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The properties of semiconductors can be crucially impacted by midgap states induced by dopants, which can be native or intentionally incorporated in the crystal lattice. For Bernal-stacked bilayer graphene (BLG), which has a tunable bandgap, the existence of midgap states induced by dopants has been conjectured, but never confirmed experimentally. Here, we report scanning tunneling microscopy and spectroscopy results, supported by tight-binding calculations, that demonstrate the existence of midgap states in BLG. We show that the midgap state in BLG -- for which we demonstrate gate-tunability -- appears when the dopant is hosted on the non-dimer sublattice sites. We further evidence the presence of narrow resonances at the onset of the high energy bands (valence or conduction, depending on the dopant type) when the dopants lie on the dimer sublattice sites. These results suggest that dopants/defects can play an important role in the transport and optical properties of multilayer graphene samples, especially at energies close to the band extrema., Includes supplementary material

Graphite crystals used to prepare graphene-based heterostructures are generally assumed to be defect free. We report here scanning tunneling microscopy results that show graphite commonly used to prepare graphene devices can contain a significant amount of native defects. Extensive scanning of the surface allows us to determine the concentration of native defects to be 6.6$\times$10$^8$ cm$^{-2}$. We further study the effects of these native defects on the electronic properties of Bernal-stacked bilayer graphene. We observe gate-dependent intravalley scattering and successfully compare our experimental results to T-matrix-based calculations, revealing a clear carrier density dependence in the distribution of the scattering vectors. We also present a technique for evaluating the spatial distribution of short-scale scattering. A theoretical analysis based on the Boltzmann transport equation predicts that the dilute native defects identified here are an important extrinsic source of scattering, ultimately setting the mobility at low temperatures., Comment: Accepted in Nano Letters

We use the exact analytical technique introduced by us [Phys. Rev. B 101, 115405 (2020)] to recover the surface Green's functions and the corresponding Fermi-arc surface states for various lattice models of Weyl semimetals. For these models we use the $T$-matrix formalism to calculate the quasiparticle interference patterns generated in the presence of impurity scattering. In particular, we consider the models introduced by Kourtis et al. [Phys. Rev. B 93, 041109(R) (2016)] (model A) and Lau et al. [Phys. Rev. Lett. 119, 076801 (2017)] (model B), and we find that, as opposed to observations previously obtained via joint density of states and spin-dependent scattering probability, the interarc scattering in the quasiparticle interference features is fully suppressed in model A, and is very small in model B. Our findings indicate that these models may not correctly describe materials such as ${\mathrm{MoTe}}_{2}$, since for such materials interarc scattering is clearly visible experimentally, e.g., by Deng et al. [Nat. Phys. 12, 1105 (2016)]. We also focus on the minimal models proposed by McCormick et al. [Phys. Rev. B 95, 075133 (2017)], which indeed recover significant interarc scattering features.

We provide here a new and exact formalism to describe the formation of end, edge or surface states through the evolution of impurity-induced states. We propose a general algorithm that consists of finding the impurity states via the T-matrix formalism and showing that they evolve into boundary modes when the impurity potential goes to infinity. We apply this technique to obtain Majorana states in 1D and 2D systems described by the Kitaev model with point-like and respectively line-like impurities. We confirm our exact analytical results by a numerical tight-binding approach. We argue that our formalism can be applied to other topological models, as well as to any model exhibiting edge states., 5 pages, 5 figures

International audience; We provide a new analytical tool to calculate the energies of Andreev bound states (ABS) in long imperfect SNS junctions, at present these can only be described by numerical tools. We model an NS junction as a delta-function "Andreev" impurity, i.e., a localized potential which scatters an electron into a hole with opposite spin. We show using the scattering matrix formalism that, quite surprisingly, an "Andreev" impurity is equivalent to an NS junction characterized by both Andreev reflection and a finite amount of normal scattering. The ABS energies are then calculated using the T-matrix formalism applied to a system with two Andreev impurities. Our results lie between those for a perfect long SNS junction limit described by the Andreev approximation (ABS energies depend linearly on the phase and are independent of the chemical potential) and the particle-inthe-box limit (bound state energies are independent of the phase and have a linear dependence on the chemical potential). Moreover, we recover a closed-form expression for the ABS energies by expanding around the particle-in-the-box limit.

In this work we provide a direct and non-numerical technique to obtain the surface Green's functions for three-dimensional systems. This technique is based on the ideas presented by V. Kaladzhyan and C. Bena [Phys. Rev. B 100, 081106(R) (2019)], in which we start with an infinite system and model the boundary using a planelike infinite-amplitude potential. Such a configuration can be solved exactly using the $T$-matrix formalism. We apply our method to calculate the surface Green's function and the corresponding Fermi-arc states for Weyl semimetals. We also apply the technique to systems of lower dimensions, such as Kane-Mele and Chern insulator models, to provide a more efficient and non-numerical method to describe the formation of edge states.

