Prof. Cristiane Morais Smith, Institute for Theoretical Physics, Utrecht University, the Netherlands

DOI: 10.65398/BDIE3082

In this work, we provide an overview of the progress in understanding and manipulating topological states of matter. These materials are insulating in the bulk, but host dissipation-free states at the edges. The experimental observation of the quantum spin Hall effect in single-layer bismuth in 2017 has shown that it is possible to realize this fascinating state of matter at room temperature, as has opened new perspectives in the field. More recently, by carefully controlling the growth conditions, it was found that Sierpinski gasket fractals of single-layer bismuth may form spontaneously on top of InSb substrates. These fractal structures with dimensions d = 1.58 were shown to be topological, and to exhibit both zero-dimensional end states, as well as dissipation-free one-dimensional edge currents. The use of topological fractal structures may be a convenient alternative to generate more one-dimensional edge currents, since fractals have basically no bulk, only edges, and may pave the way to the construction of efficient nanodevices based on topological insulators.

The ages of humanity are named according to the materials used to build tools, which determine the life standard in a certain period. After going through the Stone Age, the Bronze Age, the Iron Age, among others, we are now slowly approaching the end of the Silicon Age. Currently, our devices are based on transistors made mostly of silicon, which is a semiconductor that operates between a current carrying metallic state (1) and an insulating state (0). With a sequence of 0 and 1 bits, information is encoded and read, allowing for the construction of very useful devices, such as cell phones, computers, television, etc.

However, our communication-based society is reaching a boundary because storage of information in data centers consumes an enormous amount of energy, much beyond what is acceptable. We are literally burning our planet, and measures are necessary from scientists in all fields.

One possibility to keep the actual life standard and reduce the consumption of energy would be to minimize the waste. When the semiconductor operates in the metallic state, the moving electrons sometimes bump into impurities or defects of the material and lose energy, leading to heating and dissipation. If the heating could be avoided, much energy could be spared.

A conceivable way to generate currents without losses is to use superconductors. Those have been discovered more than a century ago, and have been used for many purposes, including Josephson junction arrays, which are at the heart of the more modern quantum computers [1]. However, the dream of finding a superconductor operating at room temperature has not yet been realized, and even the best performing high-Tc superconductors require quite a significant cooling, which also consumes energy [2]. A second alternative would be to use the so-called topological insulators [3]. This is a novel state of matter, discovered in 1980 by von Klitzing when measuring transport at very low temperatures (~ 1 Kelvin) in a two-dimensional (2D) gas of electrons (GaAs/AlGaAs quantum well) in the presence of a strong perpendicular magnetic field [4]. It was found that these materials are insulating in the bulk but metallic at the edges, and the edge currents have no losses. Since the discovery of this phenomenon, called the quantum Hall effect, several developments followed. Notably, it was understood that there is a topological invariant protecting these states, based on the symmetries and dimensionality of the system [5]. However, two factors were hampering the use of topological insulators in technology: first, these edge states could be detected only at very low temperatures; and second, they required a very strong magnetic field.

The field received strong impulse in 2005, when Kane and Mele showed that a magnetic field is not a necessary condition for realizing a topological insulator. Indeed, a strong spin-orbit coupling can lead to a quantum spin Hall effect because the intrinsic spin-orbit coupling acts as a positive magnetic field for spin up and a negative magnetic field for spin down [6]. Therefore, in the quantum spin Hall effect, no charge is transferred, but spin is [7]. This effect was observed experimentally by the group of Molenkamp in 2006, but again a very low temperature was required [8].

In the meantime, graphene, a single layer of carbon atoms, was synthesized, and it was shown that the quantum Hall effect could be measured at room temperature in graphene. However, a very large magnetic field, of about 29 Tesla, was necessary for that [9]. The field of topological materials has seen a huge progress when bismuthene, a single layer of bismuth atoms was synthesized and investigated. Since the atoms of bismuth are heavy, the spin-orbit coupling is very strong in bismuthene, leading to a very robust quantum spin Hall effect, with a room temperature gap [10]. This seems to be the long-sought material, which could possibly be used to build novel tools, since the topological edge currents arise due to intrinsic properties and the insulating gap is as large as room temperature. Currently, novel two-dimensional materials are being synthesized, and combinations of them, involving twisting of one layer on top of another, have proven to lead to novel effects [11].

