Quantum transport through MoS$_2$ constrictions defined by photodoping
Alexander Epping
Luca Banszerus
Johannes G\"uttinger
Luisa Kr\"uckeberg
Kenji Watanabe
Takashi Taniguchi
Fabian Hassler
Bernd Beschoten
Christoph~Stampfer
We present a device scheme to explore mesoscopic transport through molybdenum disulfide (MoS$_2$) constrictions using photodoping. The devices are based on van-der-Waals heterostructures where few-layer MoS$_2$ flakes are partially encapsulated by hexagonal boron nitride (hBN) and covered by a few-layer graphene flake to fabricate several electrical contacts. Since the as-fabricated devices are insulating at low temperatures, we use photo-induced remote doping in the hBN to create free charge carriers in the MoS$_2$ layer. On top of the device, we place additional metal structures, which define the shape of the constriction and act as shadow masks during photodoping of the underlying MoS$_2$. Low temperature two- and four-terminal measurements show signatures of quantized conductance through these constrictions. The conductance steps have a height of $e^2/h$ indicating the breaking of both valley and Kramers' degeneracy.
Manuscript_arxiv.tex

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\author{Alexander Epping}
\affiliation[]
{These authors contributed equally to this work}
\alsoaffiliation[]
{JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany}
\alsoaffiliation[]
{Peter Gr\"unberg Institute (PGI-9), Forschungszentrum J\"ulich, 52425 J\"ulich, Germany}

\author{Luca Banszerus}
\alsoaffiliation[]
{These authors contributed equally to this work}
\affiliation[]
{JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany}
\alsoaffiliation[]
{Peter Gr\"unberg Institute (PGI-9), Forschungszentrum J\"ulich, 52425 J\"ulich, Germany}

\author{Johannes G\"uttinger}
\alsoaffiliation[]
{JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany}
\affiliation[]
{Peter Gr\"unberg Institute (PGI-9), Forschungszentrum J\"ulich, 52425 J\"ulich, Germany}

\author{Luisa Kr\"uckeberg}
\affiliation[]
{JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany}

\author{Kenji Watanabe}
\affiliation[]
{National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan }

\author{Takashi Taniguchi}
\affiliation[]
{National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan }

\author{Fabian Hassler}
\affiliation[]
{JARA-Institute for Quantum Information, RWTH Aachen University, 52074 Aachen, Germany}

\author{Bernd Beschoten}
\affiliation[]
{JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany}

\author{Christoph~Stampfer}
\affiliation[]
{JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany}
\alsoaffiliation[]
{Peter Gr\"unberg Institute (PGI-9), Forschungszentrum J\"ulich, 52425 J\"ulich, Germany}
\email{stampfer@physik.rwth-aachen.de}


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\title
  {Quantum transport through MoS$_2$ constrictions defined by photodoping}

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\keywords{MoS$_2$, nanoconstricition, photodoping, quantized conductance}

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\begin{abstract}
  We present a device scheme to explore mesoscopic transport through molybdenum disulfide (MoS$_2$) constrictions using photodoping. The devices are based on van-der-Waals heterostructures where few-layer MoS$_2$ flakes are partially encapsulated by hexagonal boron nitride (hBN) and covered by a few-layer graphene flake to fabricate several electrical contacts. Since the as-fabricated devices are insulating at low temperatures, we use photo-induced remote doping in the hBN to create free charge carriers in the MoS$_2$ layer. On top of the device, we place additional metal structures, which define the shape of the constriction and act as shadow masks during photodoping of the underlying MoS$_2$. Low temperature two- and four-terminal measurements show signatures of quantized conductance through these constrictions. The conductance steps have a height of $e^2/h$ indicating the breaking of both valley and Kramers' degeneracy.
\end{abstract}

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Van-der-Waals heterostructures based on graphene and transition metal dichalcogenides (TMDCs) are attracting increasing attention in mesoscopic physics and solid-state research\cite{Novo16,Geim13,Rad10,Herr13,Rus08,Ter16}. One of the main challenges in this field is to find ways of confining carriers in well defined nanostructures. Achieving a high degree of control over carrier confinement is necessary for manipulating individual charges or spins\cite{Elz04,Wee88,Wep13,Joh92}, as well as for investigating mesoscopic physics phenomena such as quantized conductance or valley filtering in bilayer graphene and TMDCs\cite{Ryc07}. Moreover, good control over carrier confinement is a necessary requirement for studying fundamental material properties such as the g-factor and the charge carriers effective mass in mesoscopic devices.  

