拓撲絕緣體

拓撲絕緣體是一種內部絕緣，界面允許電荷移動的材料。

在拓撲絕緣體的內部，電子能帶結構和常規的絕緣體相似，其費米能級位於導帶和價帶之間。在拓撲絕緣體的邊界或是表面存在一些特殊的量子態，這些量子態位於塊體能帶結構的帶隙之中，從而允許導電^[1]。拓撲絕緣體的塊體可以用類似拓撲學中的虧格的整數表徵，是拓撲序的一個特例^[2]。利用體邊對應，可以預言材料在開邊界條件下拓撲邊緣態的性質。

受拓撲保護的邊緣態（一維）在碲化汞/碲化鎘量子阱中被預言於2006年^[3]，隨後於2007年由實驗觀測證實^[4]。很快，拓撲絕緣體又被預言存在於含鉍的二元化合物三維固體中^[5]^[6]。第一個實驗實現的三維拓撲絕緣體是在銻化鉍中被觀察到^[7]，隨後不同實驗組又通過角分辨光電子譜的方法，在銻，碲化鉍，硒化鉍，碲化銻中觀察到了受拓撲保護的表面量子態^[8]。現在人們相信，在其他一些材料體系中，也存在拓撲絕緣態^[9]。在這些材料中，由於自然存在的缺陷，費米能級實際上是位於導帶或是位於價帶，必須通過摻雜或者通過改變其電勢將費米能級調節到能隙之中，以觀察拓撲保護的邊緣態^[10]^[11]。

類似的邊緣效應同樣出現於量子霍爾效應之中。以整數量子霍爾效應為例，在強垂直磁場下，低溫的二維系統體態性質可以被拓撲量子數標記。在數學中，此拓撲量子數被稱作陳數（Chern numbers）。二維量子霍爾系統邊緣出現手性邊緣態，陳數對應手性邊緣態的數目和量子化電導 ${\frac {e^{2}}{h}}$ ^[12]。在拓撲材料的理論研究中，體邊對應一直扮演着重要的角色。體邊對應指的是，當真實的材料包含的原子數目非常大（數量級為 $10^{20}$ 或更多），我們可以把此材料近似於熱力學極限，並用布洛赫定理與能帶理論來描述材料體態性質，並根據體態性質來預言材料在開邊界條件下受拓撲保護的邊緣態的性質^[13]。

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[1]
Kane, C. L.; Mele, E. J. Quantum Spin Hall Effect in Graphene. Physical Review Letters (American Physical Society (APS)). 2005-11-23, 95 (22): 226801. doi:10.1103/physrevlett.95.226801.
[2]
Kane, C. L.; Mele, E. J. Z₂ Topological Order and the Quantum Spin Hall Effect. Physical Review Letters. 30 September 2005, 95 (14): 146802. doi:10.1103/PhysRevLett.95.146802.
[3]
Bernevig, B. Andrei; Taylor L. Hughes, Shou-Cheng Zhang. Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science. 2006-12-15, 314 (5806): 1757–1761 [2010-03-25]. PMID 17170299. doi:10.1126/science.1133734. （原始內容存檔於2008-06-17）.
[4]
Konig, Markus; Steffen Wiedmann, Christoph Brune, Andreas Roth, Hartmut Buhmann, Laurens W. Molenkamp, Xiao-Liang Qi, Shou-Cheng Zhang. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science. 2007-11-02, 318 (5851): 766–770 [2010-03-25]. PMID 17885096. doi:10.1126/science.1148047. （原始內容存檔於2010-05-11）.
[5]
Fu, Liang; C. L. Kane. Topological insulators with inversion symmetry. Physical Review B. 2007-07-02, 76 (4): 045302 [2010-03-26]. doi:10.1103/PhysRevB.76.045302.
[6]
Shuichi Murakami. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New Journal of Physics. 2007, 9 (9): 356–356 [2010-03-26]. ISSN 1367-2630. doi:10.1088/1367-2630/9/9/356.
[7]
Hsieh, D.; D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava & M. Z. Hasan. A Topological Dirac insulator in a 3D quantum spin Hall phase. Nature. 2008, 452 (9): 970–974 [2010]. PMID 18432240. doi:10.1038/nature06843. （原始內容存檔於2009-12-23）.
[8]
Hasan, M. Z; C. L Kane. Topological Insulators. 1002.3895. 2010-02-20 [2010-04-27]. （原始內容存檔於2021-03-08）.
[9]
Lin, Hsin; L. Andrew Wray, Yuqi Xia, Suyang Xu, Shuang Jia, Robert J. Cava, Arun Bansil, M. Zahid Hasan. Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nat Mater. 2010-07, 9 (7): 546–549 [2010-08-05]. ISSN 1476-1122. PMID 20512153. doi:10.1038/nmat2771.
[10]
Hsieh, D.; Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, M. Z. Hasan. Observation of Time-Reversal-Protected Single-Dirac-Cone Topological-Insulator States in Bi2Te3 and Sb2Te3. Physical Review Letters. 2009, 103 (14): 146401 [2010-03-25]. PMID 19905585. doi:10.1103/PhysRevLett.103.146401.
[11]
Noh, H.-J.; H. Koh, S.-J. Oh, J.-H. Park, H.-D. Kim, J. D. Rameau, T. Valla, T. E. Kidd, P. D. Johnson, Y. Hu and Q. Li. Spin-orbit interaction effect in the electronic structure of Bi2Te3 observed by angle-resolved photoemission spectroscopy. EPL Europhysics Letters. 2008, 81 (5): 57006 [2010-04-25]. doi:10.1209/0295-5075/81/57006.
[12]
Hatsugai Y. Chern number and edge states in the integer quantum Hall effect.. Phys Rev Lett. 1993, 71 (22): 3697–3700. PMID 10055049. doi:10.1103/PhysRevLett.71.3697.
[13]
Bernevig, B. Andrei; Hughes, Taylor L. Topological insulators and topological superconductors. Princeton. 2013. ISBN 978-1-4008-4673-3. OCLC 934514528.

