Even if individual two-dimensional materials own various interesting and unexpected properties, the stacking of such layers leads to van der Waals solids which unite the characteristics of two dimensions with novel features originating from the interlayer interactions. In this topical review, we cover fabrication and characterization of van der Waals heterosructures with a focus on heterobilayers made of monolayers of semiconducting transition metal dichalcogenides. Experimental and theoretical techniques to investigate those heterobilayers are introduced. Most recent findings focusing on different transition metal dichalcogenides heterostructures are presented and possible optical transitions between different valleys, appearance of moire patterns and signatures of moire excitons are discussed. The fascinating and fast growing research on van der Waals hetero-bilayers provide promising insights required for their application as emerging quantum-nano materials.
Graphene constitutes one of the key elements in many functional van der Waals heterostructures. However, it has negligible optical visibility due to its monolayer nature. Here we study the visibility of graphene in various van der Waals heterostructures and include the effects of the source spectrum, oblique incidence and the spectral sensitivity of the detector to obtain a realistic model. A visibility experiment is performed at different wavelengths, resulting in a very good agreement with our calculations. This allows us to reliably predict the conditions for better visibility of graphene i…
We consider the role of deformations in graphene heterostructures with hexagonal crystals (including strain, wrinkles and dislocations) on the geometrical properties of moire patterns characteristic for a pair of two incommensurate misaligned isostructural crystals. By, employing a phenomenological theory to describe generic moire perturbation in van der Waals heterostructures of graphene and hexagonal crystals we investigate the electronic properties of such heterostructure.
Van der Waals heterostructures of graphene and hexagonal boron nitride feature a moiré superlattice for graphene's Dirac electrons. Here, we review the effects generated by this superlattice, including a specific miniband structure featuring gaps and secondary Dirac points, and a fractal spectrum of magnetic minibands known as Hofstadter's butterfly.
We report an experimental study of excitons in a double quantum well van der Waals heterostructure made of atomically thin layers of \Mo* and hexagonal boron nitride (hBN). The emission of neutral and charged excitons is controlled by gate voltage, temperature, and both the helicity and the power of optical excitation.
Indirect excitons (IXs) in van der Waals transition-metal dichalcogenide (TMD) heterostructures are characterized by a high binding energy making them stable at room temperature and giving the opportunity for exploring fundamental phenomena in excitonic systems and developing excitonic devices operational at high temperatures. We present the observation of IXs at room temperature in van der Waals TMD heterostructures based on monolayers of MoS$_2$ separated by atomically thin hexagonal boron nitride. The IXs realized in the TMD heterostructure have lifetimes orders of magnitude longer than lif…
Van der Waals heterostructures built by vertically stacked transition metal dichalcogenides (TMDs) exhibit a rich energy landscape including interlayer and intervalley excitons. Recent experiments demonstrated an ultrafast charge transfer in TMD heterostructures. However, the nature of the charge transfer process has remained elusive. Based on a microscopic and material-realistic exciton theory, we reveal that phonon-mediated scattering via strongly hybridized intervalley excitons governs the charge transfer process that occurs on a sub-100fs timescale. We track the time-, momentum-, and energ…
We develop a theory for interlayer tunneling in van der Waals heterostructures driven under a strong electromagnetic field, using graphene/{\it h}-BN/graphene as a paradigmatic example. Our theory predicts that strong anti-resonances appear at bias voltage values equal to an integer multiple of the light frequency. These features are found to originate from photon-assisted resonant tunneling transitions between Floquet sidebands of different graphene layers, and are unique to two-band systems due to the interplay of both intraband and interband tunneling transitions. Our results point to the p…
In van der Waals heterostructures, electronic bands of two-dimensional (2D) materials, their nontrivial topology, and electron-electron interactions can be dramatically changed by a moire pattern induced by twist angles between different layers. Such process is referred to as twistronics, where the tuning of twist angle can be realized through mechanical manipulation of 2D materials. Here we demonstrate an experimental technique that can achieve in situ dynamical rotation and manipulation of 2D materials in van der Waals heterostructures. Using this technique we fabricated heterostructures whe…
