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Tuneable electronic properties in transition metal dichalcogenides (TMDs) are essential to further their use in device applications. Here, we present a comprehensive scanning tunnelling microscopy and spectroscopy study of a doping-induced charge density wave (CDW) in semiconducting bulk 1T-ZrSe$_2$. We find that atomic impurities which locally shift the Fermi level ($E_F$) into the conduction band trigger a CDW reconstruction concomitantly to the opening of a gap at $E_F$. Our findings shed new light on earlier photoemission spectroscopy and theoretical studies of bulk 1T-ZrSe$_2$, and provide a local understanding of the electron-doping mediated CDW transition observed in semiconducting TMDs., Comment: 16 pages, 4 figures

We propose an empirical extension of Yakhot's model of strong turbulence [V. Yakhot, Phys. Rev. E 57(2) (1998)] that correctly describes the statistics of longitudinal velocity increments not only in the inertial range but also for larger scales up to the system length scale L. Two additional parameters are introduced to the original equation for the probability density function of velocity increments. These parameters can be motivated by physical arguments and their values are fixed by large scale boundary conditions. The resultant model ensures correct convergence of structure functions at the system length scale, including vanishing slopes at L, and shows good agreement with experimental data., Comment: 16 pages, 9 figures

Scanning Tunneling Spectroscopy (STS) is a unique technique to probe the local density of states (LDOS) at the atomic scale by measuring the tunneling conductance between a sharp tip and a sample surface. However, the technique suffers of well-known limitations, the so-called set-point effect, which can potentially introduce artifacts in the measurements. We compare several STS imaging schemes applied to the LDOS modulations of the charge density wave state on atomically flat surfaces, and demonstrate that only constant-height STS is capable of mapping the intrinsic LDOS. In the constant-current STS, commonly used and easier-to-implement, the tip-sample distance variations imposed by the feedback loop result in set-point-dependent STS images and possibly mislead the identification of the CDW gap edges., Comment: 7 pages, 6 figures

The mechanism of high-temperature superconductivity remains one of the great challenges of contemporary physics. Here, we review efforts to image the vortex lattice in copper oxide-based high-temperature superconductors and to measure the characteristic electronic structure of the vortex core of a $d$-wave superconductor using scanning tunneling spectroscopy., Comment: Main text : 7 pages, 8 figures

Geometric phase analysis (GPA) is a widely used technique for extracting displacement and strain fields from scanning probe images. Here, we demonstrate that GPA should be implemented with caution when several fundamental lattices contribute to the image, in particular in twisted heterostructures featuring moir\'e patterns. We find that in this case, GPA is likely to suggest the presence of chiral displacement and periodic strain fields, even if the structure is completely relaxed and without distortions. These delusive fields are subject to change with varying twist angles, which could mislead the interpretation of twist angle dependent properties.

Synthetic materials and heterostructures obtained by the controlled stacking of exfoliated monolayers are emerging as attractive functional materials owing to their highly tunable properties. We present a detailed scanning tunneling microscopy and spectroscopy study of single layer MoS$_2$-on-gold heterostructures as a function of twist angle. We find that their electronic properties are determined by the hybridization of the constituent layers and are modulated at the moir\'e period. The hybridization depends on the layer alignment and the modulation amplitude vanishes with increasing twist angle. We explain our observations in terms of a hybridization between the nearest sulfur and gold atoms, which becomes spatially more homogeneous and weaker as the moir\'e periodicity decreases with increasing twist angle, unveiling the possibility of tunable hybridization of electronic states via twist angle engineering.

Inducing and controlling spin-orbit coupling (SOC) in graphene is key to create topological states of matter, and for the realization of spintronic devices. Placing graphene onto a transition metal dichalcogenide is currently the most successful strategy to achieve this goal, but there is no consensus as to the nature and the magnitude of the induced SOC. Here, we show that the presence of backscattering in graphene-on-WSe$_2$ heterostructures can be used to probe SOC and to determine its strength quantitatively, by imaging quasiparticle interference with a scanning tunneling microscope. A detailed theoretical analysis of the Fourier transform of quasiparticle interference images reveals that the induced SOC consists of a valley-Zeeman ($\lambda_{\text{vZ}}\approx 2$ meV) and a Rashba ($\lambda_\text{R}\approx 15$ meV) term, one order of magnitude larger than what theory predicts, but in excellent agreement with earlier transport experiments. The validity of our analysis is confirmed by measurements on a 30 degree twist angle heterostructure that exhibits no backscattering, as expected from symmetry considerations. Our results demonstrate a viable strategy to determine SOC quantitatively by imaging quasiparticle interference.