Because of the absence of a band gap, mesoscopic physics in graphene is mostly studied in devices where the material has been etched into the desired shape\cite{Ter16,Rus08}. These devices typically suffer from scattering and trap states on their rough edges\cite{Ter16,Tom11,Bis14}. In contrast, bilayer graphene\cite{Gos12,All12} and semiconducting \mbox{TMDCs}\cite{Son15,Ke16} allow for smooth electrostatic confinement with the help of metallic gates, similarly to what has been pioneered in two-dimensional electron gases (2DEGs) in conventional semiconductor heterostructures for the past decades\cite{Wee88,Gold1998,Petta05}.

Here, we demonstrate a new method of obtaining confinement in van-der-Waals heterostructures based on few-layers MoS$_2$ encapsulated in hexagonal boron nitride (hBN). In analogy to remote doping in semiconductor 2DEGs, we make use of the photodoping effect recently observed in hBN-graphene heterostructures\cite{Ju14,Neu16}. We use this effect to define stable and smooth sub-wavelength lateral doping profiles in MoS$_2$ employing metal shadow masks that are lithographically defined on top of an hBN/MoS$_2$/hBN heterostructure. Our photo-induced remote doping approach enables us to reach free charge carrier densities of up to $10^{13}$~cm$^{-2}$, when using an additional gate voltage during light illumination. It thus allows to create arbitrary carrier density profiles defined by the shadow mask geometry. The feasibility of our approach is demonstrated by studying quantum transport through constrictions in MoS$_2$. We expect that our method allows to fabricate more complex doping geometries such as antidot lattices or Aharonov-Bohm rings with the advantage that the metal used for shadow masking does not need to be electrically contacted, when studying mesoscopic transport.

 \begin{figure*}[t!]\centering
\includegraphics[draft=false,keepaspectratio=true,clip,%
                   width=\linewidth]%
                   {Fig1.eps}
\caption[Fig01]{ (a)-(c) Schematic illustration of the device fabrication. (a) MoS$_2$ is partially encapsulated by two hBN flakes. (b) A few-layer graphene flake is transferred over the entire heterostructure. (c) After etching the graphene into the desired shape, the shadow mask structure and the contacts are made. (d) Optical microscope image of the complete van-der-Waals heterostructure. (e) Raman spectra on a final device recorded in the lead area indicated by the white circle in panel (f) (top) and in the sandwiched part of the device indicated by the white square in panel (f) (bottom) showing the individual Raman peaks of hBN (green), graphene (purple) and MoS$_2$ (blue). (f) Raman map showing the integrated intensities of the hBN peak, the G-peak of the FLG and the A$_{1g}$ mode of the MoS$_2$, represented by the colors purple, blue and green, respectively. The black areas are covered by metal whose boundaries are indicated by the white dashed lines. (g) Schematic of a typical multi-terminal device. (h) Scanning electron microscopy image of a four-terminal device. The individual flakes are highlighted by dashed lines with the same color as in panel (e). (i) Close-up of the shadow mask that forms a 250~nm wide constriction. }
\label{fig01}
\end{figure*}



All samples are fabricated using the well-established dry van-der-Waals stacking technique\cite{Wan13,Bansz15}, but we employ only a single transfer step to fabricate all few-layer graphene contacts. A few-layer MoS$_2$ crystal gets partially encapsulated between two hBN flakes and is then placed onto a Si$^{++}$/SiO$_2$ substrate (Figure~1a). In order to electrically contact the MoS$_2$ from the top surface, the left and the right parts of the MoS$_2$ flake remain uncovered by the upper hBN crystal. At the same time, the inner fully encapsulated area of MoS$_2$ is well protected against transfer related degradation and contamination ensuring a high quality of the device. Thereafter, one large flake of few-layer graphene (FLG) is placed on top of the entire structure, which is in direct contact with MoS$_2$ in all parts that are not protected by the upper hBN flake (Figure~1b). Figure~1d shows an optical microscopy image of such a complete van-der-Waals stack. In contrast to previous reports\cite{Roy14,Yu14,Cui15}, where multiple flakes of FLG have been transferred to fabricate electrical contacts, we only transfer a single large FLG flake that is subsequently patterned (Figure~1c). Using a single flake to contact multi-terminal MoS$_2$ devices is convenient, as it saves several transfer steps or crystal phase engineering of the MoS$_2$\cite{Kap14}. Furthermore, it prevents any direct contact of solvents or wet chemistry with the MoS$_2$. When immersed in solvents, even the outer parts of the MoS$_2$ are fully encapsulated by the FLG. Electron-beam lithography and reactive ion etching (RIE) using an Ar/O$_2$ plasma allow to selectively etch the FLG and the MoS$_2$, while the etching rate is negligible for the top hBN. The graphene above the top hBN and several trenches through the MoS$_2$ are etched. The trenches are needed to separate the individual leads on each side of the multi-terminal devices (see illustration in Figure~1g). In a last step, Ti/Au contacts to the FLG, as well as the shadow mask structures for the constriction on the top hBN (see also Figure 1c) are evaporated, followed by lift-off.


