| volume = 82| issue = 4| pages = 3045–67| last = Hasan| first = M. Zahid|author2=Kane, Charles L.  | title = Topological Insulators| journal = Reviews of Modern Physics| year = 2010| bibcode=2010RvMP...82.3045H|arxiv = 1002.3895 | s2cid = 16066223| url = https://repository.upenn.edu/physics_papers/39}}</ref><ref name=Hsieh09>{{Cite journal|last1=Hsieh|first1=D.|last2=Xia|first2=Y.|last3=Qian|first3=D.|last4=Wray|first4=L. |display-authors=etal |date=2009|title=A tunable topological insulator in the spin helical Dirac transport regime|url=https://www.nature.com/articles/nature08234|journal=Nature|language=en|volume=460|issue=7259|pages=1101–5|doi=10.1038/nature08234|issn=1476-4687|pmid=19620959|arxiv=1001.1590|bibcode=2009Natur.460.1101H|s2cid=4369601}}</ref><ref>{{Cite journal|last1=Hsieh|first1=D.|last2=Xia|first2=Y.|last3=Wray|first3=L.|last4=Qian|first4=D.|last5=Pal|first5=A.|last6=Dil|first6=J. H.|last7=Osterwalder|first7=J.|last8=Meier|first8=F.|last9=Bihlmayer|first9=G.|last10=Kane|first10=C. L. |display-authors=etal |date=2009|title=Observation of Unconventional Quantum Spin Textures in Topological Insulators|url=https://www.science.org/doi/10.1126/science.1167733|journal=Science|language=en|volume=323|issue=5916|pages=919–922|doi=10.1126/science.1167733|issn=0036-8075|pmid=19213915|bibcode=2009Sci...323..919H|s2cid=118353248|arxiv=0902.2617}}</ref><ref>{{Citation|last1=Hasan|first1=M. Zahid|title=Topological Insulators, Topological Dirac semimetals, Topological Crystalline Insulators, and Topological Kondo Insulators|date=2015|work=Topological Insulators|pages=55–100|publisher=John Wiley & Sons, Ltd|language=en|doi=10.1002/9783527681594.ch4|isbn=978-3-527-68159-4|last2=Xu|first2=Su-Yang|last3=Neupane|first3=Madhab}}</ref><ref name=":3">{{Cite journal|last1=Chen|first1=Xi|last2=Ma|first2=Xu-Cun|last3=He|first3=Ke|last4=Jia|first4=Jin-Feng|last5=Xue|first5=Qi-Kun|date=2011-03-01|title=Molecular Beam Epitaxial Growth of Topological Insulators|journal=Advanced Materials|language=en|volume=23|issue=9|pages=1162–5|doi=10.1002/adma.201003855|pmid=21360770|bibcode=2011AdM....23.1162C |s2cid=33855507|issn=0935-9648|doi-access=free}}</ref> and [[bismuth selenide]].<ref name=":3" /><ref>{{Cite journal|last1=Chiatti|first1=Olivio|last2=Riha|first2=Christian|last3=Lawrenz|first3=Dominic|last4=Busch|first4=Marco|last5=Dusari|first5=Srujana|last6=Sánchez-Barriga|first6=Jaime|last7=Mogilatenko|first7=Anna|last8=Yashina|first8=Lada V.|last9=Valencia|first9=Sergio|date=2016-06-07|title=2D layered transport properties from topological insulator {{chem|Bi|2|Se|3}} single crystals and micro flakes|journal=Scientific Reports|language=En|volume=6|issue=1|pages=27483|doi=10.1038/srep27483|pmid=27270569|pmc=4895388|arxiv=1512.01442|bibcode=2016NatSR...627483C|issn=2045-2322}}</ref> Many semiconductors within the large family of [[Heusler compound|Heusler material]]s are now believed to exhibit topological surface states.<ref>{{Cite journal| doi = 10.1038/nmat2770 | volume = 9 | issue = 7 | pages = 541–5 | last = Chadov | first = Stanislav |author2=Xiao-Liang Qi |author3=Jürgen Kübler |author4=Gerhard H. 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Zahid Hasan  | title = Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena | journal = Nat Mater | date = July 2010 | doi = 10.1038/nmat2771|arxiv = 1003.0155 |bibcode = 2010NatMa...9..546L }}</ref> In some of these materials, the Fermi level actually falls in either the conduction or valence bands due to naturally-occurring defects, and must be pushed into the bulk gap by [[Semiconductor#Doping|doping]] or gating.<ref>{{Cite journal| doi = 10.1103/PhysRevLett.103.146401| volume = 103| issue = 14| pages = 146401| last = Hsieh| pmid = 19905585| first = D. |author2=Y. Xia |author3=D. Qian |author4=L. Wray |author5=F. Meier |author6=J. H. Dil |author7=J. Osterwalder |author8=L. Patthey |author9=A. V. Fedorov |author10=H. Lin |author11=A. Bansil |author12=D. Grauer |author13=Y. S. Hor |author14=R. J. Cava |author15=M. Z. Hasan| title = Observation of Time-Reversal-Protected Single-Dirac-Cone Topological-Insulator States in {{chem|Bi|2|Te|3}} and {{chem|Sb|2|Te|3}}| journal = Physical Review Letters| year = 2009| bibcode=2009PhRvL.103n6401H| url = https://www.dora.lib4ri.ch/psi/islandora/object/psi%3A17462}}</ref><ref>{{Cite journal| doi = 10.1209/0295-5075/81/57006| volume = 81| issue = 5| pages = 57006| last = Noh| first = H.-J. |author2=H. Koh |author3=S.-J. Oh |author4=J.-H. Park |author5=H.-D. Kim |author6=J. D. Rameau |author7=T. Valla |author8=T. E. Kidd |author9=P. D. Johnson |author10=Y. Hu |author11=Q. Li | title = Spin-orbit interaction effect in the electronic structure of {{chem|Bi|2|Te|3}} observed by angle-resolved photoemission spectroscopy| journal = EPL| year = 2008|arxiv = 0803.0052 |bibcode = 2008EL.....8157006N | s2cid = 9282408}}</ref> The surface states of a 3D topological insulator is a new type of [[two-dimensional electron gas]] (2DEG) where the electron's spin is locked to its linear momentum.<ref name=Hsieh09/>