Strain is attracting much interest as a mean to tune the properties of thin exfoliated two-dimensional materials and their heterostructures. Numerous devices to apply tunable uniaxial strain are proposed in the literature, but only few for biaxial strain where there is often a trade-off between maximum strain and uniformity, reversibility and device size. We present a compact device that allows the controlled application of uniform in-plane biaxial strain, with maximum deformation and uniformity comparable to those found in much larger devices. Its performance and strain uniformity over the sample area are modeled using finite element analysis and demonstrated by measuring the response of exfoliated 2H-MoS$_2$ to strain by Raman spectroscopy., Comment: 10 pages, 4 figures

Low magnetic field scanning tunneling spectroscopy of individual Abrikosov vortices in heavily overdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ unveils a clear d-wave electronic structure of the vortex core, with a zero-bias conductance peak at the vortex center that splits with increasing distance from the core. We show that previously reported unconventional electronic structures, including the low energy checkerboard charge order in the vortex halo and the absence of a zero-bias conductance peak at the vortex center, are direct consequences of short inter-vortex distance and consequent vortex-vortex interactions prevailing in earlier experiments., Comment: Main text : 5 pages, 4 figures Supplemental material : 3 pages, 2 figures

In the presence of multiple bands, well-known electronic instabilities may acquire new complexity. While multiband superconductivity is the subject of extensive studies, the possibility of multiband charge density waves (CDWs) has been largely ignored so far. Here, combining energy dependent scanning tunnelling microscopy (STM) topography with a simple model of the charge modulations and a self-consistent calculation of the CDW gap, we find evidence for a multiband CDW in 2H-NbSe$_2$. This CDW not only involves the opening of a gap on the inner band around the K-point, but also on the outer band. This leads to spatially out-of-phase charge modulations from electrons on these two bands, which we detect through a characteristic energy dependence of the CDW contrast in STM images.

Charge density waves (CDWs) are understood in great details in one dimension, but they remain largely enigmatic in two dimensional systems. In particular, numerous aspects of the associated energy gap and the formation mechanism are not fully understood. Two long standing riddles are the amplitude and position of the CDW gap with respect to the Fermi level ($E_F$) and the frequent absence of CDW contrast inversion (CI) between opposite bias scanning tunneling microscopy (STM) images. Here, we find compelling evidence that these two issues are intimately related. Combining density functional theory and STM to analyse the CDW pattern and modulation amplitude in 1$T$-TiSe$_2$, we find that CI takes place at an unexpected negative sample bias because the CDW gap opens away from $E_F$, deep inside the valence band. This bias becomes increasingly negative as the CDW gap shifts to higher binding energy with electron doping. This study shows the importance of CI in STM images to identify periodic modulations with a CDW and to gain valuable insight into the CDW gap, whose measurement is notoriously controversial., Comment: Main text 8 pages, 4 figures + Supplemental Material 7 pages, 6 figures

Many atomically thin exfoliated 2D materials degrade when exposed to ambient conditions. They can be protected and investigated by means of transport and optical measurements if they are encapsulated between chemically inert single layers in the controlled atmosphere of a glove box. Here, we demonstrate that the same encapsulation procedure is also compatible with scanning tunneling microscopy (STM) and spectroscopy (STS). To this end, we report a systematic STM/STS investigation of a model system consisting of an exfoliated 2H-NbSe2 crystal capped with a protective 2H-MoS2 monolayer. We observe different electronic coupling between MoS2 and NbSe2, from a strong coupling when their lattices are aligned within a few degrees to 2 essentially no coupling for 30{\deg} misaligned layers. We show that STM always probes intrinsic NbSe2 properties such as the superconducting gap and charge density wave at low temperature when setting the tunneling bias inside the MoS2 band gap, irrespective of the relative angle between the NbSe2 and MoS2 lattices. This study demonstrates that encapsulation is fully compatible with STM/STS investigations of 2D materials., Comment: 19 pages, 5 figures

The surface diffusion potential landscape plays an essential role in a number of physical and chemical processes such as self-assembly and catalysis. Diffusion energy barriers can be calculated theoretically for simple systems, but there is currently no experimental technique to systematically measure them on the relevant atomic length scale. Here, we introduce an atomic force microscopy based method to semiquantitatively map the surface diffusion potential on an atomic length scale. In this proof of concept experiment, we show that the atomic force microscope damping signal at constant frequency-shift can be linked to nonconservative processes associated with the lowering of energy barriers and compared with calculated single-atom diffusion energy barriers., Comment: 8 pages and 3 figures