Two-dimensional (2D) van der Waals heterostructures serve as a promising platform to exploit various physical phenomena in a diverse range of novel spintronic device applications. The efficient spin injection is the prerequisite for these devices. The recent discovery of magnetic 2D materials leads to the possibility of fully 2D van der Waals spintronics devices by implementing spin injection through magnetic proximity effect (MPE). Here, we report the investigation of magnetic proximity effect in 2D CrBr3/graphene van der Waals heterostructures, which is probed by Zeeman spin Hall effect thro…
Formation of moiré superlattices is common in Van der Waals heterostructures as a result of the mismatch between lattice constants and misalignment of crystallographic directions of the constituent two-dimensional crystals. We discuss theoretically electron transport in a Van der Waals tunnelling transistor in which one of the electrodes is made of two crystals forming a moiré superlattice at their interface. By investigating structures containing either the aligned graphene/hexagonal boron nitride heterostructure or twisted bilayer graphene, we show that negative differential resistance is po…
Investigating thermal transport in van der Waals heterostructure is of scientific interest and practical importance for their applications in a broad range. In this work, thermal conductivity of one-dimensional heterostructure consisting of carbon and boron nitride nanotubes is systematically investigated via molecular dynamics simulations. Thermal conductivity is found to have strong dependences on temperature, length and diameter. In addition, the axial strain and intensity of van der Waals interaction are demonstrated to be able to modulate thermal conductivity up to about 43% and 37%, resp…
We investigate excitons in stacked transition metal dichalcogenide (TMDC) layers under perpendicularly applied electric field, herein MoSe$_2$/WSe$_2$ van der Waals heterostructures. Band structures are obtained with density functional theory calculations, along with the electron and hole wave functions in conduction and valence bands, respectively. Although the type-II nature of the heterostructure leads to fully charge separated inter-layer excitons, charge carriers distribution among the layers is shown to be easily tunable by external field. Our results show that moderate values of electri…
Detecting single charging events in quantum devices is an important step towards realizing practical quantum circuits for quantum information processing. In this work, we demonstrate that van derWaals heterostructure devices with gated nano-constrictions in monolayer WSe2 can be used as charge detectors for nearby quantum dots. These results open the possibility of implementing charge detection schemes based on 2D materials in complex quantum circuits.
The time-dependent spin current mediated spin transfer torque behaviour has been investigated via scattering formalism within density functional theory framework supported by Green's function. Quantum magnetotransport characteristics have been revealed in a model semiconducting MoS$_2$/phosphorene van der Waals heterostructure. The dynamics of spin current channelized heterolayer transport has been studied with rotational variation in magnetization angle. It is observed that the time-dependent spin transport torque remains invariant irrespective of magnetization angle direction. The polarized …
We investigate van der Waals (vdW) heterostructures made of germanene, stanene or silicene with hexagonal Boron Nitride (h-BN). The intriguing topological properties of these buckled honeycomb materials can be maintained and further engineered in the heterostructures, where the competition between the substrate effect and external electric fields can be used to control the tunable topological phase transitions. Using such heterostructures as building blocks, various vdW topological domain walls (DW) are designed, along which there exist valley polarized quantum spin Hall edge states or valley-…
We report the fabrication of perpendicular ferromagnetic electrodes for tunneling devices consist of van der Waals heterostructure. We found MgO/Co/Pt films on Hexagonal BN shows perpendicular magnetic anisotropy (PMA) with the easy axis perpendicular to the substrate. Vacuum annealing enhances the perpendicular anisotropy. The easy axis along the perpendicular direction persists up to room temperature with the Pt layer thickness ranges from 1.5 to 5 nm. We employed the PMA electrodes to construct tunneling devices on graphene and monolayer WSe2, where spin injection characteristics and field …
Van der Waals heterostructures formed by stacking of various two-dimensional materials are promising in electronic applications. However, the performances of most reported electronic devices based on van der Waals heterostructures are far away from existing technologies'. Here we report high performance heterostructure devices based on vertically stacked tungsten diselenide and tin diselenide. Due to the unique band alignment and the atomic thickness of the material, both charge carrier transport and energy barrier can be effectively modulated by the applied electrical field. As a result, the …