Fully bulk-insulating or intrinsic 3D topological insulator states exist in Bi-based materials as demonstrated in surface transport measurements.<ref name="NaturePhysics-9">{{cite journal|title=Observation of topological surface state quantum Hall effect in an intrinsic three-dimensional topological insulator|author= Xu, Y |author2=Miotkowski, I. |author3=Liu, C. |author4=Tian, J. |author5=Nam, H. |author6=Alidoust, N. |author7=Hu, J. |author8=Shih, C.-K |author9=Hasan, M.Z. |author10=Chen, Y.-P. |doi = 10.1038/nphys3140|volume=10|issue=12|journal=Nature Physics|pages=956–963|arxiv=1409.3778|bibcode=2014NatPh..10..956X|year=2014|s2cid= 51843826 }}</ref> In a new Bi based chalcogenide (Bi<sub>1.1</sub>Sb<sub>0.9</sub>Te<sub>2</sub>S) with slightly Sn - doping, exhibits an intrinsic semiconductor behavior with Fermi energy and Dirac point lie in the bulk gap and the surface states were probed by the charge transport experiments.<ref name="NatureComm">{{cite journal|title=Sn-doped Bi<sub>1.1</sub>Sb<sub>0.9</sub>Te<sub>2</sub>S bulk crystal topological insulator with excellent properties|journal= Nature Communications|author1= Kushwaha, S. K. |author2=Pletikosić, I. |author3=Liang, T. |author4=Gyenis, A. |author5=Lapidus, S. H. |author6=Yao, Tian |author7=Zhao He |author8= Burch, K. |author9=Jingjing Lin <!-- |author11= Wudi Wang |author12= Huiwen, J. |author13=Fedorov, A. V. |author14=Ali Yazdani |author15= Ong, N. P. |author16=Cava, R. J. --> |display-authors=3 |doi = 10.1038/ncomms11456|volume=7|page=11456|arxiv=1508.03655|year=2015|pmid= 27118032|pmc= 4853473}}</ref>

It was proposed in 2008 and 2009 that topological insulators are best understood not as surface conductors per se, but as bulk 3D magnetoelectrics with a quantized [[magnetoelectric]] effect.<ref>{{cite journal | last1=Qi | first1=Xiao-Liang | last2=Hughes | first2=Taylor L. | last3=Zhang | first3=Shou-Cheng | title=Topological field theory of time-reversal invariant insulators | journal=Physical Review B | publisher=American Physical Society (APS) | volume=78 | issue=19 | date=2008-11-24 | issn=1098-0121 | doi=10.1103/physrevb.78.195424 | page=195424|arxiv=0802.3537 | bibcode=2008PhRvB..78s5424Q| s2cid=117659977 }}</ref><ref>{{cite journal | last1=Essin | first1=Andrew M. | last2=Moore | first2=Joel E. | last3=Vanderbilt | first3=David | title=Magnetoelectric Polarizability and Axion Electrodynamics in Crystalline Insulators | journal=Physical Review Letters | volume=102 | issue=14 | date=2009-04-10 | issn=0031-9007 | doi=10.1103/physrevlett.102.146805 | page=146805| pmid=19392469 |arxiv=0810.2998| bibcode=2009PhRvL.102n6805E | s2cid=1133717 }}</ref>   This can be revealed by placing topological insulators in magnetic field. The effect can be described in language similar to that of the hypothetical [[Axion|axion particle]] of particle physics.<ref>{{cite journal | last=Wilczek | first=Frank | title=Two applications of axion electrodynamics | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=58 | issue=18 | date=1987-05-04 | issn=0031-9007 | doi=10.1103/physrevlett.58.1799 | pages=1799–1802| pmid=10034541 | bibcode=1987PhRvL..58.1799W }}</ref>   The effect was reported by researchers at [[Johns Hopkins University]] and [[Rutgers University]] using [[N. Peter Armitage|THz spectroscopy]] who showed that the Faraday rotation was quantized by the fine structure constant.<ref>{{cite journal | last1 = Wu | first1 = Liang | last2 = Salehi | first2 = M. | last3 = Koirala | first3 = N. | last4 = Moon | first4 = J. | last5 = Oh | first5 = S. | last6 = Armitage | first6 = N. P. | year = 2016 | title = Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator | journal = Science | volume = 354 | issue = 6316| pages = 1124–7 | doi=10.1126/science.aaf5541 | pmid = 27934759| arxiv = 1603.04317 | doi-access = free | bibcode = 2016Sci...354.1124W }}</ref>

In 2012, topological [[Kondo effect|Kondo]] insulators were identified in [[samarium hexaboride]], which is a bulk insulator at low temperatures.<ref>{{cite journal|last=Samuel Reich|first=Eugenie|year=2012|title=Hopes surface for exotic insulator: Findings by three teams may solve a 40-year-old mystery|journal=Nature|publisher=Springer Science and Business Media LLC|volume=492|issue=7428|pages=165|doi=10.1038/492165a|issn=0028-0836|pmid=23235853|bibcode=2012Natur.492..165S |doi-access=free}}</ref><ref>{{cite journal|last1=Dzero|first1=Maxim|last2=Sun|first2=Kai|last3=Galitski|first3=Victor|author-link3=Victor Galitski|last4=Coleman|first4=Piers|author-link4=Piers Coleman|date=2010-03-12|title=Topological Kondo Insulators|journal=Physical Review Letters|volume=104|issue=10|page=106408|arxiv=0912.3750|doi=10.1103/physrevlett.104.106408|issn=0031-9007|pmid=20366446|bibcode=2010PhRvL.104j6408D|s2cid=119270507}}</ref>