The charge density wave (CDW) in solids is a collective ground state combining lattice distortions and charge ordering. It is defined by a complex order parameter with an amplitude and a phase. The amplitude and wavelength of the charge modulation are readily accessible to experiment. However, accurate measurements of the corresponding phase are significantly more challenging. Here we combine reciprocal and real space information to map the full complex order parameter based on topographic scanning tunneling microscopy (STM) images. Our technique overcomes limitations of earlier Fourier space based techniques to achieve distinct amplitude and phase images with high spatial resolution. Applying this analysis to transition metal dichalcogenides provides striking evidence that their CDWs consist of three individual charge modulations whose ordering vectors are connected by the fundamental rotational symmetry of the crystalline lattice. Spatial variations in the relative phases of these three modulations account for the different contrasts often observed in STM topographic images. Phase images further reveal topological defects and discommensurations, a singularity predicted by theory for a nearly commensurate CDW. Such precise real space mapping of the complex order parameter provides a powerful tool for a deeper understanding of the CDW ground state whose formation mechanisms remain largely unclear.

Ideal monolayers of common semiconducting transition metal dichalcogenides (TMDCs) such as MoS$_2$, WS$_2$, MoSe$_2$, and WSe$_2$ possess many similar electronic properties. As it is the case for all semiconductors, however, the physical response of these systems is strongly determined by defects in a way specific to each individual compound. Here we investigate the ability of exfoliated monolayers of these TMDCs to support high-quality, well-balanced ambipolar conduction, which has been demonstrated for WS$_2$, MoSe$_2$, and WSe$_2$, but not for MoS$_2$. Using ionic-liquid gated transistors we show that, contrary to WS$_2$, MoSe$_2$, and WSe$_2$, hole transport in exfoliated MoS$_2$ monolayers is systematically anomalous, exhibiting a maximum in conductivity at negative gate voltage (V$_G$) followed by a suppression of up to 100 times upon further increasing V$_G$. To understand the origin of this difference we have performed a series of experiments including the comparison of hole transport in MoS$_2$ monolayers and thicker multilayers, in exfoliated and CVD-grown monolayers, as well as gate-dependent optical measurements (Raman and photoluminescence) and scanning tunneling imaging and spectroscopy. In agreement with existing {\it ab-initio} calculations, the results of all these experiments are consistently explained in terms of defects associated to chalcogen vacancies that only in MoS$_2$ monolayers -- but not in thicker MoS$_2$ multilayers nor in monolayers of the other common semiconducting TMDCs -- create in-gap states near the top of the valence band that act as strong hole traps. Our results demonstrate the importance of studying systematically how defects determine the properties of 2D semiconducting materials and of developing methods to control them.

We report a detailed study of the microscopic effects of Cu intercalation on the charge density wave (CDW) in 1\textit{T}-Cu$_x$TiSe$_2$. Scanning tunneling microscopy and spectroscopy (STM/STS) reveal a unique, Cu driven spatial texturing of the charge ordered phase, with the appearance of energy dependent CDW patches and sharp $\pi$-phase shift domain walls ($\pi$DWs). The energy and doping dependencies of the patchwork are directly linked to the inhomogeneous potential landscape due to the Cu intercalants. They imply a CDW gap with unusual features, including a large amplitude, the opening below the Fermi level and a shift to higher binding energy with electron doping. Unlike the patchwork, the $\pi$DWs occur independently of the intercalated Cu distribution. They remain atomically sharp throughout the investigated phase diagram and occur both in superconducting and non-superconducting specimen. These results provide unique atomic-scale insight on the CDW ground state, questioning the existence of incommensurate CDW domain walls and contributing to understand its formation mechanism and interplay with superconductivity.

The copper oxides present the highest superconducting temperature and properties at odds with other compounds, suggestive of a fundamentally different superconductivity. In particular, the Abrikosov vortices fail to exhibit localized states expected and observed in all clean superconductors. We have explored the possibility that the elusive vortex-core signatures are actually present but weak. Combining local tunneling measurements with large-scale theoretical modeling, we positively identify the vortex states in YBa$_2$Cu$_3$O$_{7-\delta}$. We explain their spectrum and the observed variations thereof from one vortex to the next by considering the effects of nearby vortices and disorder in the vortex lattice. We argue that the superconductivity of copper oxides is conventional, but the spectroscopic signature does not look so because the superconducting carriers are a minority., Comment: Published version, including supplemental material

The capability to isolate one to few unit-cell thin layers from the bulk matrix of layered compounds opens fascinating prospects to engineer novel electronic phases. However, a comprehensive study of the thickness dependence and of potential extrinsic effects are paramount to harness the electronic properties of such atomic foils. One striking example is the charge density wave (CDW) transition temperature in layered dichalcogenides whose thickness dependence remains unclear in the ultrathin limit. Here we present a detailed study of the thickness and temperature dependences of the CDW in VSe$_2$ by scanning tunnelling microscopy (STM). We show that mapping the real-space CDW periodicity over a broad thickness range unique to STM provides essential insight. We introduce a robust derivation of the local order parameter and transition temperature based on the real space charge modulation amplitude. Both quantities exhibit a striking non-monotonic thickness dependence that we explain in terms of a 3D to 2D dimensional crossover in the FS topology. This finding highlights thickness as a true tuning parameter of the electronic ground state and reconciles seemingly contradicting thickness dependencies determined in independent transport studies.