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Moore, Joel. The Birth of Topological Insulators. Nature. 2010, 464 (7286): 194. PMID 20220837. doi:10.1038/nature08916.
Kane, C. L.; Mele, E. J. A New Spin on the Insulating State. Science. 2006, 314 (5806): 1692. PMID 17170283. doi:10.1126/science.1136573.
Kane, C.L. Topological Insulator: An Insulator with a Twist. Nature. 2008, 4 (5): 348. doi:10.1038/nphys955.
Witze, Alexandra. Topological Insulators: Physics On the Edge. Science News. 2010. （原始內容存檔於2012-09-27）.
Brumfield, Geoff. Topological insulators: Star material : Nature News. Nature. 2010, 466 (7304): 310–311 [2010-08-06]. PMID 20631773. doi:10.1038/466310a. （原始內容存檔於2017-10-12）.
Murakami, Shuichi. Focus on Topological Insulators. New Journal of Physics. 2010. （原始內容存檔於2011-03-09）.

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[Kane_Mele_2005_p.-1] [1]
Kane, C. L.; Mele, E. J. Quantum Spin Hall Effect in Graphene. Physical Review Letters (American Physical Society (APS)). 2005-11-23, 95 (22): 226801. doi:10.1103/physrevlett.95.226801.

[2] [2]
Kane, C. L.; Mele, E. J. Z₂ Topological Order and the Quantum Spin Hall Effect. Physical Review Letters. 30 September 2005, 95 (14): 146802. doi:10.1103/PhysRevLett.95.146802.

[3] [3]
Bernevig, B. Andrei; Taylor L. Hughes, Shou-Cheng Zhang. Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science. 2006-12-15, 314 (5806): 1757–1761 [2010-03-25]. PMID 17170299. doi:10.1126/science.1133734. （原始內容存檔於2008-06-17）.

[4] [4]
Konig, Markus; Steffen Wiedmann, Christoph Brune, Andreas Roth, Hartmut Buhmann, Laurens W. Molenkamp, Xiao-Liang Qi, Shou-Cheng Zhang. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science. 2007-11-02, 318 (5851): 766–770 [2010-03-25]. PMID 17885096. doi:10.1126/science.1148047. （原始內容存檔於2010-05-11）.

[5] [5]
Fu, Liang; C. L. Kane. Topological insulators with inversion symmetry. Physical Review B. 2007-07-02, 76 (4): 045302 [2010-03-26]. doi:10.1103/PhysRevB.76.045302.

[6] [6]
Shuichi Murakami. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New Journal of Physics. 2007, 9 (9): 356–356 [2010-03-26]. ISSN 1367-2630. doi:10.1088/1367-2630/9/9/356.

[7] [7]
Hsieh, D.; D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava & M. Z. Hasan. A Topological Dirac insulator in a 3D quantum spin Hall phase. Nature. 2008, 452 (9): 970–974 [2010]. PMID 18432240. doi:10.1038/nature06843. （原始內容存檔於2009-12-23）.

[8] [8]
Hasan, M. Z; C. L Kane. Topological Insulators. 1002.3895. 2010-02-20 [2010-04-27]. （原始內容存檔於2021-03-08）.

[9] [9]
Lin, Hsin; L. Andrew Wray, Yuqi Xia, Suyang Xu, Shuang Jia, Robert J. Cava, Arun Bansil, M. Zahid Hasan. Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nat Mater. 2010-07, 9 (7): 546–549 [2010-08-05]. ISSN 1476-1122. PMID 20512153. doi:10.1038/nmat2771.

[10] [10]
Hsieh, D.; Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, M. Z. Hasan. Observation of Time-Reversal-Protected Single-Dirac-Cone Topological-Insulator States in Bi2Te3 and Sb2Te3. Physical Review Letters. 2009, 103 (14): 146401 [2010-03-25]. PMID 19905585. doi:10.1103/PhysRevLett.103.146401.

[11] [11]
Noh, H.-J.; H. Koh, S.-J. Oh, J.-H. Park, H.-D. Kim, J. D. Rameau, T. Valla, T. E. Kidd, P. D. Johnson, Y. Hu and Q. Li. Spin-orbit interaction effect in the electronic structure of Bi2Te3 observed by angle-resolved photoemission spectroscopy. EPL Europhysics Letters. 2008, 81 (5): 57006 [2010-04-25]. doi:10.1209/0295-5075/81/57006.

[pmid10055049-12] [12]
Hatsugai Y. Chern number and edge states in the integer quantum Hall effect.. Phys Rev Lett. 1993, 71 (22): 3697–3700. PMID 10055049. doi:10.1103/PhysRevLett.71.3697.

[Bernevig_Hughes_2013_p.-13] [13]
Bernevig, B. Andrei; Hughes, Taylor L. Topological insulators and topological superconductors. Princeton. 2013. ISBN 978-1-4008-4673-3. OCLC 934514528.

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拓撲絕緣體

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拓撲絕緣體

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