Persistence of a two-dimensional topological insulator state in wide HgTe quantum wells

Multifractal Conductance Fluctuations of Helical Edge States

Two-dimensional topological insulators are characterized by the bulk gap and one-dimensional helical states running along the edges. The theory predicts the topological protection of the helical transport from coherent backscattering. However, the unexpected deviations of the conductance from the quantized value and localization of the helical modes are generally observed in long samples. Moreover, at millikelvin temperatures significant mesoscopic fluctuations are developed as a function of the electron energy. Here we report the results of an experimental study of the transport in a HgTe quantum well with an inverted energy spectrum that reveal a multifractality of the conductance fluctuations in the helical edge state dominated transport regime. We attribute observed multifractality to mesoscopic fluctuations of the electron wave function or local density of states at the spin quantum Hall transition. We have shown that the mesoscopic two-dimensional topological insulator provides a highly tunable experimental system in which to explore the physics of the Anderson transition between topological states.

Quantum Spin Hall Effect in Two-Monolayer-Thick InN/InGaN Coupled Multiple Quantum Wells

In this study, we present a theoretical study of the quantum spin Hall effect in InN/InGaN coupled multiple quantum wells with the individual well widths equal to two atomic monolayers. We consider triple and quadruple quantum wells in which the In content in the interwell barriers is greater than or equal to the In content in the external barriers. To calculate the electronic subbands in these nanostructures, we use the eight-band k∙p Hamiltonian, assuming that the effective spin-orbit interaction in InN is negative, which represents the worst-case scenario for achieving a two-dimensional topological insulator. For triple quantum wells, we find that when the In contents of the external and interwell barriers are the same and the widths of the internal barriers are equal to two monolayers, a topological insulator with a bulk energy gap of 0.25 meV can appear. Increasing the In content in the interwell barriers leads to a significant increase in the bulk energy gap of the topological insulator, reaching about 0.8 meV. In these structures, the topological insulator can be achieved when the In content in the external barriers is about 0.64, causing relatively low strain in quantum wells and making the epitaxial growth of these structures within the range of current technology. Using the effective 2D Hamiltonian, we study the edge states in strip structures containing topological triple quantum wells. We demonstrate that the opening of the gap in the spectrum of the edge states caused by decreasing the width of the strip has an oscillatory character regardless of whether the pseudospin-mixing elements of the effective Hamiltonian are omitted or taken into account. The strength of the finite size effect in these structures is several times smaller than that in HgTe/HgCdTe and InAs/GaSb/AlSb topological insulators. Therefore, its influence on the quantum spin Hall effect is negligible in strips with a width larger than 150 nm, unless the temperature at which electron transport is measured is less than 1 mK. In the case of quadruple quantum wells, we find the topological insulator phase only when the In content in the interwell barriers is larger than in the external barriers. We show that in these structures, a topological insulator with a bulk energy gap of 0.038 meV can be achieved when the In content in the external barriers is about 0.75. Since this value of the bulk energy gap is very small, quadruple quantum wells are less useful for realizing a measurable quantum spin Hall system, but they are still attractive for achieving a topological phase transition and a nonlocal topological semimetal phase.

Polarization-Induced Phase Transitions in Ultra-Thin InGaN-Based Double Quantum Wells

We investigate the phase transitions and the properties of the topological insulator in InGaN/GaN and InN/InGaN double quantum wells grown along the [0001] direction. We apply a realistic model based on the nonlinear theory of elasticity and piezoelectricity and the eight-band k·p method with relativistic and nonrelativistic linear-wave-vector terms. In this approach, the effective spin−orbit interaction in InN is negative, which represents the worst-case scenario for obtaining the topological insulator in InGaN-based structures. Despite this rigorous assumption, we demonstrate that the topological insulator can occur in InGaN/GaN and InN/InGaN double quantum wells when the widths of individual quantum wells are two and three monolayers (MLs), and three and three MLs. In these structures, when the interwell barrier is sufficiently thin, we can observe the topological phase transition from the normal insulator to the topological insulator via the Weyl semimetal, and the nontopological phase transition from the topological insulator to the nonlocal topological semimetal. We find that in InGaN/GaN double quantum wells, the bulk energy gap in the topological insulator phase is much smaller for the structures with both quantum well widths of 3 MLs than in the case when the quantum well widths are two and three MLs, whereas in InN/InGaN double quantum wells, the opposite is true. In InN/InGaN structures with both quantum wells being three MLs and a two ML interwell barrier, the bulk energy gap for the topological insulator can reach about 1.2 meV. We also show that the topological insulator phase rapidly deteriorates with increasing width of the interwell barrier due to a decrease in the bulk energy gap and reduction in the window of In content between the normal insulator and the nonlocal topological semimetal. For InN/InGaN double quantum wells with the width of the interwell barrier above five or six MLs, the topological insulator phase does not appear. In these structures, we find two novel phase transitions, namely the nontopological phase transition from the normal insulator to the nonlocal normal semimetal and the topological phase transition from the nonlocal normal semimetal to the nonlocal topological semimetal via the buried Weyl semimetal. These results can guide future investigations towards achieving a topological insulator in InGaN-based nanostructures.