The devices are characterized by scanning confocal Raman microscopy which enables us to spatially probe the structural (and partly electronic) quality of all individual layers of the van-der-Waals heterostructure. Figure~1e depicts two typical Raman spectra. The upper spectrum in Figure~1e is recorded in the lead area (see open circle in Figure~1f), where the MoS$_2$ is covered with FLG. The G-peak at 1583~cm$^{-1}$ (purple) originates from the FLG\cite{Dressel09}. The hBN peak is located at around 1365~cm$^{-1}$ (green)\cite{Gor11} while the A$_{1g}$ and the E$^1_{2g}$ modes of the MoS$_2$ are located at 409~cm$^{-1}$ (blue) and 386~cm$^{-1}$, respectively\cite{Sah13,Ton13}. The lower spectrum in Figure~1e is recorded on the hBN/MoS$_2$/hBN area of the device, close to the shadow masks, where only the two peaks from the hBN and the MoS$_2$ are visible (see open square in Figure~1f). Figure~1f shows a Raman map of the intensities of the characteristic Raman peaks of the 2D materials in use. Green represents the intensity of the hBN peak, purple depicts the G-peak intensity of the FLG, blue shows the intensity of the A$_{1g}$ mode of the MoS$_2$. The dark areas are the gold shadow masks, which define the constriction in the underlying MoS$_2$ layer. Figure~1h shows a scanning electron microscopy image (SEM) of a typical device. The boundaries of the individual flakes are indicated by the dashed colored lines, where we use the same color code as in Figure~1e. Figure~1i shows an SEM image of a typical shadow mask structure with a width of 250~nm and a length of 175~nm.
\begin{figure*}[tb]\centering
\includegraphics[draft=false,keepaspectratio=true,clip,%
                   width=\linewidth]%
                   {Fig2.eps}
\caption[fig02]{(a) Current through a Hall bar device as function of back gate voltage for temperatures ranging from 300~K to 5~K in steps of 25~K between 300~K and 25~K as well as steps of 5~K between 25~K and 5~K. Inset: optical microscope image of the Hall bar fabricated from a hBN/MoS$_2$/hBN/FLG van-der-Waals heterostructure. (b) Current as function of the gate voltage in a Hall bar for several consecutive photodoping steps of V$^{\mathrm{PD}}_\mathrm{g}=-10$~V. (c) Charge carrier mobility of the Hall bar as function of temperature, showing increased carrier mobility at low temperatures after photodoping at 150~K. Inset: Charge carrier mobility as function of the photodoping gate voltage for different charge carrier densities, indicating that photodoping does not influence the charge carrier mobility of the device.} \label{fig02}
\end{figure*}