It is shown that the anomalous elasticity of membranes affects the profile and thermodynamics of a bubble in van der Waals heterostructures. Our theory generalizes the non-linear plate theory as well as membrane theory of the pressurised blister test to incorporate the power-law scale dependence of the bending rigidity and Young's modulus of a two-dimensional crystalline membrane. This scale dependence caused by long-ranged interaction of relevant thermal fluctuations (flexural phonons), is responsible for the anomalous Hooke's law observed recently in graphene. It is shown that this anomalous…
Near-gate plasmons are a new type of plasma oscillations emerging in homogeneous two-dimensional electron systems where a gate provides partial screening of electron-electron interaction. Here we develop a theory of the near-gate plasmons in van der Waals heterostructures comprising a conducting layer separated by a thin insulator from an uncharged disk-shaped gate. We show that in these structures the near-gate plasmons form gate-size-quantized quasi-stationary discrete modes even in the collisionless limit. Belonging to continuum spectrum of two-dimensional plasmons outside of the disk-gate,…
Heterostructures formed between two different van der Waals materials enable interactions and functionalities absent in each component. In this work we show that vicinity to an atomically thin WSe$_2$ sheet dramatically impacts the energies of the symmetry-broken low Landau levels of bilayer graphene, possibly due to screening. We present a systematic study of the magnetic field and electrical displacement field dependences of the Landau level gaps at filling factor $ν$ = 1, 2, 3, and compare to BN encapsulated pristine bilayer graphene. The exchange-dominated energy splitting between the N = …
Ferromagnetic van der Waals (vdW) materials are in demand for spintronic devices with all-two-dimensional-materials heterostructures. Here, we demonstrate mechanical exfoliation of magnetic-atom-intercalated transition metal dichalcogenide Cr1/3TaS2 from its bulk crystal; previously such intercalated materials were thought difficult to exfoliate. Magnetotransport in exfoliated tens-of-nanometres-thick flakes revealed ferromagnetic ordering below its Curie temperature TC ~ 110 K as well as strong in-plane magnetic anisotropy; these are identical to its bulk properties. Further, van der Waals he…
In this paper, we propose van del Waals heterostructure-based thermionic devices for the applications in cooling and power generation in the temperature range of 300 to 400 K. By using two-dimensional materials of low cross-plane thermal conductivity as the barrier materials and graphene as electrodes, our calculation demonstrates that our proposed device will have a higher efficiency as compared to other methods such as thermoelectric device and the traditional thermionic devices. By using the parameters within the current technology, we predict a cooling capability at more than 50$\%$ of the…
In this study, the electrostatically induced quantum confinement structure, quantum point contact, has been realized in p-type trilayer tungsten diselenide-based van der Waals heterostructure with modified van der Waals contact method with degenerately doped transition metal dichalcogenide crystals. Clear quantized conductance and pinch-off state through the one-dimensional confinement were observed by dual-gating of split gate electrodes and top gate. Conductance plateaus were observed at step of 0.5 $\times$ 2e$^{2}$/h at zero and high magnetic field in addition to quarter plateaus at a fini…
We numerically study the general valley polarization and anomalous Hall effect in van der Waals (vdW) heterostructures based on monolayer jacutingaite family materials Pt$_{2}$AX$_{3}$ (A = Hg, Cd, Zn; X = S, Se, Te). We perform a systematic study on the atomic, electronic, and topological properties of vdW heterostructures composed of monolayer Pt$_{2}$AX$_{3}$ and two-dimensional ferromagnetic insulators. We show that four kinds of vdW heterostructures exhibit valley-polarized quantum anomalous Hall phase, i.e., Pt$_{2}$HgS$_{3}$/NiBr$_{2}$, Pt$_{2}$HgSe$_{3}$/CoBr$_{2}$, Pt$_{2}$HgSe$_{3}$/…
We introduce a method of local gating for van der Waals heterostructures, employing a few-layer graphene patterned bottom gate. Being a member of the 2D material family, few-layer graphene adapts perfectly to the commonly used stacking method. Its versatility regarding patterning as well as its flatness make it an ideal candidate for experiments on locally gated 2D materials. Moreover, in combination with ultra-thin hexagonal boron nitride as an insulating layer, sharp potential steps can be created and the quality of the investigated 2D material can be sustained. To underline the good feasibi…