Engineering topological phases in triple HgTe/CdTe quantum wells

Quantum wells formed by layers of HgTe between Hg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{1-x}$$\end{document}1-xCd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_x$$\end{document}xTe barriers lead to two-dimensional (2D) topological insulators, as predicted by the BHZ model. Here, we theoretically and experimentally investigate the characteristics of triple HgTe quantum wells. We describe such heterostructure with a three dimensional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8\times 8$$\end{document}8×8 Kane model, and use its eigenstates to derive an effective 2D Hamiltonian for the system. From these we obtain a phase diagram as a function of the well and barrier widths and we identify the different topological phases composed by zero, one, two, and three sets of edge states hybridized along the quantum wells. The phase transitions are characterized by a change of the spin Chern numbers and their corresponding band inversions. Complementary, transport measurements are experimentally investigated on a sample close to the transition line between the phases with one and two sets of edges states. Accordingly, for this sample we predict a gapless spectrum with low energy bulk conduction subbands given by one parabolic and one Dirac subband, and with edge states immersed in the bulk valence subbands. Consequently, we show that under these conditions, local and non-local transport measurements are inconclusive to characterize a sole edge state conductivity due to bulk conductivity. On the other hand, Shubnikov-de Haas (SdH) oscillations show an excellent agreement with our theory. Particularly, we show that the measured SdH oscillation frequencies agrees with our model and show clear signatures of the coexistence of a parabolic and Dirac subbands.

Reorganization of the topological surface mode of topological insulating α-Sn

α-Sn is a topologically nontrivial semimetal in its natural structure. Upon compressively strained in plane, it transforms into a topological insulator. But, up to now, a clear and systematic understanding of the topological surface mode of topological insulating α-Sn is still lacking. In the present work, first-principle simulations are employed to investigate the electronic structure evolution of Ge_1-xSn_xalloys aiming at understanding the band reordering, topological phase transition and topological surface mode of α-Sn in detail. Progressing from Ge to Sn with increasing Sn content in Ge_1-xSn_x, the conduction band inverts with the first valence band (VB) and then with the second VB sequentially, rather than inverting with the latter directly. Correspondingly, a topologically nontrivial surface mode arises in the first inverted band gap. Meanwhile, a fragile Dirac cone appears in the second inverted band gap as a result of the reorganization of the topological surface mode caused by the first VB. The reorganization of the topological surface mode in α-Sn is very similar to the HgTe case. The findings of the present work are helpful for understanding and utilizing of the topological surface mode of α-Sn.

Progress in Epitaxial Thin-Film Na3 Bi as a Topological Electronic Material

Trisodium bismuthide (Na₃ Bi) is the first experimentally verified topological Dirac semimetal, and is a 3D analogue of graphene hosting relativistic Dirac fermions. Its unconventional momentum-energy relationship is interesting from a fundamental perspective, yielding exciting physical properties such as chiral charge carriers, the chiral anomaly, and weak anti-localization. It also shows promise for realizing topological electronic devices such as topological transistors. Herein, an overview of the substantial progress achieved in the last few years on Na₃ Bi is presented, with a focus on technologically relevant large-area thin films synthesized via molecular beam epitaxy. Key theoretical aspects underpinning the unique electronic properties of Na₃ Bi are introduced. Next, the growth process on different substrates is reviewed. Spectroscopic and microscopic features are illustrated, and an analysis of semiclassical and quantum transport phenomena in different doping regimes is provided. The emergent properties arising from confinement in two dimensions, including thickness-dependent and electric-field-driven topological phase transitions, are addressed, with an outlook toward current challenges and expected future progress.

Helical Edge Transport in Millimeter-Scale Thin Films of Na3Bi

A two-dimensional topological insulator (2DTI) has an insulating bulk and helical edges robust to nonmagnetic backscattering. While ballistic transport has been demonstrated in micron-scale 2DTIs, larger samples show significant backscattering and a nearly temperature-independent resistance of unclear origin. Spin polarization has been measured, however the degree of helicity is difficult to quantify. Here, we study 2DTI few-layer Na₃Bi on insulating Al₂O₃. A nonlocal conductance measurement demonstrates edge conductance in the topological regime with an edge mean free path ∼100 nm. A perpendicular magnetic field suppresses spin-flip scattering in the helical edges, resulting in a giant negative magnetoresistance (GNMR) up to 80% at 0.9 T. Comparison to theory indicates >96% of scattering is helical spin scattering significantly exceeding the maximum (67%) expected for a nonhelical metal. GNMR, coupled with nonlocal measurements, thus provides an unambiguous experimental signature of helical edges that we expect to be generically useful in understanding 2DTIs.

