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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

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.

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 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 unidirectional 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 CDW 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.

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 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.

Exfoliation, namely, the peeling of layered materials down to a single unit-cell thin foil, opens promising avenues to fabricate novel electronic materials. New properties and original functionalities emerge in the single and few layer configurations of a number of layered compounds, in particular in transition metal dichalcogenides. However, many of these thin exfoliated materials are very sensitive to ambient conditions impeding the exploration of this new and fascinating parameter space. Here we describe a method of mechanical exfoliation in ultra-high vacuum (UHV). This technique is easily adaptable to any UHV system and allows preparing and studying air sensitive nanoflakes in situ. We present the basic design and proof-of-concept scanning tunneling microscopy imaging of VSe2 nanoflakes.