Universal conductance fluctuations in Sierpinski carpets

Abstract

We theoretically investigate the conductance fluctuation of two-terminal device in Sierpinski carpets. We find that, for the circular orthogonal ensemble (COE), the conductance fluctuation does not display a universal feature; but for circular unitary ensemble (CUE) without time-reversal symmetry or circular symplectic ensemble (CSE) without spin-rotational symmetry, the conductance fluctuation can reach an identical universal value of 0.74±0.01(e²/h). We further find that the conductance distributions around the critical disorder strength for both CUE and CSE systems share the similar distribution forms. Our findings provide a better understanding of the electronic transport properties of the regular fractal structure.

Keywords

electronic transport / conductance fluctuation

References

[1] W. Ji, H. Q. Xu, and H. Guo, Quantum description of transport phenomena: Recent progress, Front. Phys. 9(6), 671 (2014)

[2] J. Zhuang, Y. Wang, Y. Zhou, J. Wang, and H. Guo, Impurity-limited quantum transport variability in magnetic tunnel junctions, Front. Phys. 12(4), 127304 (2017)

[3] H. Z. Lu and S. Q. Shen, Quantum transport in topological semimetals under magnetic fields, Front. Phys. 12(3), 127201 (2017)

[4] L. B. Altshuler, Fluctuations in the extrinsic conductivity of disordered conductors, JETP Lett. 41, 648 (1985)

[5] P. A. Lee and A. D. Stone, Universal conductance fluctuations in metals, Phys. Rev. Lett. 55(15), 1622 (1985)

[6] P. A. Lee, A. D. Stone, and H. Fukuyama, Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field, Phys. Rev. B 35(3), 1039 (1987)

[7] C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69(3), 731 (1997)

[8] S. B. Kaplan and A. Hartstein, Universal conductance fluctuations in narrow Si accumulation layers, Phys. Rev. Lett. 56(22), 2403 (1986)

[9] A. García-Martín and J. J. Sáenz, Universal conductance distributions in the crossover between diffusive and localization regimes, Phys. Rev. Lett. 87(11), 116603 (2001)

[10] W. Ren, Z. Qiao, J. Wang, Q. Sun, and H. Guo, Universal spin-Hall conductance fluctuations in two dimensions, Phys. Rev. Lett. 97(6), 066603 (2006)

[11] Z. Qiao, J. Wang, Y. Wei, and H. Guo, Universal quantized spin-Hall conductance fluctuation in graphene, Phys. Rev. Lett. 101(1), 016804 (2008)

[12] M. Yu. Kharitonov and K. B. Efetov, Universal conductance fluctuations in graphene, Phys. Rev. B 78(3), 033404 (2008)

[13] J. Wurm, A. Rycerz, I. Adagideli, M. Wimmer, K. Richter, and H. U. Baranger, Symmetry classes in graphene quantum dots: Universal spectral statistics, weak localization, and conductance fluctuations, Phys. Rev. Lett. 102(5), 056806 (2009)

[14] Z. Qiao, Y. Xing, and J. Wang, Universal conductance fluctuation of mesoscopic systems in the metal-insulator crossover regime, Phys. Rev. B 81(8), 085114 (2010)

[15] Z. M. Liao, B. H. Han, H. Z. Zhang, Y. B. Zhou, Q. Zhao, and D. P. Yu, Current regulation of universal conductance fluctuations in bilayer graphene, New J. Phys. 12(8), 083016 (2010)

[16] Z. Qiao, W. Ren, and J. Wang, Universal spin-Hall conductance fluctuations in two-dimensional mesoscopic systems, Mod. Phys. Lett. B 25(06), 359 (2011)

[17] Z. Li, T. Chen, H. Pan, F. Song, B. Wang, J. Han, Y. Qin, X. Wang, R. Zhang, J. Wan, D. Xing, and G. Wang, Twodimensional universal conductance fluctuations and the electron-phonon interaction of surface states in Bi₂Te₂Se microflakes, Sci. Rep. 2(1), 595 (2012)

[18] E. Rossi, J. H. Bardarson, M. S. Fuhrer, and S. Das Sarma, Universal conductance fluctuations in Dirac materials in the presence of long-range disorder, Phys. Rev. Lett. 109(9), 096801 (2012)