The observation by scanning tunnelling spectroscopy (STS) of Abrikosov vortex cores in the high-temperature superconductor YBa$_2$Cu$_3$O$_{7-\delta}$ (Y123) has revealed a robust pair of electron-hole symmetric states at finite subgap energy. Their interpretation remains an open question because theory predicts a different signature in the vortex cores, characterised by a strong zero-bias conductance peak. We present STS data on very homogeneous Y123 at 0.4 K revealing that the subgap features do not belong to vortices: they are actually observed everywhere along the surface with high spatial and energy reproducibility, even in the absence of magnetic field. Detailed analysis and modelling show that these states remain unpaired in the superconducting phase and belong to an incoherent channel which contributes to the tunnelling signal in parallel with the superconducting density of states., Comment: Final version with supplementary material

Semiconducting transition metal dichalcogenides (TMDs) have emerged as materials that can be used to realize two-dimensional (2D) crystals possessing rather unique transport and optical properties. Most research has so far focused on sulfur and selenium compounds, while tellurium-based materials attracted little attention so far. As a first step in the investigation of Te-based semiconducting TMDs in this context, we have studied MoTe$_2$ crystals with thicknesses above 4 nm, focusing on surface transport and a quantitative determination of the gap structure. Using ionic-liquid gated transistors, we show that ambipolar transport at the surface of the material is reproducibly achieved, with hole and electron mobility values between 10 and 30 cm$^2$/Vs at room temperature. The gap structure is determined through three different techniques: ionic-liquid gated transistors and scanning tunneling spectroscopy, that allow the measurement of the indirect gap ($E_{ind}$), and optical transmission spectroscopy on crystals of different thickness, that enables the determination of both the direct ($E_{dir}$) and the indirect gap. We find that at room temperature $E_{ind}$ = 0.88 eV and $E_{dir}$ = 1.02 eV. Our results suggest that thin MoTe$_2$ layers may exhibit a transition to a direct gap before mono-layer thickness. They should also drastically extend the range of direct gaps accessible in 2D semiconducting TMDs., Comment: 14 pages, 4 figures. To appear in 2D Materials

Manganites are technologically important materials, used widely as solid oxide fuel cell cathodes: they have also been shown to exhibit electroresistance. Oxygen bulk diffusion and surface exchange processes are critical for catalytic action, and numerous studies of manganites have linked electroresistance to electrochemical oxygen migration. Direct imaging of individual oxygen defects is needed to underpin understanding of these important processes. It is not currently possible to collect the required images in the bulk, but scanning tunnelling microscopy could provide such data for surfaces. Here we show the first atomic resolution images of oxygen defects at a manganite surface. Our experiments also reveal defect dynamics, including oxygen adatom migration, vacancy-adatom recombination and adatom bistability. Beyond providing an experimental basis for testing models describing the microscopics of oxygen migration at transition metal oxide interfaces, our work resolves the long-standing puzzle of why scanning tunnelling microscopy is more challenging for layered manganites than for cuprates., Comment: 7 figures

Tunneling spectroscopy played a central role in the experimental verification of the microscopic theory of superconductivity in the classical superconductors. Initial attempts to apply the same approach to high-temperature superconductors were hampered by various problems related to the complexity of these materials. The use of scanning tunneling microscopy/spectroscopy (STM/STS) on these compounds allowed to overcome the main difficulties. This success motivated a rapidly growing scientific community to apply this technique to high-temperature superconductors. This paper reviews the experimental highlights obtained over the last decade. We first recall the crucial efforts to gain control over the technique and to obtain reproducible results. We then discuss how the STM/STS technique has contributed to the study of some of the most unusual and remarkable properties of high-temperature superconductors: the unusual large gap values and the absence of scaling with the critical temperature; the pseudogap and its relation to superconductivity; the unprecedented small size of the vortex cores and its influence on vortex matter; the unexpected electronic properties of the vortex cores; the combination of atomic resolution and spectroscopy leading to the observation of periodic local density of states modulations in the superconducting and pseudogap states, and in the vortex cores., Comment: To appear in RMP; 65 pages, 62 figures

It is shown that prize changes of the US dollar - German Mark exchange rates upon different delay times can be regarded as a stochastic Marcovian process. Furthermore we show that from the empirical data the Kramers-Moyal coefficients can be estimated. Finally, we present an explicite Fokker-Planck equation which models very precisely the empirical probabilitiy distributions., Comment: 3 figures