In 2014, it was shown that magnetic components, like the ones in [[spin-transfer torque|spin-torque computer memory]], can be manipulated by topological insulators.<ref>{{cite magazine|url=https://www.sciencenews.org/article/weird-materials-could-make-faster-computers |title=Weird materials could make faster computers |magazine=Science News |accessdate=2014-07-23 }}</ref><ref>{{cite journal | volume=511| issue=7510|doi=10.1038/nature13534 | pmid=25056062| title=Spin-transfer torque generated by a topological insulator | journal=Nature | pages=449–451| arxiv=1402.1124| bibcode=2014Natur.511..449M| last1=Mellnik| first1=A. R| last2=Lee| first2=J. S| last3=Richardella| first3=A| last4=Grab| first4=J. L| last5=Mintun| first5=P. J| last6=Fischer| first6=M. H| last7=Vaezi| first7=A| last8=Manchon| first8=A| last9=Kim| first9=E. -A| last10=Samarth| first10=N| last11=Ralph| first11=D. C| year=2014| s2cid=205239604}}</ref> The effect is related to [[metal–insulator transition]]s ([[Bose–Hubbard model]]).{{Citation needed|date=January 2018}}

=== Floquet topological insulators ===
Topological insulators are challenging to synthesize, and limited in topological phases accessible with solid-state materials.<ref>{{Cite journal |last=Ando |first=Yoichi |date=2013-10-15 |title=Topological Insulator Materials |url=http://dx.doi.org/10.7566/jpsj.82.102001 |journal=Journal of the Physical Society of Japan |volume=82 |issue=10 |pages=102001 |doi=10.7566/jpsj.82.102001 |arxiv=1304.5693 |bibcode=2013JPSJ...82j2001A |s2cid=55912821 |issn=0031-9015}}</ref> This has motivated the search for topological phases on the systems that simulate the same principles underlying topological insulators. Discrete time quantum walks (DTQW) have been proposed for making Floquet topological insulators (FTI). This&nbsp; periodically driven system simulates an effective ([[Floquet theory|Floquet]]) Hamiltonian that is topologically nontrivial.<ref>{{Cite journal |last1=Cayssol |first1=Jérôme |last2=Dóra |first2=Balázs |last3=Simon |first3=Ferenc |last4=Moessner |first4=Roderich |date=2013-01-28 |title=Floquet topological insulators |url=http://dx.doi.org/10.1002/pssr.201206451 |journal=Physica Status Solidi RRL |volume=7 |issue=1–2 |pages=101–108 |doi=10.1002/pssr.201206451 |arxiv=1211.5623 |bibcode=2013PSSRR...7..101C |s2cid=52082807 |issn=1862-6254}}</ref> This system replicates the effective Hamiltonians from all universal classes of 1- to 3-D topological insulators.<ref>{{Cite journal |last1=Kitaev |first1=Alexei |last2=Lebedev |first2=Vladimir |last3=Feigel'man |first3=Mikhail |date=2009 |title=Periodic table for topological insulators and superconductors |url=http://dx.doi.org/10.1063/1.3149495 |journal=AIP Conference Proceedings |volume=1134 |issue=1 |pages=22–30 |publisher=AIP |doi=10.1063/1.3149495|arxiv=0901.2686 |bibcode=2009AIPC.1134...22K |s2cid=14320124 }}</ref><ref>{{Cite journal |last1=Panahiyan |first1=S. |last2=Fritzsche |first2=S. |date=2021-01-05 |title=Toward simulation of topological phenomena with one-, two-, and three-dimensional quantum walks |url=http://dx.doi.org/10.1103/physreva.103.012201 |journal=Physical Review A |volume=103 |issue=1 |page=012201 |doi=10.1103/physreva.103.012201 |arxiv=2005.08720 |bibcode=2021PhRvA.103a2201P |s2cid=218674364 |issn=2469-9926}}</ref><ref>{{Cite journal |last1=Kitagawa |first1=Takuya |last2=Rudner |first2=Mark S. |last3=Berg |first3=Erez |last4=Demler |first4=Eugene |date=2010-09-24 |title=Exploring topological phases with quantum walks |url=http://dx.doi.org/10.1103/physreva.82.033429 |journal=Physical Review A |volume=82 |issue=3 |page=033429 |doi=10.1103/physreva.82.033429 |arxiv=1003.1729 |bibcode=2010PhRvA..82c3429K |s2cid=21800060 |issn=1050-2947}}</ref><ref name=":9">{{Cite journal |last=Khazali |first=Mohammadsadegh |date=2022-03-03 |title=Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction |url=https://quantum-journal.org/papers/q-2022-03-03-664/ |journal=Quantum |language=en-GB |volume=6 |pages=664 |doi=10.22331/q-2022-03-03-664|arxiv=2101.11412 |bibcode=2022Quant...6..664K |s2cid=246635019 |doi-access=free }}</ref> Interestingly, topological properties of Floquet topological insulators could be controlled via an external periodic drive&nbsp; rather than an external magnetic field. An atomic lattice empowered by distance selective Rydberg interaction could simulate different classes of FTI over a couple of hundred sites and steps in 1, 2 or 3 dimensions.<ref name=":9" /> The long-range interaction allows designing topologically ordered periodic boundary conditions, further enriching the realizable topological phases.<ref name=":9" />

