Signatures of Topological Superconductors

A topological superconductor is a new state of matter that attract a lot of interest for its potential application in quantum computers. However, there is no single material known to host this state of matter. In this thesis, combinations of superconductors and semiconductors are investigated experimentally with the goal to engineer such a topological superconductor. The materials chosen combine spin-orbit interaction, superconductivity and onedimensionality. Then, under influence of a magnetic field, the hybrid superconductor semiconductor system is predicted to become topological. First, the theoretical background of the experiments is presented, with special attention to the superconducting quantum interference in semiconducting Josephson junctions. In addition, a description of the different materials used and the fabrication of the devices, is provided. In the first experiment we explore hole transport through GeSi core-shell nanowires. Electronic measurements reveal two transport channels only, which underlines the onedimensionality of the nanowire. On top of that, high-quality induced superconductivity is observed in both the tunneling and open regime, and evidence for strong spin-orbit interaction is presented. Then, we switch materials to a two-dimensional electron and hole gas in an InAs/GaSb double quantum well. The spin-orbit interaction is studied by measuring the difference between the densities of electrons with opposite spin orientation. Two types of spin-orbit interaction are identified by tuning the magnitude of one of them, with an applied electric field. InAs quantum wells are known to exhibit enhanced conduction at their edges. We find supercurrent through these edges in Josephson junction devices using superconducting quantum interference measurements. The interference pattern reveals a flux periodicity of h/e. Interestingly, while this periodicity is observed in the trivial regime, it was considered a signature of topological superconductivity before. We argue and show that nonlocal processes lead to the h/e effect in our devices. The correlated occurence of enhanced edge conduction and the h/e periodicity is confirmed in Josephson junctions made of InSb flakes. The final experimental chapter considers a superconducting quantum interference device, fabricated in an InAs quantum well. This geometry allows for control of the superconducting phase difference of the Josephson junction, potentially reducing the magnetic field needed for the device to become topological. Unfortunately, in the measurements we do not observe signatures of topological superconductivity. At last, we describe what device geometry and material combination could be used to do reach the topological regime. In addition, we discuss ideas for future research of the othermaterial systems used in this thesis.

Using the tight binding Kane-Mele model including the self-consistent on-site Coulomb interactions (O-CIs), we study the influence of transverse electric field in the narrow zigzag graphene nanoribbon (ZGNR) plane on the edge band structure in order to investigate the way to control the type of quantum spin Hall (QSH) system in the ZGNR. The theoretical results show that when applying weak electric field intensity, the direction of electric field can adjust these two spin-down edge bands moving along the different directions in one-dimensional q space, which leads to the two different types of degenerative breakdown of two pure spin-down edge states at q=0.5. When applying positive electric field the energy of spin-down edge band at edge site 1 is higher than that at edge site 8. On the contrary, when applying negative electric field the energy of spin-down edge band at edge site 8 is higher than that at edge site 1. It shows that we can use the direction of electric field to control the two spin-down edge currents occurring at two different energies. Further, when the electric field intensity increases above 0.69 V/nm, the increased large band gap between the two spin-down edge bands leads to the inversion of these two spin-down edge bands. That is to say, there is a spin-down band gap, however, there is not a band gap for spin-up edge band in the region of spin-down band gap. Thus the system becomes half-metallic, and the QSH does not belong in the type B any longer. Specially, when the electric field intensity reaches 1.17 V/nm in the region of spin-down band gap, the pure spin-up edge state appears at q=0.5, which shows that the strong pure spin-up edge current along the edge site 8 can occur. With increasing the intensity of electric field, the QSH system undergoes three processes from the type B to the type C. When the electric field intensity is more than 1.42 V/nm, the two spin-up edge bands also present band inversion and turn into the conduction band and the valence band, respectively. Thus the system becomes semiconducting and the QSH system does not belong in the type C, ordinary quantum Hall system. Finally, according to the results discussed above, we can expect that using the direction and the intensity of the transverse electric field in plane we can adjust the properties of edge current, and control the type of QSH system varying from the type B to the type C.