An interesting question involving these topological edge states is that the part in which we are interested, namely the dissipation-free currents, are precisely at the edge, and miniaturization would require us to engineer materials which have as many edges as possible. A possible alternative would be to design quantum fractals.

Fractals are structures that lack translational invariance but exhibit scale invariance. By zooming in on a fractal, one sees always the same structure [12]. Usually, fractals also have a non-integer dimension, the so-called Hausdorff dimension, thus allowing us to investigate quantum effects at non-integer dimensions. Let us consider a simple mathematical fractal, the Sierpinski triangle (see Fig. 1). We start with a triangle in two dimensions and cut an inverted triangle (white) in the center of the original black triangle. We obtain 3 smaller triangles, identical to the original one. Then, we repeat the procedure and obtain 9 smaller triangles after cutting an inverted triangle in the center of the 3 smaller ones. After repeating this procedure many times, the final structure is no longer a surface, as the original triangle, neither a line, but something in between. Its Hausdorff dimension is d = ln3/ln2 = 1.58, which is given by the log of the number of copies obtained after scaling.

Fractals are closely related to Nature and humankind. They are found in the shape of rivers, coastlines of countries, roots and branches of trees, ferns, etc. In the human body, the circulatory system, the intestines, the lungs, the neuronal system, and even the heartbeat are fractal. Fractal structures are very convenient when there are exchanges between the system and the environment. Since Nature is fractal, our senses perceive fractals as beautiful. Therefore, they have been for long used in decorative arts, much before they were understood and classified by mathematicians. One example are mosaics in Roman Churches, as shown in Fig. 2. In addition, several painters, like Escher and Pollock, used fractals in their artwork. Fractals are also very suitable for technology. Indeed, an antenna made of a fractal can send and receive simultaneously small and large frequencies because of the different length scales in the fractal. In addition, solar cells made of fractals can store more energy upon increasing the complexity of the fractal, in the same 2D area.

In the eighties, fractals were thoroughly investigated. The works of Mandelbroot, De Gennes, Orbach, among others, allowed for much progress in the field [12, 14]. However, those were classical fractals. Now, the time has come to understand quantum fractals. In 2015, the group of Kai Wu has shown how to build fractals using aromatic bromo compounds as the building blocks on a Ag(111) surface [15]. Then, in 2019 the first electronic quantum fractal was realized (see Fig. 3), and it was shown that electrons placed in a fractal can perceive the fractal dimension d = 1.58 [16]. Moreover, it was found that diffusion of photons through a fractal leads to a different scaling behavior of the mean-square displacement than in the classical case [17]. Finally, topological properties were investigated in several types of fractals, in a variety of quantum simulator platforms, from photonics [18] to acoustic [19, 20] ones.

Recently, it was found that fractals of single-layer bismuth can form spontaneously upon depositing bismuth on the surface of InAs, see Fig. 4 [21]. Moreover, these fractals at dimension 1.58 display topological properties of 2D and 1D materials, then reconciling the best of the two neighboring dimensions. Indeed, the Sierpinski fractals of bismuth were shown to exhibit corner states, which could be used as qubits for quantum computers, as well as 1D propagating edge states, characteristic of 2D materials. In addition, the fractal has internal and external edges, and both types of edge states were detected [22]. The detection of topological properties in these real materials opens the path to their use in novel, more eco-friend technological devices.

In conclusion, topological insulators have emerged as a possible platform to realize dissipation-free currents, hence preventing energy waste and leading to better performance in quantum nanotechnology. Although the first discovered topological states required high magnetic fields and low temperatures, the field has evolved considerably during the last decades, and it is now possible to realize the topological quantum spin Hall effect, with dissipation-free currents, at room temperature in bismuthene. Moreover, Sierpinski gaskets of single-layer bismuth were spontaneously formed in InSb substrates upon manipulating the growth conditions. Theoretical studies and spectroscopy showed that these fractals, with dimension d = 1.58, combine the best topological features of the two-dimensional and one-dimensional topological materials, namely, one-dimensional edge states, as well as zero-dimensional end states, potentially useful for building qubits for quantum computers. The use of topological fractals in nanotechnology would allow us to avoid the energy losses that occur in silicon-based devices, thus leading to a more sustainable life in the Anthropocene. These studies indicate that our search for new materials should not be restricted to integer dimensions. Instead, inspired by Nature and by the human body, new quantum materials will be designed and engineered in a way to increase performance and harmony with the environment. In this realm, quantum topological fractals hold promises to play a prominent role in realizing technological devices with little or no energy loss.