We start by discussing charge transport in a few-layer MoS$_2$ Hall bar structure and explore the method of photodoping before discussing transport through the MoS$_2$ constrictions. The Hall bar device was fabricated the same way as described above. In Figure~2a we show the current $I$ flowing through the Hall bar (see inset of Figure~2a), which is measured  in standard four-terminal geometry in longitudinal configuration as function of the applied gate voltage $V_\mathrm{g}$ for temperatures ranging between 300~K to 5~K at a constant bias of V$_\mathrm{b}$~=~100~mV. Prior to photodoping (PD), the overall current strongly decreases with decreasing temperature and falls below our detection limit for temperatures less than 10~K (see corresponding green curve taken at 2~K in the inset of Figure~2b). The strong decrease of current at low temperatures is due to (i) a strong increase of the MoS$_2$ resistance and partially (ii) to an increase of the contact resistance (see supplementary materials, Figures S1a and S1b). The increase in resistivity of the MoS$_2$ is most likely due to localization effects. At higher temperatures, we observe a gate dependent insulator-to-metal transition seen by the onset of the current flow in Figure~2a, which continuously shifts to smaller gate voltages with increasing temperature. Similar observations of a charge carrier density dependent insulator-to-metal transition in MoS$_2$ have been recently reported\cite{Rad13,Bau13,Sch14}. Metallic conductance at low temperatures can be observed for high charge carrier densities around $n\approx10^{13}$~cm$^{-2}$. In order to adjust the charge carrier density in our devices we make use of the recently reported photodoping mechanism across a hBN-to-graphene interface\cite{Ju14,Neu16,Won15}. Nitrogen vacancies and/or carbon impurities in the hBN crystal can be optically activated and act as a charge reservoir in gated graphene.
By illuminating our hBN/MoS$_2$/hBN Hall bar using a high energy light emitting diode (LED) with a center wavelength of 470~nm, these hBN defect states get optically excited. By applying a gate voltage, the activated charges are transferred into the MoS$_2$ layer leaving behind oppositely charged states in the hBN layer. The charge transfer continues until the applied (back gate induced) electric field gets fully screened by the charged hBN layer. When turning off the LED, the charges become trapped in the hBN resulting in a constant carrier doping of the illuminated area of MoS$_2$. In agreement with earlier work on similar photodoping of graphene\cite{Neu16}, we observe that the doping is stable over the entire measurement period (up to several weeks) in the temperature range between 2~K and 250~K. Figure~2b shows a series of gate characteristics of the Hall bar at 150~K before and after photodoping. The device was illuminated for 10 minutes at constant gate voltages varying between V$^{\mathrm{PD}}_\mathrm{g}=-10$~V and V$^{\mathrm{PD}}_\mathrm{g}=-80$~V in steps of $\Delta$V$_\mathrm{g}=-10$~V. The gate characteristics show a respective overall increase in current, which corresponds well to shifts of $\Delta$V$_\mathrm{g}=-10$~V in gate voltage. At the same time the gate characteristics  remain identical in slope and shape indicating a constant charge carrier mobility and a homogeneous and efficient doping process throughout the entire device. After illuminating the Hall bar at V$^{\mathrm{PD}}_\mathrm{g}=-80$~V, the device shows metallic behavior, i.e. it remains well conductive at 2~K when applying moderate gate voltages (see inset of Figure~2b and supplementary materials Figure~S2). Charge carrier mobilities extracted from Hall measurements on a typical device show values on the order of $\mu$~=~600~cm$^2$/(Vs) at low temperatures (2~K), which decreases to a value of around 45~cm$^2$/(Vs) at room temperature most likely due to electron-phonon scattering\cite{Rad13} (see Figure~2c). In the inset of Figure~2c, the charge carrier mobility at 2~K is plotted for different photodoping steps. For each photodoping step the device is warmed up to 150~K (as the LED is freezing out at low temperatures), the photodoping is performed and the device is cooled down again to 2~K to extract the low temperature mobility. The different colors represent different charge carrier densities. Similar to remote doping in 2DEGs\cite{Ding1978}, the mobility of the device is not affected by the process of photodoping, as the charge that is introduced in the hBN flake is spatially separated from the MoS$_2$ (see colored dashed lines in the inset of Figure~2c). In order to underline the high electronic quality of the heterostructures, we also measure the quantum Hall effect in a MoS$_2$ Hall bar\cite{Cui15}, where we extract quantum mobilities on the order of 2,000~cm$^2$/(Vs) (see supplementary material Figure~S3).
\\


\begin{figure}[t!]\centering
\includegraphics[draft=false,keepaspectratio=true,clip,%
                   width=\linewidth]%
                   {Fig3.eps}
\caption[fig03]{(a) Current through the constriction as function of gate voltage for different temperatures in steps of 2~K. Step-like features in the current are observed at low temperatures and get smoothed out with increasing temperature. The features vanish around 15~K. (b) Two-terminal conductance through the constriction at 2~K, showing kinks in the conductance (see blue arrows). (c) Current alongside (black trace) and through a four-terminal constriction (blue trace) as function of gate voltage after photodoping. The conductance trace alongside the constriction are smooth, whereas only the trace through the constriction exhibits step-like features in the current. Inset: schematic of the current flow corresponding to the different conductance traces.}
\label{fig03}
\end{figure}