Two-dimensional (2D) topological insulators (TIs) hold promise for\napplications in spintronics based on the fact that the propagation direction of\nedge electrons of a 2D TI is robustly linked to their spin origination. Here,\nwith the use of first-principles calculations, we predict a family of robust 2D\nTIs in monolayer square transition metal dichalcogenides (MoS2, MoSe2, MoTe2,\nWS2, WSe2, and WTe2). Sizeable intrinsic nontrivial bulk band gaps ranging from\n24 to 187 meV are obtained, guarantying the quantum spin Hall (QSH) effect\nobservable at room temperature in these new 2D TIs. Significantly different\nfrom most known 2D TIs with comparable band gaps, these sizeable energy gaps\noriginate from the strong spin-orbit interaction related to the pure d\nelectrons of the Mo/W atoms around the Fermi level. A single pair of\ntopologically protected helical edge states is established for the edge of\nthese systems with the Dirac point locating in the middle of the bulk band gap,\nand their topologically nontrivial states are also confirmed with nontrivial\ntopological invariant Z2 = 1. More interestingly, by controlling the applied\nstrain, a topological quantum phase transition between a QSH phase and a\nmetallic phase or a trivial insulating phase can be realized in these 2D\nmaterials, and the detailed topological phase diagram is established.\n

One of the most fascinating challenges in Physics is the realization of an electron-based counterpart of quantum optics, which requires the capability to generate and control single electron wave packets. The edge states of quantum spin Hall (QSH) systems, i.e., two-dimensional (2D) topological insulators realized in HgTe/CdTe and InAs/GaSb quantum wells, may turn the tide in the field, as they do not require the magnetic field that limits the implementations based on quantum Hall effect. However, the band structure of these topological states, described by a massless Dirac fermion Hamiltonian, prevents electron photoexcitation via the customary vertical electric dipole transitions of conventional optoelectronics. So far, proposals to overcome this problem are based on magnetic dipole transitions induced via Zeeman coupling by circularly polarised radiation, and are limited by the g-factor. Alternatively, optical transitions can be induced from the edge states to the bulk states, which are not topologically protected though. Here we show that an electric pulse, localized in space and/or time and applied at a QSH edge, can photoexcite electron wavepackets by intra-branch electrical transitions, without invoking the bulk states or the Zeeman coupling. Such wavepackets are spin-polarised and propagate in opposite directions, with a density profile that is independent of the initial equilibrium temperature and that does not exhibit dispersion, as a result of the linearity of the spectrum and of the chiral anomaly characterising massless Dirac electrons. We also investigate the photoexcited energy distribution and show how, under appropriate circumstances, minimal excitations (Levitons) are generated. Furthermore, we show that the presence of a Rashba spin–orbit coupling can be exploited to tailor the shape of photoexcited wavepackets. Possible experimental realizations are also discussed.

Topological insulators(TIs) constitute a novel state of quantum matter which possesses non-trivial topological properties. Although discovered only in the recent few years, TIs have attracted intensive interest among the community of condensed matter physics and material science. TIs are insulating in the bulk but have conductive gapless edge or surface states on the boundaries, which have their origin in the nontrivial bulk band topology that is induced by the strong spin-orbital interactions in the materials. Existing in all dimensions, TIs exhibit a variety of exotic physics such as quantum spin Hall effect, momentum-spin locked surface states, Dirac fermion transport, quantized anomalous Hall effect, Majorana fermions, etc. In this thesis,\n\nI study the transport properties of 2D and 3D TIs by numerical approaches. As an introduction, a brief review of TIs is given. A detailed description of the numerical methods is also presented. The results can be summarized in four aspects. First, disorder is found be able to induce a non-trivial TI from an originally trivial band insulator, where the conductance of a two terminal device drops to nearly zero and then rises to form an anomalous plateau as disorder strength is increased, and finally all the states become localized. The real space Chern number calculation as well as the effective medium theory suggests that disorder is fundamentally responsible for the emerging of the extended helical edge states in this system. We also present a levitation and pair annihilation picture of the extended states for this model. Second, by making the 2D TIs into singly connected quantum point contacts(QPCs), I show a coherent and fast Aharonov-Bohm oscillation of conductance caused by the quantum interference of the helical edge states. This oscillation not only happens against weak magnetic field but also against the gate voltage in the zero-field condition.\n\nThis results in a giant edge magnetoresistance of the device in weak magnetic fields. The amplitude of the magnetoresistance is controllable by adjusting either the\n\nQPCs' slit width or the interference loop size in the device. The oscillation is found robust against disorder. Third, by applying a uniform spin-splitting Zeeman field in the bulk of the 3D TI whose surface states can be viewed as massless Dirac fermions,\n\nI find chiral edge states on the gapped surfaces of the 3D TI, which can be considered as interface states between domains of massive and massless Dirac fermions.\n\nEffectively these states are result of splitting of a perfect interface conducting channel. This picture is confirmed by the Landauer-B?ttiker calculations in four-terminal Hall bars. Finally, I propose the concept of topological semi-metals. By calculating the local density of states on the surfaces, I demonstrate that surface states and the gapless\n\nDirac cone already exist in the system although the bulk is not gapped. We show how the uni-axial strain induces an insulating band gap and turn the semi-metal into true TI. We predict existence of quantum spin Hall effect in the thin films made of these materials, which can be significantly enhanced by disorders.