==Properties and applications==
Spin-momentum locking<ref name=Hsieh09/> in the topological insulator allows symmetry-protected surface states to host [[Majorana particle]]s if superconductivity is induced on the surface of 3D topological insulators via proximity effects.<ref>{{Cite journal| doi = 10.1103/PhysRevLett.100.096407| last = Fu | first = L. |author2=C. L. Kane  | title = Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator| journal = Phys. Rev. Lett. | volume = 100 | issue =  9| pages = 096407| year = 2008 | pmid = 18352737| bibcode=2008PhRvL.100i6407F|arxiv = 0707.1692 | s2cid = 7618062 | url = https://repository.upenn.edu/physics_papers/99 }}</ref> (Note that Majorana zero-mode can also appear without topological insulators.<ref name="Potter">{{cite journal | last1=Potter | first1=Andrew C. | last2=Lee | first2=Patrick A. | title=Topological superconductivity and Majorana fermions in metallic surface states | journal=Physical Review B | volume=85 | issue=9 | pages=094516 | date=23 March 2012 | issn=1098-0121 | doi=10.1103/physrevb.85.094516 | arxiv=1201.2176 | bibcode=2012PhRvB..85i4516P | s2cid=59462024 }}</ref>) The non-trivialness of topological insulators is encoded in the existence of a gas of [[helical Dirac fermion]]s. Dirac particles which behave like massless relativistic fermions have been observed in 3D topological insulators. Note that the gapless surface states of topological insulators differ from those in the [[quantum Hall effect]]: the gapless surface states of topological insulators are symmetry-protected (i.e., not topological), while the gapless surface states in quantum Hall effect are topological (i.e., robust against any local perturbations that can break all the symmetries). The <math>\mathbb{Z}_2</math> topological invariants cannot be measured using traditional transport methods, such as spin Hall conductance, and the transport is not quantized by the <math>\mathbb{Z}_2</math> invariants. An experimental method to measure <math>\mathbb{Z}_2</math> topological invariants was demonstrated which provide a measure of the <math>\mathbb{Z}_2</math> topological order.<ref>{{Cite journal| doi = 10.1126/science.1167733 | last = Hsieh | first = D. |author2=D. Hsieh |author3=Y. Xia |author4=L. Wray |author5=D. Qian |author6=A. Pal |author7=J. H. Dil |author8=F. Meier |author9=J. Osterwalder |author10=C. L. Kane |author11=G. Bihlmayer |author12=Y. S. Hor |author13=R. J. Cava |author14=M. Z. Hasan | title = Observation of Unconventional Quantum Spin Textures in Topological Insulators| journal = Science | volume = 323 | issue = 5916 | pages = 919–922 | year = 2009 | pmid = 19213915|bibcode = 2009Sci...323..919H | arxiv = 0902.2617 | s2cid = 118353248 }}</ref>  (Note that the term [[Spin liquid|<math>\mathbb{Z}_2</math> topological order]] has also been used to describe the [[topological order]] with emergent <math>\mathbb{Z}_2</math> [[gauge theory]] discovered in 1991.<ref>{{cite journal | last1 = Read | first1 = N. | last2 = Sachdev | first2 = Subir | year = 1991 | title = Large-N expansion for frustrated quantum antiferromagnets | journal = Phys. Rev. Lett. | volume = 66 | issue = 13| pages = 1773–6 | doi=10.1103/physrevlett.66.1773|bibcode = 1991PhRvL..66.1773R | pmid=10043303}}</ref><ref>{{cite journal | last1 = Wen | first1 = Xiao-Gang | year = 1991 | title = Mean Field Theory of Spin Liquid States with Finite Energy Gaps | journal = Phys. Rev. B | volume = 44 | issue = 6| pages = 2664–2672 | doi=10.1103/physrevb.44.2664| pmid = 9999836 |bibcode = 1991PhRvB..44.2664W }}</ref>) More generally (in what is known as the ''ten-fold way'') for each spatial dimensionality, each of the ten Altland—Zirnbauer symmetry classes of [[Random matrix|random Hamiltonians]] labelled by the type of discrete symmetry (time-reversal symmetry, particle-hole symmetry, and chiral symmetry) has a corresponding group of topological invariants (either <math>\mathbb{Z}</math>, <math>\mathbb{Z}_2</math> or trivial) as described by the [[periodic table of topological invariants]].<ref>{{Cite journal| doi = 10.1103/RevModPhys.88.035005 | last = Chiu| first = C. |author2=J. Teo|author3=A. Schnyder |author4=S. Ryu  | title = Classification of topological quantum matter with symmetries| journal = Rev. Mod. Phys.| volume = 88 | issue = 35005 | pages = 035005| year = 2016 | arxiv = 1505.03535| bibcode = 2016RvMP...88c5005C| s2cid = 119294876}}</ref>

The most promising applications of topological insulators are spintronic devices and dissipationless [[transistor]]s for [[quantum computer]]s based on the [[quantum spin Hall effect|quantum Hall effect]]<ref name="König07"/> and [[quantum anomalous Hall effect]].<ref>{{Cite journal|last1=Chang|first1=Cui-Zu|last2=Zhang|first2=Jinsong|last3=Feng|first3=Xiao|last4=Shen|first4=Jie|last5=Zhang|first5=Zuocheng|last6=Guo|first6=Minghua|last7=Li|first7=Kang|last8=Ou|first8=Yunbo|last9=Wei|first9=Pang|date=2013-04-12|title=Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator|journal=Science|language=en|volume=340|issue=6129|pages=167–170|doi=10.1126/science.1234414|issn=0036-8075|pmid=23493424|arxiv=1605.08829|bibcode=2013Sci...340..167C|s2cid=29455044}}</ref> In addition, topological insulator materials have also found practical applications in advanced [[Magnetoelectronics|magnetoelectronic]] and [[optoelectronic]] devices.<ref>{{Cite journal|last1=Yue|first1=Zengji|last2=Cai|first2=Boyuan|last3=Wang|first3=Lan|last4=Wang|first4=Xiaolin|last5=Gu|first5=Min|date=2016-03-01|title=Intrinsically core-shell plasmonic dielectric nanostructures with ultrahigh refractive index|journal=Science Advances|language=en|volume=2|issue=3|pages=e1501536|doi=10.1126/sciadv.1501536|issn=2375-2548|pmc=4820380|pmid=27051869|bibcode=2016SciA....2E1536Y}}</ref><ref>{{Cite journal|last1=Yue|first1=Zengji|last2=Xue|first2=Gaolei|last3=Liu|first3=Juan|last4=Wang|first4=Yongtian|last5=Gu|first5=Min|date=2017-05-18|title=Nanometric holograms based on a topological insulator material|journal=Nature Communications|language=en|volume=8|pages=ncomms15354|doi=10.1038/ncomms15354|pmc=5454374|pmid=28516906|bibcode=2017NatCo...815354Y}}</ref>

