Abstract
We explore a combined effect of hexagonal warping and a finite effective mass on both the tunneling density of electronic surface states and the structure of Landau levels of 3D topological insulators. We find the increasing warping to transform the square-root van Hove singularity into a logarithmic one. For moderate warping, an additional logarithmic singularity and a jump in the tunneling density of surface states appear. By combining the perturbation theory and the WKB approximation, we calculate the Landau levels in the presence of hexagonal warping. We predict that due to the degeneracy removal, the evolution of Landau levels in the magnetic field is drastically modified.
Similar content being viewed by others
References
M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013).
L. Fu, Phys. Rev. Lett. 103, 266801 (2009).
C.-X. Liu, X.-L. Qi, H. J. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Phys. Rev. B: Condens. Matter 82, 045122 (2010).
Y. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S.-C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science (Washington) 325, 178 (2009).
Zh. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, and A. Kapitulnik, Phys. Rev. B: Condens. Matter 84, 041104 (2011).
K. Kuroda, M. Arita, K. Miyamoto, M. Ye, J. Jiang, A. Kimura, E. E. Krasovskii, E. V. Chulkov, H. Iwasawa, T. Okuda, K. Shimada, Y. Ueda, H. Namatame, and M. Taniguchi, Phys. Rev. Lett. 105, 076802 (2010).
M. Nomura, S. Souma, A. Takayama, T. Sato, T. Takahashi, K. Eto, K. Segawa, and Y. Ando, Phys. Rev. B: Condens. Matter 89, 045134 (2014).
C. M. Wang and F. J. Yu, Phys. Rev. B: Condens. Matter 84, 155440 (2011).
X. Xiao and W. Wen, Phys. Rev. B: Condens. Matter 88, 045442 (2013).
S. Smirnov, Phys. Rev. B: Condens. Matter 88, 205301 (2013).
Z.-G. Fu, F. Zheng, Z. Wang, and P. Zhang, Prog. Theor. Exp. Phys., 103I01 (2013).
S. Urazhdin, D. Bilc, S. D. Mahanti, S. H. Tessmer, Th. Kyratsi, and M. G. Kanazidis, Phys. Rev. B: Condens. Matter 69, 085313 (2004).
Zh. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, Phys. Rev. Lett. 104, 016401 (2010).
P. Sessi, M. M. Otrokov, T. Bathon, M. G. Vergniory, S. S. Tsirkin, K. A. Kokh, O. E. Tereshchenko, E. V. Chulkov, and M. Bode, Phys. Rev. B: Condens. Matter 88, 161407(R) (2013).
V. S. Stolyarov, T. Cren, C. Brun, S. I. Bozhko, L. V. Yashina, and D. Roditchev, (in preparation).
T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa, Phys. Rev. B: Condens. Matter 82, 081305(R) (2010).
T. Zhang, N. Levy, J. Ha, Y. Kuk, and J. A. Stroscio, Phys. Rev. B: Condens. Matter 87, 115410 (2013).
Y.-S. Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, and T. Sasagawa, arXiv:1408.0873 (unpublished).
Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Q.-K. Xue, Phys. Rev. Lett. 108, 016401 (2012).
A. Yu. Dmitriev, N. I. Fedotov, V. F. Nasretdinova, and S. V. Zaitsev-Zotov, arXiv:1408.4991 (unpublished).
K. Saha, S. Das, K. Sengupta, and D. Sen, Phys. Rev. B: Condens. Matter 84, 165439 (2011).
P. Schwab and M. Dzierzawa, Phys. Rev. B: Condens. Matter 85, 155403 (2012).
M. M. Vazifeh and M. Franz, Phys. Rev. B: Condens. Matter 86, 045451 (2012).
Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39 (2), 78 (1984); J. Phys. C: Solid State Phys. 17, 6039 (1984).
P. G. Silvestrov, P. W. Brouwer, and E. G. Mishchenko, Phys. Rev. B: Condens. Matter 86, 075302 (2012).
F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. B: Condens. Matter 86, 081303(R) (2012).
S. Basak, H. Lin, L. A. Wray, S.-Y. Xu, L. Fu, M. Z. Hasan, and A. Bansil, Phys. Rev. B: Condens. Matter 84, 121401(R) (2011).
Z. Yang and J. H. Han, Phys. Rev. B: Condens. Matter 83, 045415 (2011).
For a review, see, e.g., I. M. Lifshitz, M. Y. Azbel, and M. I. Kaganov, Electron Theory of Metals (Nauka, Moscow, 1971; Consultants Bureau, New York, 1973).
L. A. Falkovsky, Sov. Phys. JETP 22, 423 (1965)
A. Yu. Ozerin and L. A. Falkovsky, Phys. Rev. B: Condens. Matter 85, 205143 (2012).
M. R. Scholz, J. Sánchez-Barriga, J. Braun, D. Marchenko, A. Varykhalov, M. Lindroos, Y. J. Wang, H. Lin, A. Bansil, J. Minár, H. Ebert, A. Volykhov, L. V. Yashina, and O. Rader, Phys. Rev. Lett. 110, 216801 (2013).
A. Wolos, S. Szyszko, A. Drabinska, M. Kaminska, S. G. Strzelecka, A. Hruban, A. Materna, and M. Piersa, Phys. Rev. Lett. 109, 247604 (2012).
A. A. Schafgans, K. W. Post, A. A. Taskin, Y. Ando, X.-Liang Qi, B. C. Chapler, and D. N. Basov, Phys. Rev. B: Condens. Matter 85, 195440 (2012).
Author information
Authors and Affiliations
Corresponding author
Additional information
The article is published in the original.
Rights and permissions
About this article
Cite this article
Repin, E.V., Burmistrov, I.S. Surface states in a 3D topological insulator: The role of hexagonal warping and curvature. J. Exp. Theor. Phys. 121, 509–520 (2015). https://doi.org/10.1134/S1063776115100192
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063776115100192