We study the formation of Majorana states in superconductors using the Majorana polarization, which can locally evaluate the Majorana character of a given state. We introduce the definition of the Majorana polarization vector and the corresponding criterion to identify a Majorana state, and we apply it to some simple cases such as a one-dimensional wire with spin-orbit coupling, subject to a Zeeman magnetic field, and proximitized by a superconductor, as well as to an NS junction made with such a wire. We also apply this criterion to two-dimensional finite-size strips and squares subject to the same physical conditions. Our analysis demonstrates the necessity of using the Majorana polarization local order parameter to characterize the Majorana states, particularly in finite-size systems., Comment: 16 pages, 11 figures, review article published in Comptes Rendus Physique

We study the spin-resolved spectral properties of the impurity states associated to the presence of magnetic impurities in two-dimensional, as well as one-dimensional systems with Rashba spin-orbit coupling. We focus on Shiba bound states in superconducting materials, as well as on impurity states in metallic systems. Using a combination of a numerical T-matrix approximation and a direct analytical calculation of the bound state wave function, we compute the local density of states (LDOS) together with its Fourier transform (FT). We find that the FT of the spin-polarized LDOS, a quantity accessible via spin-polarized STM, allows to accurately extract the strength of the spin-orbit coupling. Also we confirm that the presence of magnetic impurities is strictly necessary for such measurement, and that non-spin-polarized experiments cannot have access to the value of the spin-orbit coupling., 26 pages, 6 figures

International audience; We study the formation of Majorana states in quasi-one-dimensional (quasi-1D) and two-dimensional square lattices with open boundary conditions, with general anisotropic Rashba coupling, in the presence of an applied Zeeman field and in the proximity of a superconductor. For systems in which the length of the system is very large (quasi-1D) we calculate analytically the exact topological invariant, and we find a rich corresponding phase diagram which is strongly dependent on the width of the system. We compare our results with previous results based on a few-band approximation. We also investigate numerically open two-dimensional systems of finite length in both directions. We use the recently introduced generalized Majorana polarization, which can locally evaluate the Majorana character of a given state. We find that the formation of Majoranas depends strongly on the geometry of the system: for a very elongated wire the finite-size numerical phase diagram reproduces the analytical phase diagram for infinite systems, while if the length and the width are comparable, no Majorana states can form; however, one can show the formation of “quasi-Majorana” states that have a local Majorana character but no global Majorana symmetry.

We have measured the lifetime of spin imbalances in the quasiparticle population of a superconductor ($\tau_s$) in the frequency domain. A time-dependent spin imbalance is created by injecting spin-polarised electrons at finite excitation frequencies into a thin-film mesoscopic superconductor (Al) in an in-plane magnetic field (in the Pauli limit). The time-averaged value of the spin imbalance signal as a function of excitation frequency, $f_{RF}$ shows a cut-off at $f_{RF} \approx 1/(2\pi\tau_s)$. The spin imbalance lifetime is relatively constant in the accessible ranges of temperatures, with perhaps a slight increase with increasing magnetic field. Taking into account sample thickness effects, $\tau_s$ is consistent with previous measurements and of the order of the electron-electron scattering time $\tau_{ee}$. Our data are qualitatively well-described by a theoretical model taking into account all quasiparticle tunnelling processes from a normal metal into a superconductor., Comment: Includes Supplementary Information

In this paper we demonstrate under what conditions a pseudo-spin degree of freedom or character can be ascribed to the Majorana bound states (MBS) which can be created at the end of one dimensional non-interacting systems, corresponding to D, DIII and BDI in the usual classification scheme. We have found that such a character is directly related to the class of the topological superconductor and its description by a $\mathbb{Z}$, rather than a $\mathbb{Z}_2$, invariant which corresponds to the BDI class. We have also found that the DIII case with mirror symmetry, which supports multiple MBS, is in fact equivalent to the BDI class with an additional time-reversal symmetry. In all cases where a character can be given to the Majorana states we show how to construct the appropriate operator explicitly in various examples. We also examine the consequences of the Majorana character by considering possible hybridization of MBS brought into proximity and find that two MBS with the same character do not hybridize. Finally, we show that having this character or not has no consequence on the braiding properties of MBS., Comment: 10 pages, 1 figure

13 pages, 17 figures; International audience; In this paper we show that for a range of configurations of inhomogeneous magnetic fields it is possible to create flat bands of Majorana states localized on the edges of 2-d lattices. Majorana bound states have been predicted to exist in both one dimensional and two dimensional systems with Rashba spin-orbit coupling, magnetic fields, and placed in proximity to a superconductor. For the proposed systems we present the bulk topological phase diagrams, and we study the conditions for weak topology which predicts the formation of bands of Majorana states. The Majorana bands are demonstrated to be relatively stable with respect to a variety of different perturbations on both square and hexagonal lattices.