=== Thermoelectrics ===
Some of the most well-known topological insulators are also [[thermoelectric materials]], such as Bi<sub>2</sub>Te<sub>3</sub> and its alloys with Bi<sub>2</sub>Se<sub>3</sub> (n-type thermoelectrics) and Sb<sub>2</sub>Te<sub>3</sub> (p-type thermoelectrics).<ref>{{Cite journal |last1=Witting |first1=Ian T. |last2=Chasapis |first2=Thomas C. |last3=Ricci |first3=Francesco |last4=Peters |first4=Matthew |last5=Heinz |first5=Nicholas A. |last6=Hautier |first6=Geoffroy |last7=Snyder |first7=G. Jeffrey |date=June 2019 |title=The Thermoelectric Properties of Bismuth Telluride |journal=Advanced Electronic Materials |language=en |volume=5 |issue=6 |doi=10.1002/aelm.201800904 |doi-access=free |issn=2199-160X}}</ref> High thermoelectric power conversion efficiency is realized in materials with low thermal conductivity, high electrical conductivity, and high [[Seebeck coefficient]] (i.e., the incremental change in voltage due to an incremental change in temperature). Topological insulators are often composed of heavy atoms, which tends to lower thermal conductivity and are therefore beneficial for thermoelectrics. A recent study also showed that good electrical characteristics (i.e., high electrical conductivity and Seebeck coefficient) can arise in topological insulators due to warping of the bulk band structure, which is driven by band inversion.<ref>{{Citation |last1=Toriyama |first1=Michael |title=Are Topological Insulators Promising Thermoelectrics? |date=2023-11-06 |url=https://chemrxiv.org/engage/chemrxiv/article-details/6545ba97c573f893f1c4221d |access-date=2024-01-07 |language=en |doi=10.26434/chemrxiv-2023-3nvl3 |last2=Snyder |first2=G. Jeffrey}}</ref> Often, the electrical conductivity and Seebeck coefficient are conflicting properties of thermoelectrics and difficult to optimize simultaneously. Band warping, induced by band inversion in a topological insulator, can mediate the two properties by reducing the effective mass of electrons/holes and increasing the valley degeneracy (i.e., the number of electronic bands that are contributing to charge transport). As a result, topological insulators are generally interesting candidates for thermoelectric applications.

==Synthesis==
Topological insulators can be grown using different methods such as [[Metal organic chemical vapor deposition|metal-organic chemical vapor deposition]] (MOCVD),<ref>{{Cite journal|last1=Alegria|first1=L. D.|last2=Schroer|first2=M. D.|last3=Chatterjee|first3=A.|last4=Poirier|first4=G. R.|last5=Pretko|first5=M.|last6=Patel|first6=S. K.|last7=Petta|first7=J. R.|date=2012-08-06|title=Structural and Electrical Characterization of {{chem|Bi|2|Se|3}} Nanostructures Grown by Metal–Organic Chemical Vapor Deposition|journal=Nano Letters|language=EN|volume=12|issue=9|pages=4711–4|doi=10.1021/nl302108r|pmid=22827514|issn=1530-6984|arxiv=1108.4978|bibcode=2012NanoL..12.4711A|s2cid=28030427}}</ref>
[[physical vapor deposition]] (PVD),<ref>{{Cite journal|date= 2017  |title=Large-Area and Transferred High-Quality Three-Dimensional Topological Insulator Bi2–x Sb x Te3–y Se y Ultrathin Film by Catalyst-Free Physical Vapor Deposition|journal= Nano Letters|language=en|volume=17|issue=4|pages=2354–60|doi= 10.1021/acs.nanolett.6b05260|last1=Tu, Ngoc Han|first1=Tanabe, Yoichi|last2=Satake, Yosuke|first2=Huynh, Khuong Kim|last3=Le, Phuoc Huu|first3=Matsushita, Stephane Yu|last4=Tanigaki, Katsumi|pmid=28337910|arxiv=1601.06541|bibcode=2017NanoL..17.2354T|s2cid=206738534}}</ref> solvothermal synthesis,<ref>{{Cite journal|date=2003-06-01|title=Preparation and characterization of wire-like {{chem|Sb|2|Se|3}} and flake-like {{chem|Bi|2|Se|3}} nanocrystals|journal=Journal of Crystal Growth|language=en|volume=253|issue=1–4|pages=445–451|doi=10.1016/S0022-0248(03)01019-4|issn=0022-0248|last1=Wang|first1=Debao|last2=Yu|first2=Dabin|last3=Mo|first3=Maosong|last4=Liu|first4=Xianming|last5=Qian|first5=Yitai|bibcode=2003JCrGr.253..445W}}</ref>  sonochemical technique<ref>{{Cite journal|date=2004-11-15|title=Sonochemical synthesis of bismuth selenide nanobelts at room temperature|journal=Journal of Crystal Growth|language=en|volume=271|issue=3–4|pages=456–461|doi=10.1016/j.jcrysgro.2004.08.015|issn=0022-0248|last1=Cui|first1=Hongmei|last2=Liu|first2=Hong|last3=Wang|first3=Jiyang|last4=Li|first4=Xia|last5=Han|first5=Feng|last6=Boughton|first6=R.I.|bibcode=2004JCrGr.271..456C}}</ref>  and [[molecular beam epitaxy]][[File:Molecular Beam Epitaxy.png|thumb|250x250px|Schematic of the components of a MBE system]]
(MBE).<ref name=":3" />  MBE has so far been the most common experimental technique. The growth of thin film topological insulators is governed by weak [[van der Waals interactions]].<ref>{{Cite journal|last1=Jerng|first1=Sahng-Kyoon|last2=Joo|first2=Kisu|last3=Kim|first3=Youngwook|last4=Yoon|first4=Sang-Moon|last5=Lee|first5=Jae Hong|last6=Kim|first6=Miyoung|last7=Kim|first7=Jun Sung|last8=Yoon|first8=Euijoon|last9=Chun|first9=Seung-Hyun|date=2013|title=Ordered growth of topological insulator {{chem|Bi|2|Se|3}} thin films on dielectric amorphous SiO2 by MBE|journal=Nanoscale|language=en|volume=5|issue=21|pages=10618–22|doi=10.1039/C3NR03032F|pmid=24056725|issn=2040-3364|arxiv=1308.3817|bibcode=2013Nanos...510618J|s2cid=36212915}}</ref> The weak interaction allows to exfoliate the thin film from bulk crystal with a clean and perfect surface. The van der Waals interactions in epitaxy also known as van der Waals epitaxy (VDWE), is a phenomenon governed by weak van der Waals interactions between layered materials of different or same elements<ref name=":4">{{Cite journal|last1=Geim|first1=A. K.|last2=Grigorieva|first2=I. V.|date=2013|title=Van der Waals heterostructures|journal=Nature|language=En|volume=499|issue=7459|pages=419–425|doi=10.1038/nature12385|pmid=23887427|issn=0028-0836|arxiv=1307.6718|s2cid=205234832}}</ref> in which the materials are stacked on top of each other. This approach allows the growth of layered topological insulators on other substrates for [[heterostructure]] and [[integrated circuit]]s.<ref name=":4" />

'''MBE growth of topological insulators'''

Molecular beam epitaxy (MBE) is an [[epitaxy]] method for the growth of a crystalline material on a crystalline substrate to form an ordered layer.  MBE is performed in [[Vacuum|high vacuum]] or [[ultra-high vacuum]], the elements are heated in different electron beam evaporators until they [[Sublimation (phase transition)|sublime]]. The gaseous elements then condense on the wafer where they react with each other to form [[single crystal]]s.