The paper systematically study topological superconducting (TSC) phases in monolayer NbSe2 by constructing the hybrid paring tight-binding model of mixing on-site s-wave pairing (ps ) and long-range pairing (p A1) for the first time. We observe rich phases with both fixed and sensitive Chern numbers (CNs) depending on the chemical potential (μ) and out-of-plane magnetic field (Vz ). As p A1 increases, the TSC phase manifests matching and mismatching features according to whether the CNs match with the number of topological edge states (TESs). Strikingly, the introduction of long-range pairing significantly reduces the critical Vz to form TSC phases compared with the pure on-site s-wave paring. Moreover, the TSC phases can be modulated even at Vz = 0 under appropriate μ and p A1, which is identified by the robust TESs of ribbons. Additionally, the long-range pairing influences the hybridization of bulk and edge states, resulting in a matching/mismatching bulk-boundary correspondence with localized/oscillating TESs on the ribbons. Our findings are helpful for realizing TSC states through compressive strain experimentally to strengthen long-range pairings, as well as designing and regulating TSC materials.

We investigate transport through a normal-superconductor (NS) junction made from a quantum spin Hall (QSH) system with helical edge states and a two-dimensional (2D) chiral topological superconductor (TSC) having a chiral Majorana edge mode. We employ a two-dimensional extended four-band model for HgTe-based quantum wells in a magnetic (Zeeman) field and subject to s-wave superconductivity. We show using the Bogoliubov-de Gennes scattering formalism that this structure provides a striking transport signal of a 2D TSC. As a function of the sample width (or Fermi energy) the conductance resonances go through a sequence of $2e^2/h$ (non-trivial phase) and $4e^2/h$ plateaux (trivial phase) which fall within the region of a non-zero Chern number (2D limit) as the sample width becomes large. These signatures are a manifestation of the topological nature of the QSH effect and the TSC.

We theoretically study the superconducting properties of multi-band two-dimensional transition metal oxide superconductors by analyzing not only the role played by conventional singlet pairings, but also by the triplet order parameters, favored by the spin-orbit couplings present in these ma- terials. In particular, we focus on the two-dimensional electron gas at the (001) interface between LaAlO3 and SrTiO3 band insulators where the low electron densities and the sizeable spin-orbit couplings affect the superconducting features. Our theoretical study is based on an extended su- perconducting mean-field analysis of the typical multi-band tight-binding Hamiltonian, as well as on a parallel analysis of the effective electronic bands in the low-momentum limit, including static on-site and inter-site intra-band attractive potentials under applied magnetic fields. The presence of triplet pairings is able to strongly reduce the singlet order parameters which, as a result, are no longer a monotonic function of the charge density. The interplay between the singlet and the triplet pairings affects the dispersion of quasi-particle excitations in the Brillouin zone and also induces anisotropy in the superconducting behavior under the action of an in-plane and of an out- of-plane magnetic fields. Finally, non-trivial topological superconducting states become stable as a function of the charge density, as well as of the magnitude and of the orientation of the magnetic field. In addition to the chiral, time-reversal breaking, topological superconducting phase, favored by the linear Rashba couplings and by the on-site attractive potentials in the presence of an out- of-plane magnetic field, we find that a time-reversal invariant topological helical superconducting phase is promoted by not-linear spin-orbit couplings and by the inter-site attractive interactions in the absence of magnetic field.