Next we discuss constriction devices. For observing 
size-quantization effects, including quantized conductance,
the Fermi wavelength of the carriers has to be on the order 
of the device size, e.g. the width of the constriction.
For reasonable carrier densities this requires devices with
feature sizes on the order of 100~nm (or below).
In order to harness photodoping for such small device feature sizes, we employ a shadow masking technique using non-transparent metal structures. 
%
Notably, this approach is in contrast to previous work, where some of us demonstrated photodoping with micron-scale spatial resolution using a confocal laser set-up\cite{Neu16}. The metallic shadow masks, predefined by electron beam lithography, allow to define long-lasting doping profiles in arbitrary geometries (see e.g. Figures~1g-1i). Using this shadow masking technique, we illuminate only the source and drain contact areas (leads) and the constriction, while everything covered by metal is unaffected by the photodoping process. As the MoS$_2$ is non-conductive at 2~K prior to the photodoping no depletion is necessary to form a constriction between the metallic shadow masks. In contrast to conventional constrictions in 2DEGs, we neither have to etch our sample nor have to use electrostatic gates to define the structure as free carriers are only available in the regions, where the sample has been photodoped. 

Figure~3a shows the two-terminal conductance through a 275~nm wide constriction at constant bias of V$_\mathrm{b}$~=~3~mV for temperatures ranging from 2~K to 24~K (after photodoping the device at 50~K and V$^{\mathrm{PD}}_\mathrm{g}=-60$~V). The conductance trace at 2~K shows step-like features suggesting size quantization effects in the constriction (see Figure~3b). A very similar behavior is observed for all four measured constriction devices (see supplementary material).

The step-like features in the conductance traces start to smear out with increasing temperature and remain visible up to 15~K (Figure~2a). The smearing may come from temperature broadening as one would expect that the conductance steps are washed out at a temperature $T\approx \Delta/k_B \pi $ with $\Delta$ the mode spacing. As we will see later $\Delta = 2-4\,$meV for a similar device, which corresponds to $T= 7-15\,$K. 

 In order to verify that the conductance steps originate from the constriction, we perform two-terminal measurements in different contact configurations on a separate multi-terminal device (see Figure~3c) with a constriction width of 250~nm. The black trace in Figure~3c shows the conductance between two contacts on one lead, i.e. on one side of the constriction of a four-terminal device, (see the black arrow in the inset of Figure~3c). The measurements were performed at 2~K after photodoping the device at V$^\mathrm{PD}_\mathrm{g}=-60$~V and 50~K. The trace shows a smooth and monotonic increase of the conductance with the applied gate voltage except for a feature close to the onset of the conductance at around V$_\mathrm{g}=-5$~V and is similar in shape to the traces obtained from several Hall bar devices. In contrast, the conductance through the constriction (blue trace in Figure~3c) shows the same step-like features as they were observed for the two-terminal device.



\begin{figure}[b!]\centering
\includegraphics[draft=false,keepaspectratio=true,clip,%
                   width=\linewidth]%
                   {Fig4.eps}
\caption[fig04]{(a) Four-terminal conductance through the constriction as function of the applied gate voltage. Equidistant steps appear at multiples of $e^2/h$. (b) Derivative with respect to the applied gate voltage of the trace shown in (a), showing dips that correspond to the steps in (a). (c) Bias spectroscopy measurement showing diamonds at the positions of the steps, indicated by the black dashed lines.}
\label{fig04}
\end{figure}


The total resistance from the two-terminal data shown in Figure~3 heavily depends on the particular device and ranges between 10 and 100~k$\Omega$ and is thus significantly larger than the conductance quantum, as the contact resistances heavily contributes to the overall resistance. A subtraction of this parasitic resistances proves problematic as both are strongly gate voltage and bias dependent. In order to reduce the influence of these additional device resistances on the measurement, we perform four-terminal measurements on the multi-terminal device (see inset of Figure~4a). It is important to point out that this four-terminal measurement allows to extract an effective two-terminal conductance as the voltage is probed roughly 4~$\mu m$ away from the constriction. This length is significantly larger than the elastic mean free path in the MoS$_2$. Figure~4a shows the four-terminal conductance as function of the applied gate voltage at 2~K. The conductance ranges from 0 to 10~$e^2/h$ and shows %nearly equidistant 
step-like features with a step height close to $e^2/h$ without any correction for an additional serial resistance. Notably, the feature near the onset of the conduction, which is seen in the conductance alongside the constriction is again visible at around V$_\mathrm{g}$~=~0~V (compare with the black arrow Figure~3c). In Figure~4b, we present the derivative of the conductance trace shown in Figure~4a with respect to gate voltage. The transconductance shows clear dips corresponding to the steps in the conductance. To extract an energy scale for these step-like features, we perform finite bias spectroscopy measurements at 2~K (see Figure~4c). Here, the bias axis corresponds to the measured bias at the voltage probes in the four-terminal configuration. The data shows diamonds at the positions of the steps,  
as one would expect for quantized conductance steps.
The data is rather noisy for $V_\mathrm{g}$ up to 30~V which might originate from potential disorder. Above this value, we observe diamonds whose energy extend $\Delta_n$  is decreasing with increasing gate voltage.