MBE is an appropriate technique for the growth of high quality single-crystal films.  In order to avoid a huge [[Lattice constant#Lattice matching|lattice mismatch]] and [[Crystallographic defect|defects]] at the interface, the substrate and thin film are expected to have similar lattice constants.  MBE has an advantage over other methods due to the fact that the synthesis is performed in high vacuum hence resulting in less contamination.  Additionally,  lattice defect is reduced due to the ability to influence the growth rate and the ratio of species of source materials present at the substrate interface.<ref name=":5">{{Cite journal|last1=Heremans|first1=Joseph P.|last2=Cava|first2=Robert J.|last3=Samarth|first3=Nitin|date=2017-09-05|title=Tetradymites as thermoelectrics and topological insulators|journal=Nature Reviews Materials|language=En|volume=2|issue=10|pages=17049|doi=10.1038/natrevmats.2017.49|issn=2058-8437|bibcode=2017NatRM...217049H}}</ref> Furthermore,  in MBE,  samples can be grown layer by layer which results in flat surfaces with smooth interface for engineered heterostructures.  Moreover,  MBE synthesis technique benefits from the ease of moving a topological insulator sample from the growth chamber to a characterization chamber such as angle-resolved photoemission spectroscopy (ARPES) or [[scanning tunneling microscopy]] (STM) studies.<ref name=":7">{{Cite web|url=https://www.wiley.com/en-us/Topological+Insulators:+Fundamentals+and+Perspectives-p-9783527337026|title=Topological Insulators: Fundamentals and Perspectives|date=2015-06-29|website=Wiley.com|language=en-us|access-date=2018-07-29}}</ref>

Due to the weak van der Waals bonding, which relaxes the lattice-matching condition, TI can be grown on a wide variety of substrates<ref name=":8">{{Cite journal|last1=He|first1=Liang|last2=Kou|first2=Xufeng|last3=Wang|first3=Kang L.|date=2013-01-31|title=Review of 3D topological insulator thin-film growth by molecular beam epitaxy and potential applications|journal=Physica Status Solidi RRL|language=en|volume=7|issue=1–2|pages=50–63|doi=10.1002/pssr.201307003|issn=1862-6254|bibcode=2013PSSRR...7...50H|s2cid=97544002|doi-access=free}}</ref> such as Si(111),<ref>{{Cite journal|date=2011-10-31|title=Epitaxial growth of topological insulator {{chem|Bi|2|Se|3}} film on Si(111) with atomically sharp interface|journal=Thin Solid Films|language=en|volume=520|issue=1|pages=224–9|doi=10.1016/j.tsf.2011.07.033|issn=0040-6090|last1=Bansal|first1=Namrata|last2=Kim|first2=Yong Seung|last3=Edrey|first3=Eliav|last4=Brahlek|first4=Matthew|last5=Horibe|first5=Yoichi|last6=Iida|first6=Keiko|last7=Tanimura|first7=Makoto|last8=Li|first8=Guo-Hong|last9=Feng|first9=Tian|last10=Lee|first10=Hang-Dong|last11=Gustafsson|first11=Torgny|last12=Andrei|first12=Eva|last13=Oh|first13=Seongshik|arxiv=1104.3438|bibcode=2011TSF...520..224B|s2cid=118512981}}</ref><ref>{{Cite journal|last1=Zhang|first1=Guanhua|last2=Qin|first2=Huajun|last3=Teng|first3=Jing|last4=Guo|first4=Jiandong|last5=Guo|first5=Qinlin|last6=Dai|first6=Xi|last7=Fang|first7=Zhong|last8=Wu|first8=Kehui|date=2009-08-03|title=Quintuple-layer epitaxy of thin films of topological insulator {{chem|Bi|2|Se|3}}|journal=Applied Physics Letters|language=en|volume=95|issue=5|pages=053114|doi=10.1063/1.3200237|issn=0003-6951|arxiv=0906.5306|bibcode=2009ApPhL..95e3114Z}}</ref>  {{chem|link=Aluminium oxide|Al|2|O|3}}, [[GaAs]](111),<ref>{{Cite journal|last1=Richardella|first1=A.|last2=Zhang|first2=D. M.|last3=Lee|first3=J. S.|last4=Koser|first4=A.|last5=Rench|first5=D. W.|last6=Yeats|first6=A. L.|last7=Buckley|first7=B. B.|last8=Awschalom|first8=D. D.|last9=Samarth|first9=N.|date=2010-12-27|title=Coherent heteroepitaxy of {{chem|Bi|2|Se|3}} on GaAs (111)B|journal=Applied Physics Letters|language=en|volume=97|issue=26|pages=262104|doi=10.1063/1.3532845|arxiv=1012.1918|bibcode=2010ApPhL..97z2104R|issn=0003-6951|doi-access=free}}</ref>

[[InP]](111), [[CdS]](0001) and {{chem|link=Yttrium iron garnet|Y|3|Fe|5|O|12}} .