[19] Z.-G. Li, S. Zhang, and F.-Q. Song, Universal conductance fluctuations of topological insulators, Acta Physica Sinica 64(8), 97202 (2015)

[20] L.-X. Wang, S. Wang, J.-G. Li, C.-Z. Li, D. P. Yu, and Z.-M. Liao, Universal conductance fluctuations in Dirac semimetal Cd₃As₂ nanowires, Phys. Rev. B 94, 161402(R) (2016)

[21] Y. Hu, H. Liu, H. Jiang, and X. C. Xie, Numerical study of universal conductance fluctuations in threedimensional topological semimetals, Phys. Rev. B 96(13), 134201 (2017)

[22] Y. Q. Li, K. H. Wu, J. R. Shi, and X. C. Xie, Electron transport properties of three-dimensional topological insulators, Front. Phys. 7(2), 165 (2012)

[23] B. B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 156(3775), 636 (1967)

[24] B. B. Mandelbrot, Self-affine fractals and fractal dimension, Phys. Scr. 32(4), 257 (1985)

[25] R. Rammal, Nature of eigenstates on fractal structures, Phys. Rev. B 28(8), 4871 (1983)

[26] A. Chakrabarti and B. Bhattacharyya, Sierpinski gasket in a magnetic field: Electron states and transmission characteristics, Phys. Rev. B 56(21), 13768 (1997)

[27] X. R. Wang, Localization in fractal spaces: Exact results on the Sierpinski gasket, Phys. Rev. B 51(14), 9310 (1995)

[28] Y. Asada, K. Slevin, and T. Ohtsuki, Possible Anderson transition below two dimensions in disordered systems of noninteracting electrons, Phys. Rev. B 73(4), 041102 (2006)

[29] M. K. Schwalm and W. A. Schwalm, Length scaling of conductance distribution for random fractal lattices, Phys. Rev. B 54(21), 15086 (1996)

[30] Y. Liu, Z. Hou, P. M. Hui, and W. Sritrakool, Electronic transport properties of Sierpinski lattices, Phys. Rev. B 60(19), 13444 (1999)

[31] Z. Lin, Y. Cao, Y. Liu, and P. M. Hui, Electronic transport properties of Sierpinski lattices in a magnetic field, Phys. Rev. B 66(4), 045311 (2002)

[32] C. Y. Ho and C. R. Chang, Spin transport in fractal conductors with the Rashba spin–orbit coupling, Spin 02(02), 1250008 (2012)

[33] B. R. Lee, C. R. Chang, and I. Klik, Spin transport in multiply connected fractal conductors, Spin 04(03), 1450007 (2014)

[34] E. van Veen, S. Yuan, M. I. Katsnelson, M. Polini, and A. Tomadin, Quantum transport in Sierpinski carpets, Phys. Rev. B 93(11), 115428 (2016)

[35] M. Brzezińska, A. M. Cook, and T. Neupert, Topology in the Sierpiński–Hofstadter problem, Phys. Rev. B 98(20), 205116 (2018)

[36] J. Shang, Y. Wang, M. Chen, J. Dai, X. Zhou, J. Kuttner, G. Hilt, X. Shao, J. M. Gottfried, and K. Wu, Assembling molecular Sierpiński triangle fractals, Nat. Chem. 7(5), 389 (2015)

[37] S. N. Kempkes, M. R. Slot, S. E. Freeney, S. J. M. Zevenhuizen, D. Vanmaekelbergh, I. Swart, and C. M. Smith, Design and characterization of electrons in a fractal geometry, Nat. Phys. 15(2), 127 (2019)

[38] Z. Qiao, W. Ren, J. Wang, and H. Guo, Low-field phase diagram of the spin Hall effect in the mesoscopic regime, Phys. Rev. Lett. 98(19), 196402 (2007)

[39] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge: Cambridge University Press, England, 1995

[40] It should be noted that the rms(G) in Figs. 3 and 4 seem not to be exactly same mainly due to discrete distribution of disorder strength W in numerical method. In order to reasonably describe the universal conductance fluctuations, we use a range of [–0.1, 0.1] (e²/h) based on the UCF value. The same method also used in other papers (Refs. [10–12]).

[41] In circular orthogonal ensemble, there does not exist a metal-insulator transition when system dimension below 3 according to the scaling theory of localization. Therefore, there is not UCF in circular orthogonal ensemble.

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature