The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional semiconductors that have a quantized spin-Hall conductance and a vanishing charge-Hall conductance. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, and that does not require the application of a large magnetic field. The quantum spin Hall state does not break charge conservation symmetry and spin- conservation symmetry (in order to have well defined Hall conductances).
The first proposal for the existence of a quantum spin Hall state was developed by Charles Kane and Gene Mele[1] who adapted an earlier model for graphene by F. Duncan M. Haldane[2] which exhibits an integer quantum Hall effect. The Kane and Mele model is two copies of the Haldane model such that the spin up electron exhibits a chiral integer quantum Hall Effect while the spin down electron exhibits an anti-chiral integer quantum Hall effect. A relativistic version of the quantum spin Hall effect was introduced in the 1990s for the numerical simulation of chiral gauge theories;[3][4] the simplest example consisting of a parity and time reversal symmetric U(1)gauge theory with bulk fermions of opposite sign mass, a massless Dirac surface mode, and bulk currents that carry chirality but not charge (the spin Hall current analogue). Overall the Kane-Mele model has a charge-Hall conductance of exactly zero but a spin-Hall conductance of exactly (in units of ). Independently, a quantum spin Hall model was proposed by Andrei Bernevig and Shoucheng Zhang[5] in an intricate strain architecture which engineers, due to spin-orbit coupling, a magnetic field pointing upwards for spin-up electrons and a magnetic field pointing downwards for spin-down electrons. The main ingredient is the existence of spin–orbit coupling, which can be understood as a momentum-dependent magnetic field coupling to the spin of the electron.
Real experimental systems, however, are far from the idealized picture presented above in which spin-up and spin-down electrons are not coupled. A very important achievement was the realization that the quantum spin Hall state remains to be non-trivial even after the introduction of spin-up spin-down scattering,[6] which destroys the quantum spin Hall effect. In a separate paper, Kane and Mele introduced a topological invariant which characterizes a state as trivial or non-trivial band insulator (regardless if the state exhibits or does not exhibit a quantum spin Hall effect). Further stability studies of the edge liquid through which conduction takes place in the quantum spin Hall state proved, both analytically and numerically that the non-trivial state is robust to both interactions and extra spin-orbit coupling terms that mix spin-up and spin-down electrons. Such a non-trivial state (exhibiting or not exhibiting a quantum spin Hall effect) is called a topological insulator, which is an example of symmetry-protected topological order protected by charge conservation symmetry and time reversal symmetry. (Note that the quantum spin Hall state is also a symmetry-protected topological state protected by charge conservation symmetry and spin- conservation symmetry. We do not need time reversal symmetry to protect quantum spin Hall state. Topological insulator and quantum spin Hall state are different symmetry-protected topological states. So topological insulator and quantum spin Hall state are different states of matter.)
In HgTe quantum wellsedit
Since graphene has extremely weak spin-orbit coupling, it is very unlikely to support a quantum spin Hall state at temperatures achievable with today's technologies. Two-dimensional topological insulators (also known as the quantum spin Hall insulators) with one-dimensional helical edge states were predicted in 2006 by Bernevig, Hughes and Zhang to occur in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride,[7] and were observed in 2007. [8]
Different quantum wells of varying HgTe thickness can be built. When the sheet of HgTe in between the CdTe is thin, the system behaves like an ordinary insulator and does not conduct when the Fermi level resides in the band-gap. When the sheet of HgTe is varied and made thicker (this requires the fabrication of separate quantum wells), an interesting phenomenon happens. Due to the inverted band structure of HgTe, at some critical HgTe thickness, a Lifshitz transition occurs in which the system closes the bulk band gap to become a semi-metal, and then re-opens it to become a quantum spin Hall insulator.
In the gap closing and re-opening process, two edge states are brought out from the bulk and cross the bulk-gap. As such, when the Fermi level resides in the bulk gap, the conduction is dominated by the edge channels that cross the gap. The two-terminal conductance is in the quantum spin Hall state and zero in the normal insulating state. As the conduction is dominated by the edge channels, the value of the conductance should be insensitive to how wide the sample is. A magnetic field should destroy the quantum spin Hall state by breaking time-reversal invariance and allowing spin-up spin-down electron scattering processes at the edge. All these predictions have been experimentally verified in an experiment [9] performed in the Molenkamp labs at Universität Würzburg in Germany.
^Kane, C.L.; Mele, E.J. (25 November 2005). "Quantum Spin Hall Effect in Graphene". Physical Review Letters. 95 (22): 226081. arXiv:cond-mat/0411737. Bibcode:2005PhRvL..95v6801K. doi:10.1103/PhysRevLett.95.226801. PMID 16384250. S2CID 6080059.
^Haldane, F.D.M. (31 October 1988). "Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"". Physical Review Letters. 61 (18): 2015–2018. Bibcode:1988PhRvL..61.2015H. doi:10.1103/PhysRevLett.61.2015. PMID 10038961.
^Kaplan, David B. (1992). "A method for simulating chiral fermions on the lattice". Physics Letters B. 288 (3–4): 342–347. arXiv:hep-lat/9206013. Bibcode:1992PhLB..288..342K. CiteSeerX10.1.1.286.587. doi:10.1016/0370-2693(92)91112-m. S2CID 14161004.
^Golterman, Maarten F.L.; Jansen, Karl; Kaplan, David B. (1993). "Chern-Simons currents and chiral fermions on the lattice". Physics Letters B. 301 (2–3): 219–223. arXiv:hep-lat/9209003. Bibcode:1993PhLB..301..219G. doi:10.1016/0370-2693(93)90692-b. S2CID 9265777.
^Bernevig, B. Andrei; Zhang, Shou-Cheng (14 March 2006). "Quantum Spin Hall Effect". Physical Review Letters. 96 (10): 106802. arXiv:cond-mat/0504147. Bibcode:2006PhRvL..96j6802B. doi:10.1103/PhysRevLett.96.106802. PMID 16605772. S2CID 2618285.
^Kane, C.L.; Mele, E.J. (28 September 2005). "Z2 Topological Order and the Quantum Spin Hall Effect". Physical Review Letters. 95 (14): 146802. arXiv:cond-mat/0506581. Bibcode:2005PhRvL..95n6802K. doi:10.1103/PhysRevLett.95.146802. PMID 16241681. S2CID 1775498.
^Bernevig, B. Andrei; Hughes, Taylor L.; Zhang, Shou-Cheng (2006-12-15). "Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells". Science. 314 (5806): 1757–1761. arXiv:cond-mat/0611399. Bibcode:2006Sci...314.1757B. doi:10.1126/science.1133734. ISSN 0036-8075. PMID 17170299. S2CID 7295726.
Maciejko, J.; Hughes, T. L.; Zhang, S. C. (2011). "The Quantum Spin Hall Effect". Annual Review of Condensed Matter Physics. 2: 31–53. Bibcode:2011ARCMP...2...31M. doi:10.1146/annurev-conmatphys-062910-140538.
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The quantum spin Hall state is a state of matter proposed to exist in special two dimensional semiconductors that have a quantized spin Hall conductance and a vanishing charge Hall conductance The quantum spin Hall state of matter is the cousin of the integer quantum Hall state and that does not require the application of a large magnetic field The quantum spin Hall state does not break charge conservation symmetry and spin S z displaystyle S z conservation symmetry in order to have well defined Hall conductances Contents 1 Description 2 In HgTe quantum wells 3 See also 4 References 5 Further readingDescription editThe first proposal for the existence of a quantum spin Hall state was developed by Charles Kane and Gene Mele 1 who adapted an earlier model for graphene by F Duncan M Haldane 2 which exhibits an integer quantum Hall effect The Kane and Mele model is two copies of the Haldane model such that the spin up electron exhibits a chiral integer quantum Hall Effect while the spin down electron exhibits an anti chiral integer quantum Hall effect A relativistic version of the quantum spin Hall effect was introduced in the 1990s for the numerical simulation of chiral gauge theories 3 4 the simplest example consisting of a parity and time reversal symmetric U 1 gauge theory with bulk fermions of opposite sign mass a massless Dirac surface mode and bulk currents that carry chirality but not charge the spin Hall current analogue Overall the Kane Mele model has a charge Hall conductance of exactly zero but a spin Hall conductance of exactly s x y spin 2 displaystyle sigma xy text spin 2 nbsp in units of e 4 p displaystyle frac e 4 pi nbsp Independently a quantum spin Hall model was proposed by Andrei Bernevig and Shoucheng Zhang 5 in an intricate strain architecture which engineers due to spin orbit coupling a magnetic field pointing upwards for spin up electrons and a magnetic field pointing downwards for spin down electrons The main ingredient is the existence of spin orbit coupling which can be understood as a momentum dependent magnetic field coupling to the spin of the electron Real experimental systems however are far from the idealized picture presented above in which spin up and spin down electrons are not coupled A very important achievement was the realization that the quantum spin Hall state remains to be non trivial even after the introduction of spin up spin down scattering 6 which destroys the quantum spin Hall effect In a separate paper Kane and Mele introduced a topological Z 2 displaystyle mathbb Z 2 nbsp invariant which characterizes a state as trivial or non trivial band insulator regardless if the state exhibits or does not exhibit a quantum spin Hall effect Further stability studies of the edge liquid through which conduction takes place in the quantum spin Hall state proved both analytically and numerically that the non trivial state is robust to both interactions and extra spin orbit coupling terms that mix spin up and spin down electrons Such a non trivial state exhibiting or not exhibiting a quantum spin Hall effect is called a topological insulator which is an example of symmetry protected topological order protected by charge conservation symmetry and time reversal symmetry Note that the quantum spin Hall state is also a symmetry protected topological state protected by charge conservation symmetry and spin S z displaystyle S z nbsp conservation symmetry We do not need time reversal symmetry to protect quantum spin Hall state Topological insulator and quantum spin Hall state