==PVD growth of topological insulators==

The physical vapor deposition (PVD) technique does not suffer from the disadvantages of the exfoliation method and, at the same time, it is much simpler and cheaper than the fully controlled growth by molecular-beam epitaxy. The PVD method enables a reproducible synthesis of single crystals of various layered quasi-two-dimensional materials including topological insulators (i.e., {{chem|Bi|2|Se|3}}, {{chem|Bi|2|Te|3}}).<ref>{{Cite journal|last1=Kong|first1=D.|last2=Dang|first2=W.|last3=Cha|first3=J.J.|last4=Li|first4=H.|last5=Meister|first5=S.|last6=Peng|first6=H. K.|last7=Cui|first7=Y|date=2010|title=SFew-layer nanoplates of {{chem|Bi|2|Se|3}} and {{chem|Bi|2|Te|3}} with highly tunable chemical potential|journal=Nano Letters|language=EN|volume=10|issue=6|pages=2245–50|doi = 10.1021/nl101260j|pmid=20486680|arxiv=1004.1767|bibcode=2010NanoL..10.2245K|s2cid=37687875}}</ref> The resulted single crystals have a well-defined crystallographic orientation; their composition, thickness, size, and the surface density on the desired substrate can be controlled.
The thickness control is particularly important for 3D TIs in which the trivial (bulky) electronic channels usually dominate the transport properties and mask the response of the topological (surface) modes. By reducing the thickness, one lowers the contribution of trivial bulk channels into the total conduction, thus forcing the topological modes to carry the electric current.<ref>{{Cite journal|last1=Stolyarov|first1=V.S.|last2=Yakovlev|first2=D.S.|last3=Kozlov|first3=S.N.|last4=Skryabina|first4=O.V.|last5=Lvov|first5=D.S.|date=2020|title=Josephson current mediated by ballistic topological states in Bi2Te2.3Se0.7 single nanocrystals|journal=Communications Materials |language=EN|volume=1|issue=1 |page=38| doi = 10.1038/s43246-020-0037-y|bibcode=2020CoMat...1...38S|s2cid=220295733|doi-access=free}} [[File:CC-BY icon.svg|50px]]  Text was copied from this source, which is available under a [https://creativecommons.org/licenses/by/4.0/  Creative Commons Attribution 4.0 International License].</ref>

=== Bismuth-based topological insulators ===
Thus far, the field of topological insulators has been focused on bismuth and antimony [[chalcogenide]] based materials such as {{chem|Bi|2|Se|3}}, {{chem|Bi|2|Te|3}}, {{chem|Sb|2|Te|3}} or [[Bismuth antimonide|Bi<sub>1 − x</sub>Sb<sub>x</sub>]], Bi<sub>1.1</sub>Sb<sub>0.9</sub>Te<sub>2</sub>S.<ref name="NatureComm"/>  The choice of chalcogenides is related to the van der Waals relaxation of the lattice matching strength which restricts the number of materials and substrates.<ref name=":5" />  Bismuth chalcogenides have been studied extensively for TIs and their applications in [[thermoelectric materials]].  The van der Waals interaction in TIs exhibit important features due to low surface energy.  For instance, the surface of {{chem|link=Bismuth telluride|Bi|2|Te|3}} is usually terminated by Te due to its low surface energy.<ref name=":3" />

Bismuth chalcogenides have been successfully grown on different substrates. In particular, Si has been a good substrate for the successful growth of  {{chem|Bi|2|Te|3}} . However, the use of sapphire as substrate has not been so encouraging due to a large mismatch of about 15%.<ref name=":6">{{Cite journal|last1=Ginley|first1=Theresa P.|last2=Wang|first2=Yong|last3=Law|first3=Stephanie|date=2016-11-23|title=Topological Insulator Film Growth by Molecular Beam Epitaxy: A Review|journal=Crystals|language=en|volume=6|issue=11|pages=154|doi=10.3390/cryst6110154|doi-access=free|bibcode=2016Cryst...6..154G }}</ref> The selection of appropriate substrate can improve the overall properties of TI. The use of buffer layer can reduce the lattice match hence improving the electrical properties of TI.<ref name=":6" /> {{chem|Bi|2|Se|3}} can be grown on top of various Bi<sub>2 − x</sub>In<sub>x</sub>Se<sub>3</sub> buffers.  Table 1 shows {{chem|link=Bismuth selenide|Bi|2|Se|3}}, {{chem|Bi|2|Te|3}}, {{chem|link=Antimony telluride|Sb|2|Te|3}}  on different substrates and the resulting lattice mismatch. Generally,  regardless of the substrate used, the resulting films have a textured surface that is characterized by pyramidal single-crystal domains with quintuple-layer steps. The size and relative proportion of these pyramidal domains vary with factors that include film thickness, lattice mismatch with the substrate and interfacial chemistry-dependent film nucleation. The synthesis of thin films have the stoichiometry problem due to the high vapor pressures of the elements. Thus, binary tetradymites are extrinsically doped as n-type ({{chem|Bi|2|Se|3}}, {{chem|Bi|2|Te|3}} ) or p-type ({{chem|Sb|2|Te|3}} ).<ref name=":5" /> Due to the weak van der Waals bonding, graphene is one of the preferred substrates for TI growth despite the large lattice mismatch.

{| class="wikitable"
|+Lattice mismatch of different substrates<ref name=":8" /> 
!Substrate
!{{chem|Bi|2|Se|3}} %
!{{chem|Bi|2|Te|3}} %
!{{chem|Sb|2|Te|3}} %
|-
|graphene
| −40.6
| −43.8
| −42.3
|-
|Si
| −7.3
| −12.3
| −9.7
|-
|{{chem|CaF|2}}
| −6.8
| −11.9
| −9.2
|-
|GaAs
| −3.4
| −8.7
| −5.9
|-
|CdS
| −0.2
| −5.7
| −2.8
|-
|InP
|0.2
| −5.3
| −2.3
|-
|{{chem|BaF|2}}
|5.9
|0.1
|2.8
|-
|CdTe
|10.7
|4.6
|7.8
|-
|{{chem|Al|2|O|3}}
|14.9
|8.7
|12.0
|-
|{{chem|SiO|2}}
|18.6
|12.1
|15.5
|}

== Identification ==
The first step of topological insulators identification takes place right after synthesis, meaning without breaking the vacuum and moving the sample to an atmosphere. That could be done by using angle-resolved photoemission spectroscopy (ARPES) or scanning tunneling microscopy (STM) techniques.<ref name=":7" /> Further measurements includes structural and chemical probes such as X-ray diffraction and energy-dispersive spectroscopy but depending on the sample quality, the lack of sensitivity could remain. Transport measurements cannot uniquely pinpoint the <math>\mathbb{Z}_2</math> topology by definition of the state.