In the first part of this thesis we propose interferometric schemes to probe the properties of edge states of topological insulators and superconductors. First, we consider two helical liquids on opposite edges of a narrow two-dimensional topological insulator, which are connected by one or several local tunnel junctions. In the presence of spatially inhomogeneous Rashba spin-orbit coupling, the spin textures of the helical states on opposite edges are different. We demonstrate that this has a strong impact on the electron transport between the edges. In particular, in the case of many random tunnel contacts, the localization length depends strongly on the spin textures of the edge states. We also propose to realize a Fabry-P\\'erot interferometer to measure the spin texture.
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\nSecond, we consider domain walls between superconducting and magnetic regions placed on top of a topological insulator, that were predicted support transport channels for Majorana fermions. We propose to study noise correlations in a Hanbury Brown-Twiss type interferometer and find three signatures of the Majorana nature of the channels. First, the average charge current in the outgoing leads vanishes. Furthermore, we predict an anomalously large shot noise in the output ports for a vanishing average current signal. Adding a quantum point contact to the setup, we find a surprising absence of partition noise which can be traced back to the Majorana nature of the carriers. Finally, we calculate the full counting statistics of this structure. At zero bias, we find an interpretation of Majorana-mediated charge transport in terms of two independent half-charge processes.
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\nIn the second part of this thesis, we explain how the quantum theory of weak measurements inspired a new method for the measurement of small effects and precision metrology. Many successful implementations of the weak-value amplification scheme have been recently reported. We review this scheme in some details with an emphasis on its benefits and limitations. We then generalize the method, and propose to use weak measurements away from the weak-value amplification regime to carry out precision measurements of time delays of light. Our scheme is robust to several sources of noise that are shown to only limit the relative precision of the measurement. Thus, they do not set a limit on the smallest measurable phase shift contrary to standard interferometry and weak-value based measurement techniques. Our idea is not restricted to phase-shift measurements and could be used to measure other small effects using a similar protocol.

In the past decade, Majorana Fermions (MFs) has been getting much interest because of their non-Abelian statistics and their application for topological quantum computation. One of the promising system hosting MFs is the hybrid structures between topological insulators (TIs) and s-wave superconductors. As the probe of MFs, measurements of a.c. Josephson effect, i.e. the Shapiro steps and Josephson radiation, is a useful tool. We fabricated Josephson junction on the 3D TI (Bi 0.2 Sb 0.8 ) 2 Te 3 grown on SrTiO 3 (111) substrate. We found that the supercurrent and the normal resistance of Josephson junction is gate-tunable and that the dominant carriers in the sample are from the topological surface states. Next, we measured the a.c. Josephson effects. In the Shapiro step measurement, we observed the both even and odd integer multiple of hf/2e steps. This contradicts with the existence of MFs in which odd multiple steps are expected to disappear. Furthermore, we also observed Josephson radiation with frequency of 2eV/h but no emission with frequency eV/h, a signature of MFs, is observed. We speculate that these results are due to huge dielectric constant of the substrate. Our experiment is the first report of a.c. Josephson effect on the gate-tunable Josephson junction of 3D TI film.

The transversal propagation of the edge states in a two-dimensional quantum spin Hall (QSH) system is classified by the characteristic parameter λ. There are two different types of helical edge states, the normal and special edge states, exhibiting distinct behaviors. The penetration depth of the normal edge state is momentum dependent, and the finite gap for edge bands decays monotonously with sample width, leading to the normal finite size effect. In contrast, the penetration depth maintains a uniform minimal value in the special edge states, and consequently the finite gap decays non-monotonously with sample width, leading to the anomalous finite size effect. To demonstrate their difference explicitly, we compared the real materials in phase diagrams. An intuitive way to search for the special edge states in the two-dimensional QSH system is also proposed.