We calculate the density of states for an interacting quantum wire in the presence two impurities of arbitrary potential strength. To perform this calculation, we describe the Coulomb interactions in the wire within the Tomonaga-Luttinger liquid theory. After establishing and solving the Dyson equation for the fermionic retarded Green's functions, we study how the profile of the local density of states is affected by the interactions in the entire range of impurity potentials. Same as in the non-interacting case, when increasing the impurity strength, the central part of the wire becomes more and more disconnected from the semi-infinite leads, and discrete localized states begin to form; the width of the corresponding peaks in the spectrum depends on the interaction strength. As expected from the Luttinger liquid theory, impurities also induce a reduction of the local density of states at small energies. Two other important aspects are highlighted: the appearance of an extra-modulation in the density of states at non-zero Fermi momentum when interactions are present, and the fact that forward scattering must be taken into account in order to recover the Coulomb blockade regime for strong impurities., Comment: 12 pages, 12 figures

We study the proximity effect in a topological nanowire tunnel coupled to an s-wave superconducting substrate. We use a general Green's function approach that allows us to study the evolution of the Andreev bound states in the wire into Majorana fermions. We show that the strength of the tunnel coupling induces a topological transition in which the Majorana fermionic states can be destroyed when the coupling is very strong. Moreover, we provide a phenomenologial study of the effects of disorder in the superconductor on the formation of Majorana fermions. We note a non-trivial effect of a quasiparticle broadening term which can take the wire from a topological into a non-topological phase in certain ranges of parameters. Our results have also direct consequences for a nanowire coupled to an inhomogenous superconductor.

We study a one-dimensional wire with strong Rashba and Dresselhaus spin-orbit coupling (SOC), which supports Majorana fermions when subject to a Zeeman magnetic field and in proximity of a superconductor. Using both analytical and numerical techniques we calculate the electronic spin texture of the Majorana end states. We find that the spin polarization of these states depends on the relative magnitude of the Rashba and Dresselhaus SOC components. Moreover, we define and calculate a local "Majorana polarization" and "Majorana density" and argue that they can be used as order parameters to characterize the topological transition between the trivial system and the system exhibiting Majorana bound modes. We find that the local "Majorana polarization" is correlated to the transverse spin polarization, and we propose to test the presence of Majorana fermions in a 1D system by a spin-polarized density of states measurement., Comment: 6 pages, 5 figures, RevTeX4; added analytical results section and supplementary material; new references added

We study the density of states in graphene at high magnetic field, when the physics is dominated by strong correlations between electrons. In particular we use the method of Haldane pseudopotentials to focus on almost empty or almost filled Landau levels. We find that, besides the usual Landau level peaks, additional peaks ("sashes") appear in the spectrum. The energies of these peaks are determined by the strength of Haldane's pseudopotentials, but as opposed to the usual two-dimensional gas, when there is a one-to-one correspondence between a Haldane pseudopotential and a peak in the spectrum, the energy of each peak is determined in general by a combination of more than one pseudopotential values. An eventual measure of these peak in the density of states spectrum of graphene would allow one to determine the value of the pseudopotentials in graphene, and thus test the strength of the interactions in this system., 9 pages, 1 table

International audience; We calculate the AC conductance and the finite-frequency non-symmetrized noise in interacting quantum wires and single-wall carbon nanotubes in the presence of an impurity. We observe a strong asymmetry in the frequency spectrum of the non-symmetrized excess noise, even in the presence of the metallic leads. We find that this asymmetry is proportional to the differential excess AC conductance of the system, defined as the difference between the AC differential conductances at finite and zero voltage, and thus disappears for a linear system. In the quantum regime, for temperatures much smaller than the frequency and the applied voltage, we find that the emission noise is exactly equal to the impurity partition noise. For the case of a weak impurity we expand our results for the AC conductance and the noise perturbatively. In particular, if the impurity is located in the middle of the wire or at one of the contacts, our calculations show that the noise exhibits oscillations with respect to frequency, whose period is directly related to the value of the interaction parameter $g$.

We show that the impurity-induced Friedel oscillations allow one to probe in an unexpected and quite remarkable manner the electronic properties of two-dimensional systems such as graphene or high-temperature superconductors. In particular, we show that by studying these oscillations, one can get access not only to the constant-energy maps, but also to more hidden information such as the chiral properties of Dirac electrons in graphene, which cannot be observed directly by other methods. For graphene, this hidden information is revealed by comparing the theoretical predictions with scanning tunneling microscopy experimental measurements of the local density of states.