== Classification ==
{{Main articles|Periodic table of topological invariants}}
[[Bloch's theorem]] allows a full characterization of the wave propagation properties of a material by assigning a matrix to each wave vector in the [[Brillouin zone]].

Mathematically, this assignment creates a [[vector bundle]]. Different materials will have different wave propagation properties, and thus different vector bundles. If we consider all insulators (materials with a band gap), this creates a space of vector bundles. It is the topology of this space (modulo trivial bands) from which the "topology" in topological insulators arises.<ref name=":12" />

Specifically, the number of [[Connected component (topology)|connected components]] of the space indicates how many different "islands" of insulators exist amongst the metallic states. Insulators in the connected component containing the vacuum state are identified as "trivial", and all other insulators as "topological". The connected component in which an insulator lies can be identified with a number, referred to as a "topological invariant".<ref name=":12" />

This space can be restricted under the presence of symmetries, changing the resulting topology. Although [[Unitary matrix|unitary]] symmetries are usually significant in quantum mechanics, they have no effect on the topology here.<ref>{{Cite web |title=10 symmetry classes and the periodic table of topological insulators |url=https://topocondmat.org/w8_general/classification.html |access-date=2022-10-11 |website=topocondmat.org}}</ref> Instead, the three symmetries typically considered are time-reversal symmetry, particle-hole symmetry, and chiral symmetry (also called sublattice symmetry). Mathematically, these are represented as, respectively: an [[Antiunitary operator|anti-unitary operator]] which commutes with the [[Hamiltonian (quantum mechanics)|Hamiltonian]]; an anti-unitary operator which [[Anticommutative property|anti-commutes]] with the Hamiltonian; and a unitary operator which anti-commutes with the Hamiltonian. All combinations of the three together with each spatial dimension result in the so-called [[Periodic table of topological invariants|periodic table of topological insulators]].<ref name=":12" />

== Future developments ==
The field of topological insulators still needs to be developed. The best bismuth chalcogenide topological insulators have about 10 meV bandgap variation due to the charge. Further development should focus on the examination of both: the presence of high-symmetry electronic bands and simply synthesized materials. One of the candidates is [[Heusler compound|half-Heusler compounds]].<ref name=":7" /> These crystal structures can consist of a large number of elements. Band structures and energy gaps are very sensitive to the valence configuration; because of the increased likelihood of intersite exchange and disorder, they are also very sensitive to specific crystalline configurations. A nontrivial band structure that exhibits band ordering analogous to that of the known 2D and 3D TI materials was predicted in a variety of 18-electron half-Heusler compounds using first-principles calculations.<ref>{{Cite journal|date=2013-04-01|title=Prediction of topological insulating behavior in inverse Heusler compounds from first principles|journal=Computational Materials Science|language=en|volume=70|pages=145–149|doi=10.1016/j.commatsci.2012.12.013|issn=0927-0256|last1=Zhang|first1=X.M.|last2=Liu|first2=E.K.|last3=Liu|first3=Z.Y.|last4=Liu|first4=G.D.|last5=Wu|first5=G.H.|last6=Wang|first6=W.H.|arxiv=1210.5816|s2cid=53506226}}</ref> These materials have not yet shown any sign of intrinsic topological insulator behavior in actual experiments.

==See also==
* [[Topological order]]
* [[Topological quantum computer]]
* [[Topological quantum field theory]]
* [[Topological quantum number]]
* [[Quantum Hall effect]]
* [[Quantum spin Hall effect]]
* [[Periodic table of topological invariants]]
* [[Bismuth selenide]]
* [[Photonic topological insulator]]

==References==
<references/>

==Further reading==
* {{Cite journal| last = Hasan | first = M. Zahid |author2=Kane, Charles L.  | year = 2010 | title = Topological Insulators | journal = [[Reviews of Modern Physics]] | volume =  82 | issue =   4| pages = 3045–67 | doi = 10.1103/RevModPhys.82.3045 | bibcode=2010RvMP...82.3045H|arxiv = 1002.3895 | s2cid = 16066223 }}
* {{Cite journal| last = Kane | first = Charles L. |author2=Moore, Joel E.  | year = 2011 | title = Topological Insulators | journal = [[Physics World]] | volume =  24 | issue =   2| pages = 32–36 | doi =  10.1088/2058-7058/24/02/36| url = http://physics.gmu.edu/~pnikolic/articles/Topological%20insulators%20(Physics%20World,%20February%202011).pdf | bibcode = 2011PhyW...24b..32K }}
*{{cite book |last1=Hasan |first1=M. Zahid |last2=Xu |first2=Su-Yang |last3=Neupane |first3=M  |chapter=Topological Insulators, Topological Dirac semimetals, Topological Crystalline Insulators, and Topological Kondo Insulators |title=Topological Insulators|editor1-first=F.|editor1-last= Ortmann|editor2-first=S. |editor2-last=Roche|editor3-first=S. O. |editor3-last=Valenzuela |publisher=Wiley  |date=2015 |doi=10.1002/9783527681594.ch4 |pages=55–100|isbn=9783527681594 }}
* {{Cite journal| doi = 10.1038/466310a | volume = 466 | issue = 7304 | pages = 310–1 | last = Brumfiel | pmid = 20631773| first = G. | title = Topological insulators: Star material |type=Nature News | journal = Nature | year = 2010 | doi-access = free }}
* {{Cite journal| last = Murakami | first = Shuichi | year = 2010 | title = Focus on Topological Insulators | journal = [[New Journal of Physics]] | url = http://iopscience.iop.org/1367-2630/focus/Focus%20on%20Topological%20Insulators }}
* {{cite web |first=Joel E. |last=Moore |url=https://spectrum.ieee.org/topological-insulators |title=Topological Insulators |work=IEEE Spectrum |date=July 2011}}
*{{cite journal |title=Topological insulators promise computing advances, insights into matter itself |journal=Proceedings of the National Academy of Sciences |year=2016 |volume=113 |issue=37 |pages=10223–4 |doi =10.1073/pnas.1611504113 |pmid=27625422|last1=Ornes |first1=S. |pmc=5027448 |doi-access=free }}
* {{Cite web|url=https://www.scientificamerican.com/article/the-strange-topology-that-is-reshaping-physics/|title=The Strange Topology That Is Reshaping Physics |work=Scientific American |year=2017 }}

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