Topological superconductor is described by a full pairing gap in the bulk with nonzero topological number and gapless surface states consisting of Majorana fermion that is a hypothetical particle of its own. Here, being its own means that a Majorana fermion should be an equal superposition of an electron and a hole state. The emergence of Majorana fermions is the most prominent characteristic of topological superconductors. In particle physics, it is still unclear if there are some elementary particles that are Majorana fermions, but importantly, they are likely to exist as quasiparticle excitations in certain condensed matter systems. It has drawn much attention in the content of Majorana fermions recently in particular the condensed matter physics area, since these Majorana states are ideal platform for non-Abelian statistics studies and can be used to fabricate as topological qubit, thus having great potential application in fault-tolerant topological quantum computation. A general introduction to the remarkable properties of Majorana fermions in condensed matter systems will be introduced following the story from Dirac equation to Majorana equation. Then this review elaborates a variety of routes to topological superconductivity in the realm of condensed mater physics, from Majorana modes in zero dimension to one dimension, that is, Majorana bound state and Majorana edge state. These Majorana states are localized states and propagating states respectively, but still obey non-Abelian statistics and remain their topological properties. Specifically, one-dimensional tight-binding model of a p-wave superconductor and a magnetic Fe chain with superconducting pairing are representative platforms for hosting Majorana bound states at the wires ends. Some emergent topological material systems, such as topological insulator, quantum spin Hall insulator, and quantum anomalous Hall insulator are two-dimensional material systems that can not only accommodate Majorana bound states but also Majorana edge states. Different material systems, from one-dimension quantum wire to two-dimension material system, from hybridized system to intrinsic system, will also be discussed specifically. Relevant theoretical studies and experimental results that show possible signatures of topological superconductivity and Majorana states are summarized as well. Particularly, experimental signatures of these Majorana states, such as zero-bias conductance peaks in tunneling spectra due to Majorana bound states, quantization of conductance in magneto-electric transport measurements due to chiral Majorana edge states, and some unconventional superconductivities in intrinsic topological superconductors will be briefly reviewed. Insights about methods to perspectives to realize topological quantum computation is provided at the end, emphasizing the braiding using different Majorana states. The goal of this review is to provide a general introduction to the subject for either experimentalists or theorists who are new to this field, focusing on the aspects and current progresses which are most important for understanding these basic physics.

The physics of a junction composed of a normal metal, quantum dot and 2D topological insulator (in a quantum spin Hall state) is elucidated. It maifests a subtle combination of Kondo correlations and quantum spin Hall edge states moving on the opposite sides of the 2D topological insulator. In a narrow strip geometry these edge states interact and a gap opens in the edge state spectrum. Consequently, Kondo screening is less effective and that affects electron transport through the junction. Specifically, when edge state coupling is strong enough, the tunneling differential conductance develops a dip at zero temperature instead of the standard zero bias Kondo peak.

Two dimensional (2D) topological insulators (TIs) and topological superconductors (TSCs) have been intensively studied for recent years due to its great potential for dissipationless electron transportation and fault-tolerant quantum computing, respectively. Here we focus on stanene, the tin analogue of graphene, to give a brief review of its development as a candidate for both 2D TI and TSC. Stanene is proposed to be a TI with a large gap of 0.3 eV, and its topological properties are sensitive to various factors, e.g., the lattice constants, chemical functionalization and layer thickness, which offer various methods for phase tunning. Experimentally, the inverted gap and edge states are observed recently, which are strong evidence for TI. In addition, stanene is also predicted to be a time reversal invariant TSC by breaking inversion symmetry, supporting helical Majorana edge modes. The layer-dependent superconductivity of stanene is recently confirmed by both transport and scanning tunneling microscopy measurements. This review gives a detailed introduction to stanene and its topological properties and some prospects are also discussed.

In this work, we theoretically investigate the light-induced topological phases and finite-size crossovers in a paradigmatic quantum spin Hall (QSH) system with high-frequency pumping optics. Taking the HgTe quantum well for an example, our numerical results show that circularly polarized light can break time-reversal symmetry and induce the quantum anomalous Hall (QAH) phase. In particular, the coupling between the edge states is spin dependent and is related not only to the size of the system, but also to the strength of the polarized pumping optics. By tuning the two parameters (system width and optical pumping strength), we obtain four transport regimes, namely, QSH, QAH, edge conducting, and normal insulator. These four different transport regimes have contrasting edge conducting properties, which will feature prominently in transport experiments on various topological materials.