References:

[1] A.F. Kockum and F. Nori, “Quantum Bits with Josephson Junctions”. In: Tafuri, F. (eds) Fundamentals and Frontiers of the Josephson Effect. Springer Series in Materials Science, vol 286, Springer, Cham. (2019).

[2] A. Schilling, M. Cantoni, J.D. Guo, H.R. Ott, “Superconductivity above 130 K in the Hg–Ba–Ca–Cu–O system”, Nature 363, 56 (1993).

[3] M.Z. Hasan, and C.L. Kane, “Colloquium: Topological insulators”, Rev. Mod. Phys. 82, 3045 (2010).

[4] K. von Klitzing, G. Dorda, and M. Pepper, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance”, Phys. Rev. Lett. 45, 494 (1980).

[5] D.J. Thouless, M. Kohmoto, M.P. Nightingale, and M. Den Nijs, “Quantized Hall Conductance in a Two Dimensional Periodic Potential”, Phys. Rev. Lett. 49, 405 (1982).

[6] C.L. Kane and E.J. Mele, “Quantum Spin Hall Effect in Graphene”, Phys. Rev. Lett. 95, 226801 (2005).

[7] C.L. Kane and E.J. Mele, “Z₂ Topological Order and the Quantum Spin Hall Effect”, Phys. Rev. Lett. 95, 146802 (2005).

[8] M. Koenig, S. Wiedmann, C. Brne, A. Roth, H. Buhmann, L.W. Molenkamp, X.L. Qi and S.C. Zhang, “Quantum Spin Hall Insulator State in HgTe Quantum Wells”, Science 318, 766 (2007).

[9] K.S. Novoselov, et al. “Room-temperature quantum Hall effect in graphene”, Science 315, 1379 (2007).

[10] F. Reis, et al., “Bismuthene on a SiC substrate: a candidate for a high-temperature quantum spin Hall material”, Science 357, 287 (2017).

[11] S. Carr et al., “Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle”, Phys. Rev. B 95, 075420 (2017).

[12] B.B. Mandelbrot, “Fractals: Form, Chance and Dimension”, (W.H. Freeman 1977); “The Fractal Geometry of Nature” (W.H. Freeman, 1983).

[13] E. Conversano, L. Tedeschini Lalli, “Sierpinski triangles in stone on medieval floors in Rome”, Journal of Applied Mathematics IV, 113 (2011).

[14] R. Orbach, “Dynamics of fractal networks”, Science 231, 814 (1986).

[15] J. Shang et al. “Assembling molecular Sierpiński triangle fractals”, Nat. Chem. 7, 389 (2015).

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

[17] X.-Y. Xu, X.-W. Wang, D.-Y. Chen, C. Morais Smith, and X.-M. Jin, “Quantum transport in fractal networks”, Nat. Photonics 15, 703 (2021).

[18] T. Biesenthal, L.J. Maczewsky, Z. Yang, M. Kremer, M. Segev, A. Szameit, and M. Heinrich, “Fractal photonic topological insulators”, Science 376, 1114 (2022).

[19] S. Zheng et al., “Observation of fractal topological states in acoustic metamaterials”, Sci. Bull. 67, 2069 (2022).

[20] J. Li, Q. Mo, J.-H. Jiang, and Z. Yang, “Higher-order topological phase in an acoustic fractal lattice”, Sci. Bull. 67, 2040 (2022).

[21] C. Liu et al., “Sierpiński structure and electronic topology in Bi thin films on InSb(111)B surfaces”, Phys. Rev. Lett. 126, 176102 (2021).

[22] R. Canyellas, C. Liu, R. Arouca, L. Eek, G. Wang, Y. Yin, D. Guan, Y. Li, S. Wang, H. Zheng, C. Liu, J. Jia, and C. Morais Smith, “Topological edge and corner states in bismuth fractal nanostructures”, Nat. Phys. 20, 1421 (2024).