are different symmetry protected topological states So topological insulator and quantum spin Hall state are different states of matter In HgTe quantum wells editSince graphene has extremely weak spin orbit coupling it is very unlikely to support a quantum spin Hall state at temperatures achievable with today s technologies Two dimensional topological insulators also known as the quantum spin Hall insulators with one dimensional helical edge states were predicted in 2006 by Bernevig Hughes and Zhang to occur in quantum wells very thin layers of mercury telluride sandwiched between cadmium telluride 7 and were observed in 2007 8 Different quantum wells of varying HgTe thickness can be built When the sheet of HgTe in between the CdTe is thin the system behaves like an ordinary insulator and does not conduct when the Fermi level resides in the band gap When the sheet of HgTe is varied and made thicker this requires the fabrication of separate quantum wells an interesting phenomenon happens Due to the inverted band structure of HgTe at some critical HgTe thickness a Lifshitz transition occurs in which the system closes the bulk band gap to become a semi metal and then re opens it to become a quantum spin Hall insulator In the gap closing and re opening process two edge states are brought out from the bulk and cross the bulk gap As such when the Fermi level resides in the bulk gap the conduction is dominated by the edge channels that cross the gap The two terminal conductance is G x x 2 e 2 h displaystyle G xx 2 frac e 2 h nbsp in the quantum spin Hall state and zero in the normal insulating state As the conduction is dominated by the edge channels the value of the conductance should be insensitive to how wide the sample is A magnetic field should destroy the quantum spin Hall state by breaking time reversal invariance and allowing spin up spin down electron scattering processes at the edge All these predictions have been experimentally verified in an experiment 9 performed in the Molenkamp labs at Universitat Wurzburg in Germany See also editSpin Hall effect Quantum Hall effectReferences edit Kane C L Mele E J 25 November 2005 Quantum Spin Hall Effect in Graphene Physical Review Letters 95 22 226081 arXiv cond mat 0411737 Bibcode 2005PhRvL 95v6801K doi 10 1103 PhysRevLett 95 226801 PMID 16384250 S2CID 6080059 Haldane F D M 31 October 1988 Model for a Quantum Hall Effect without Landau Levels Condensed Matter Realization of the Parity Anomaly Physical Review Letters 61 18 2015 2018 Bibcode 1988PhRvL 61 2015H doi 10 1103 PhysRevLett 61 2015 PMID 10038961 Kaplan David B 1992 A method for simulating chiral fermions on the lattice Physics Letters B 288 3 4 342 347 arXiv hep lat 9206013 Bibcode 1992PhLB 288 342K CiteSeerX 10 1 1 286 587 doi 10 1016 0370 2693 92 91112 m S2CID 14161004 Golterman Maarten F L Jansen Karl Kaplan David B 1993 Chern Simons currents and chiral fermions on the lattice Physics Letters B 301 2 3 219 223 arXiv hep lat 9209003 Bibcode 1993PhLB 301 219G doi 10 1016 0370 2693 93 90692 b S2CID 9265777 Bernevig B Andrei Zhang Shou Cheng 14 March 2006 Quantum Spin Hall Effect Physical Review Letters 96 10 106802 arXiv cond mat 0504147 Bibcode 2006PhRvL 96j6802B doi 10 1103 PhysRevLett 96 106802 PMID 16605772 S2CID 2618285 Kane C L Mele E J 28 September 2005 Z2 Topological Order and the Quantum Spin Hall Effect Physical Review Letters 95 14 146802 arXiv cond mat 0506581 Bibcode 2005PhRvL 95n6802K doi 10 1103 PhysRevLett 95 146802 PMID 16241681 S2CID 1775498 Bernevig B Andrei Hughes Taylor L Zhang Shou Cheng 2006 12 15 Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells Science 314 5806 1757 1761 arXiv cond mat 0611399 Bibcode 2006Sci 314 1757B doi 10 1126 science 1133734 ISSN 0036 8075 PMID 17170299 S2CID 7295726 Konig Markus Wiedmann Steffen Brune Christoph Roth Andreas Buhmann Hartmut Molenkamp Laurens W Qi Xiao Liang Zhang Shou Cheng 2007 11 02 Quantum Spin Hall Insulator State in HgTe Quantum Wells Science 318 5851 766 770 arXiv 0710 0582 Bibcode 2007Sci 318 766K doi 10 1126 science 1148047 ISSN 0036 8075 PMID 17885096 S2CID 8836690 Konig Markus Wiedmann Steffen Brune Christoph Roth Andreas Buhmann Hartmut Molenkamp Laurens W Qi Xiao Liang Zhang Shou Cheng November 2 2007 Quantum Spin Hall Insulator State in HgTe Quantum Wells Science 318 5851 766 770 arXiv 0710 0582 Bibcode 2007Sci 318 766K doi 10 1126 science 1148047 PMID 17885096 S2CID 8836690 Further reading editMaciejko J Hughes T L Zhang S C 2011 The Quantum Spin Hall Effect Annual Review of Condensed Matter Physics 2 31 53 Bibcode 2011ARCMP 2 31M doi 10 1146 annurev conmatphys 062910 140538 Qi X L and Zhang S C 2011 Rev of Mod Phys https journals aps org rmp abstract 10 1103 RevModPhys 83 1057 Retrieved from https en wikipedia org w index php title Quantum spin Hall effect amp oldid 1155453034, wikipedia, wiki, book, books, library,