Topological Protection Brought to Light by the Time-Reversal Symmetry Breaking

Recent topological band theory distinguishes electronic band insulators with respect to various symmetries and topological invariants, most commonly, the time reversal symmetry and the Z_{2} invariant. The interface of two topologically distinct insulators hosts a unique class of electronic states-the helical states, which shortcut the gapped bulk and exhibit spin-momentum locking. The magic and so far elusive property of the helical electrons, known as topological protection, prevents them from coherent backscattering as long as the underlying symmetry is preserved. Here we present an experiment that brings to light the strength of topological protection in one-dimensional helical edge states of a Z_{2} quantum spin-Hall insulator in HgTe. At low temperatures, we observe the dramatic impact of a tiny magnetic field, which results in an exponential increase of the resistance accompanied by giant mesoscopic fluctuations and a gap opening. This textbook Anderson localization scenario emerges only upon the time-reversal symmetry breaking, bringing the first direct evidence of the topological protection strength in helical edge states.

Unrestricted Electron Bunching at the Helical Edge

A quantum magnetic impurity of spin S at the edge of a two-dimensional time reversal invariant topological insulator may give rise to backscattering. We study here the shot noise associated with the backscattering current for arbitrary S. Our full analytical solution reveals that for S>1/2 the Fano factor may be arbitrarily large, reflecting bunching of large batches of electrons. By contrast, we rigorously prove that for S=1/2 the Fano factor is bounded between 1 and 2, generalizing earlier studies.

Electronic thermal conductivity in 2D topological insulator in a HgTe quantum well

We have measured the differential resistance in a two-dimensional topological insulator (2DTI) in a HgTe quantum well, as a function of the applied dc current. The transport near the charge neutrality point is characterized by a pair of counter propagating gapless edge modes. In the presence of an electric field, the energy is transported by counter propagating channels in the opposite direction. We test a hot carrier effect model and demonstrate that the energy transfer complies with the Wiedemann Franz law near the charge neutrality point in the edge transport regime.

Topological insulator with negative spin-orbit coupling and transition between Weyl and Dirac semimetals in InGaN-based quantum wells

We study the influence of negative spin-orbit coupling on the topological phase transition and properties of the topological insulator state in InGaN-based quantum wells grown along c axis of the wurtzite lattice. The realistic eight-band k·p method with relativistic and nonrelativistic linear-k terms is employed. Our calculations show that the negative spin-orbit coupling in InN is not an obstacle to obtain the topological insulator phase in InN/InGaN and InGaN/GaN quantum wells. The bulk energy gap in the topological insulator state can reach 2 meV, which allows experimental verification of the edge state transport in these materials. The topological phase transition occurs due to the band inversion between the highest light hole subband and the lowest conduction subband, and almost always is mediated by the two-dimensional Weyl semimetal, arising from an anticrossing of these subbands at zero in-plane wave vector. However, for certain InGaN/GaN quantum wells, we find that the magnitude of this anticrossing vanishes, leading to the appearance of the Dirac semimetal. The novel transition between the Weyl and Dirac semimetals originates from vanishing of the average in-plane spin-orbit interaction parameter, which decouples the conduction subband from the light hole subband at zero in-plane wave vector.

A 50/50 electronic beam splitter in graphene nanoribbons as a building block for electron optics

Based on the investigation of the multi-terminal conductance of a system composed of two graphene nanoribbons, in which one is on top of the other and rotated by [Formula: see text], we propose a setup for a 50/50 electronic beam splitter that neither requires large magnetic fields nor ultra low temperatures. Our findings are based on an atomistic tight-binding description of the system and on the Green function method to compute the Landauer conductance. We demonstrate that this system acts as a perfect 50/50 electronic beam splitter, in which its operation can be switched on and off by varying the doping (Fermi energy). We show that this device is robust against thermal fluctuations and long range disorder, as zigzag valley chiral states of the nanoribbons are protected against backscattering. We suggest that the proposed device can be applied as the fundamental element of the Hong-Ou-Mandel interferometer, as well as a building block of many devices in electron optics.

Low field magnetoresistance in a 2D topological insulator based on wide HgTe quantum well

Low field magnetoresistance is experimentally studied in a two-dimensional topological insulator (TI) in both diffusive and quasiballistic samples fabricated on top of a wide (14 nm) HgTe quantum well. In all cases a pronounced quasi-linear positive magnetoresistance is observed similar to that found previously in diffusive samples based on a narrow (8 nm) HgTe well. The experimental results are compared with the main existing theoretical models based on different types of disorder: sample edge roughness, nonmagnetic disorder in an otherwise coherent TI and metallic puddles due to locally trapped charges that act like local gate on the sample. The quasiballistic samples with resistance close to the expected quantized values also show a positive low-field magnetoresistance but with a pronounced admixture of mesoscopic effects.