Second-order topological insulator, which has (d-2)-dimensional topological hinge or corner states, has been observed in three-dimensional materials, but has yet not been observed in two-dimensional system. In this Letter, we theoretically propose the realization of second-order topological insulator in the van der Waals heterostructure of CoBr$_2$/Pt$_2$HgSe$_3$/CoBr$_2$. Pt$_2$HgSe$_3$ is a large gap $\mathbb{Z}_2$ topological insulator. With in-plane exchange field from neighboring CoBr$_2$, a large band gap above 70 meV opens up at the edge. The corner states, which are robust against edge…
We directly measure the electronic structure of twisted graphene/MoS2 van der Waals heterostructures, in which both graphene and MoS2 are monolayers. We use cathode lens microscopy and microprobe angle-resolved photoemission spectroscopy measurements to image the surface, determine twist angle, and map the electronic structure of these artificial heterostructures. For monolayer graphene on monolayer MoS2, the resulting band structure reveals the absence of hybridization between the graphene and MoS2 electronic states. Further, the graphene-derived electronic structure in the heterostructures r…
Throughout the years, strongly correlated coherent states of excitons have been the subject of intense theoretical and experimental studies. This topic has recently boomed due to new emerging quantum materials such as van der Waals (vdW) bound atomically thin layers of transition metal dichalcogenides (TMDs). We analyze the collective properties of charged interlayer excitons observed recently in bilayer TMD heterostructures. We predict new strongly correlated phases - crystal and Wigner crystal - that can be selectively realized with TMD bilayers of properly chosen electron-hole effective mas…
Van der Waals (vdW) heterostructures made of two-dimensional materials have been demonstrated to be versatile architectures for optoelectronic applications due to strong light-matter interactions. However, most of light-controlled phenomena and applications in the vdW heterostructures rely on positive photoconductance (PPC). Negative photoconductance (NPC) has not yet been reported in vdW heterostructures. Here we report the observation of the NPC in ReS2/h-BN/MoS2 vdW heterostructures-based floating gate phototransistor. The fabricated devices exhibit excellent performance of nonvolatile memo…
It is shown that chiral plasmons, characterized by a longitudinal magnetic moment accompanying the longitudinal charge plasmon, lead to electromagnetic near-fields that are also chiral. For twisted bilayer graphene, we estimate that the near field chirality of screened plasmons can be several orders of magnitude larger than that of the related circularly polarized light. The chirality also manifests itself in a deflection angle that is formed between the direction of the plasmon propagation and its Poynting vector. Twisted van der Waals heterostructures might thus provide a novel platform to p…
In the last six years, Indium selenide (InSe) has appeared as a new van der Waals heterostructure platform which has been extensively studied due to its unique electronic and optical properties. Such as transition metal dichalcogenides (TMDCs), the considerable bandgap and high electron mobility can provide a potential optoelectronic application. Here we present low-temperature transport measurements on a few-layer InSe van der Waals heterostructure with graphene-gated contacts. For high magnetic fields, we observe magnetoresistance minima at even filling factors related to two-fold spin degen…
We investigated experimentally the valley Zeeman splitting of excitonic peaks in the photoluminescence (PL) spectra of high-quality hBN/WS2/MoSe2/hBN heterostructures at near-zero twist angles under perpendicular magnetic fields up to 20 T. We identify two neutral exciton peaks in the PL spectra: the lower energy one exhibits a reduced g-factor relative to that of the higher energy peak, and much lower than the recently reported values for interlayer excitons in other van der Waals (vdW) heterostructures. We provide evidence that such a discernible g-factor stems from the spatial confinement o…
Despite a rich choice of two-dimensional materials, which exists these days, heterostructures, both vertical (van der Waals) and in-plane, offer an unprecedented control over the properties and functionalities of the resulted structures. Thus, planar heterostructures allow p-n junctions between different two-dimensional semiconductors and graphene nanoribbons with well-defined edges; and vertical heterostructures resulted in the observation of superconductivity in purely carbon-based systems and realisation of vertical tunnelling transistors. Here we demonstrate simultaneous use of in-plane an…
In transition metal dichalcogenides, defects have been found to play an important role, affecting doping, spin-valley relaxation dynamics, and assisting in proximity effects of spin-orbit coupling. Here, we study localized states in $\mathrm{WS}_2$ and how they affect tunneling through van der Waals heterostructures of h-BN/graphene/$\mathrm{WS}_2$/metal. The obtained conductance maps as a function of bias and gate voltage reveal single-electron transistor behavior (Coulomb blockade) with a rich set of transport features including excited states and negative differential resistance regimes. Ap…
Multilayer van der Waals (vdWs) heterostructures assembled by diverse atomically thin layers have demonstrated a wide range of fascinating phenomena and novel applications. Understanding the interlayer coupling and its correlation effect is paramount for designing novel vdWs heterostructures with desirable physical properties. Using a detailed theoretical study of 2D MoS2-graphene (GR)-based heterostructures based on state-of-the-art hybrid density functional theory, we reveal that for 2D few-layer heterostructures, vdWs forces between neighboring layers depend on the number of layers. Compare…
The concept of swapping the two most important spin interactions -- exchange and spin-orbit coupling -- is proposed based on two-dimensional multilayer van der Waals heterostructures. Specifically, we show by performing realistic ab initio simulations, that a single device consisting of a bilayer graphene sandwiched by a 2D ferromagnet Cr$_2$Ge$_2$Te$_6$ (CGT) and a monolayer WS$_2$, is able not only to generate, but also to swap the two interactions. The highly efficient swapping is enabled by the interplay of gate-dependent layer polarization in bilayer graphene and short-range spin-orbit an…
Long lifetimes of spatially indirect excitons (IXs), also known as interlayer excitons, make possible long-range IX propagation. Van der Waals heterostructures composed of atomically thin layers of transition-metal dichalcogenides (TMDs) give an opportunity to realize excitons with high binding energies and provide a materials platform for the realization of both excitonic quantum phenomena and excitonic devices. Propagation of IXs in TMD heterostructures is intensively studied. However, in spite of long IX lifetimes, orders of magnitude longer than lifetimes of spatially direct excitons (DXs)…
We have investigated the vibrational properties of van der Waals heterostructures of monolayer transition metal dichalcogenides (TMDs), specifically MoS2/WSe2 and MoSe2/MoS2 heterobilayers as well as twisted MoS2 bilayers, by means of ultralow-frequency Raman spectroscopy. We discovered Raman features (at 30 ~ 40 cm-1) that arise from the layer-breathing mode (LBM) vibrations between the two incommensurate TMD monolayers in these structures. The LBM Raman intensity correlates strongly with the suppression of photoluminescence that arises from interlayer charge transfer. The LBM is generated on…
Van der Waals heterostructures, vertical stacks of layered materials, offer newopportunities for novel quantum phenomena which are absent in their constituent components. Here we report the emergence of polaron quasiparticles at the interface of graphene/hexagonal boron nitride (h-BN) heterostructures. Using nanospot angle-resolved photoemission spectroscopy, we observe zone-corner replicas of h-BN valence band maxima, with energy spacing coincident with the highest phonon energy of the heterostructure|an indication of Frohlich polaron formation due to forward scattering electron-phonon coupl…
Van der Waals heterostructures comprise a new class of artificial materials formed by stacking atomically-thin planar crystals. Here, we demonstrate band structure engineering of a van der Waals heterostructure composed of a monolayer graphene flake coupled to a rotationally-aligned hexagonal boron nitride substrate. The spatially-varying interlayer atomic registry results both in a local breaking of the carbon sublattice symmetry and a long-range moiré superlattice potential in the graphene. This interplay between short- and long-wavelength effects results in a band structure described by iso…
Monolayer (ML) transition metal dichalcogenides (TMDs) integrated in optical microcavities host exciton-polaritons as a hallmark of the strong light-matter coupling regime. Analogous concepts for hybrid light-matter systems employing spatially indirect excitons with a permanent electric dipole moment in heterobilayer (HBL) crystals promise realizations of exciton-polariton gases and condensates with immanent dipolar interactions. Here, we identify optical signatures of spatially indirect momentum-bright and momentum-dark interlayer excitons in vertical MoSe$_2$-WSe$_2$ heterostructures and imp…
Indirect excitons (IX) in semiconductor heterostructures are bosons, which can cool below the temperature of quantum degeneracy and can be effectively controlled by voltage and light. IX quantum Bose gases and IX devices were explored in GaAs heterostructures where an IX range of existence is limited to low temperatures due to low IX binding energies. IXs in van der Waals transition-metal dichalcogenide (TMD) heterostructures are characterized by large binding energies giving the opportunity for exploring excitonic quantum gases and for creating excitonic devices at high temperatures. TMD hete…
The large variety of 2D materials and their co-integration in van der Waals (vdW) heterostructures enable innovative device engineering. In addition, their atomically-thin nature promotes the design of artificial materials by proximity effects that originate from short-range interactions. Such a designer approach is particularly compelling for spintronics, which typically harnesses functionalities from thin layers of magnetic and non-magnetic materials and the interfaces between them. Here, we overview recent progress on 2D spintronics and opto-spintronics using vdW heterostructures. After an …
We predict a new mechanism of surface plasmon amplification in graphene-insulator-graphene van der Waals heterostructures. The amplification occurs upon the stimulated interlayer electron tunneling accompanied by the emission of a coherent plasmon. The quantum-mechanical calculations of the non-local high-frequency tunnel conductivity show that a relative smallness of the tunneling exponent can be compensated by a strong resonance due to the enhanced tunneling between electron states with collinear momenta in the neighboring graphene layers. With the optimal selection of the barrier layer, the…
Stacking monolayers of transition metal dichalcogenides into a heterostructure with a finite twist-angle gives rise to artificial moiré superlattices with a tunable periodicity. As a consequence, excitons experience a periodic potential, which can be exploited to tailor optoelectronic properties of these materials. While recent experimental studies have confirmed twist-angle dependent optical spectra, the microscopic origin of moiré exciton resonances has not been fully clarified yet. Here, we combine first principle calculations with the excitonic density matrix formalism to study transitions…
Two-dimensional (2D) crystals, such as graphene and transition metal dichalcogenides (TMDs), present a collection of unique and complementary optoelectronic properties. Assembling different 2D materials in vertical heterostructures enables the combination of these properties in one device, thus creating multi-functional optoelectronic systems with superior performance. Here we demonstrate that graphene/WSe2/graphene heterostructures ally the high photodetection efficiency of TMDs with a picosecond photoresponse comparable to that of graphene, thereby optimizing both speed and efficiency in a s…
Long-lived interlayer excitons with distinct spin-valley physics in van der Waals heterostructures based on transition metal dichalcogenides make them promising for information processing in next-generation devices. While the emission characteristics of interlayer excitons in different types of hetero stacks have been extensively studied, the manipulation of these excitons required to alter the valley-state or tune the emission energy and intensity is still lacking. Here, we demonstrate such control over interlayer excitons in MoSe2/WSe2 heterostructures. The encapsulation of our stack with h-…
Exciton binding energies of hundreds of meV and strong light absorption in the optical frequency range make transition metal dichalcogenides (TMDs) promising for novel optoelectronic nanodevices. In particular, atomically thin TMDs can be stacked to heterostructures enabling the design of new materials with tailored properties. The strong Coulomb interaction gives rise to interlayer excitons, where electrons and holes are spatially separated in different layers. In this work, we reveal the microscopic processes behind the formation, thermalization and decay of these fundamentally interesting a…
Indirect excitons (IXs), also known as interlayer excitons, can form the medium for excitonic devices whose operation is based on controlled propagation of excitons. A proof of principle for excitonic devices was demonstrated in GaAs heterostructures where the operation of excitonic devices is limited to low temperatures. IXs in van der Waals transition-metal dichalcogenide (TMD) heterostructures are characterized by high binding energies making IXs robust at room temperature and offering an opportunity to create excitonic devices operating at high temperatures suitable for applications. Howev…
Stacking atomic monolayers of semiconducting transition metal dichalcogenides (TMDs) has emerged as an effective way to engineer their properties. In principle, the staggered band alignment of TMD heterostructures should result in the formation of inter-layer excitons with long lifetimes and robust valley polarization. However, these features have been observed simultaneously only in MoSe$_2$/WSe$_2$ heterostructures. Here we report on the observation of long lived inter-layer exciton emission in a MoS$_2$/MoSe$_2$/MoS$_2$ trilayer van der Waals heterostructure. The inter-layer nature of the o…
The existence of gapless Dirac surface band of a three dimensional (3D) topological insulator (TI) is guaranteed by the non-trivial topological character of the bulk band, yet the surface band dispersion is mainly determined by the environment near the surface. In this Letter, through in-situ angle-resolved photoemission spectroscopy (ARPES) and the first-principles calculation on 3D TI-based van der Waals heterostructures, we demonstrate that one can engineer the surface band structures of 3D TIs by surface modifications without destroying their topological non-trivial property. The result pr…