%%%%%%%%%%%%
% DISCUSSION
%%%%%%%%%%%%


  For discussing these data and in particular for estimating an effective constriction width from the extracted energy scale, we follow a simple model assuming an electron in a potential well with the effective mass $m^*$ close to the one of single-layer MoS$_2$\cite{Ros13,Kor14}. The first observation is that the height of the conductance steps is given by $e^2/h$ indicating a broken Kramers' and valley degeneracy even at zero magnetic field. Similar conductance steps of $e^2/h$ have been observed in 1D wires, where interaction effects are more prominent\cite{Bie05,Mos05,Scap06}. In the present situation, we could speculate that the broken time-reversal symmetry arises due to localized spins trapped at impurity sites due to the potential disorder or at molybdenum antisites \cite{Hon15}. 
  
  In the model, the energy extend $\Delta_n$ of the $n$-th conductance step is associated with the mode spacing. As the height decreases with $n$, the transverse confinement is rather smooth (in fact smoother than parabolic as the latter leads to a constant energy spacing).  We fit the data to a model of the form $\Delta_n = (2V_0/s^2) (s- n)$ that is derived for a P\"oschel-Teller potential well in the limit $s \gg 1$ \cite{Lan}. A schematic of the potential can be seen in Figure~5a. Fitting a straight line (see Figure~5b), we obtain the depth $V_0 \approx 50\,$meV of the potential and the number of bound states $s\approx 19$ (that corresponds to 19 confined modes). These values are connected to the width $w$ at half maximum and the mass $m$ of the electron via $2V_0/s^2 = 4\hbar^2/ m^* w^2 \approx 0.27\,$meV. This value can be now used to obtain an estimate for either $w$ or $m^*$ given the knowledge of the other.
  
We first assume a value $m^{*}\approx 0.4~m_{e}$ for the effective mass of a single-layer MoS$_2$ as predicted by tight-binding models (see Refs.~\citenum{Ros13,Kor14}).  If we assume that the Kramers' and valley degeneracy is broken on a scale $\simeq 20\,$meV such that all the modes in Figure~\ref{fig04} correspond to the same valley and spin index, we arrive at a value $w\approx 55\,$nm. In the opposite extreme, we consider the case, where the Kramers' and valley-splitting is in the order of $2-5\,$meV such that the 4 symmetry broken modes succeed. In this case only every 4-th step corresponds to a higher mode in the confining potential thus $V_0/s^2$ is effectively a factor 4 larger and we obtain $w\approx 25\,$nm. Note that the two estimates highlight the extreme cases and that the value of neither the Kramers' nor the valley-splitting can be smaller than a few meV as otherwise, we would observe steps of height $2e^2/h$.

We compare the scale ($25-55\,$nm) with numerical simulations\cite{Note1} of the (lateral) light intensity profile responsible for forming the constriction during photodoping.  Figure~5a shows the simulated intensity distribution in the constriction region defined by a 250~nm spaced shadow mask. The simulation indicates that the light intensity underneath the shadow mask is suppressed, with a lateral extend of the intensity profile (at full-width-at-half-maximum) of 220~nm (see arrows in Figure~5d).  We consider this value as an upper limit as we do not know (i) the intensity at which  photodoping becomes efficient and (ii) to which extend electrostatic effects from the metallic shadow mask may affect the potential landscape in the constriction region. We believe that the factor 4 difference between the estimate above and the value as given by the simulation can be very well attributed to the effects (i) and/or (ii).  

As an alternative explanation of the observed mode spacing, we investigate the possibility that the effective mass in few-layer MoS$_2$ is different from the one quoted above. In particular in few-layer MoS$_2$, the conduction band edge shifts from the K-point to the Q-point\cite{Capp13}, which may result in a slightly different effective mass. To investigate this possibility, we assume that $w=220\,$nm and calculate the value of the effective mass $m^*$ that corresponds to the extracted value of $2V_0^2/s^2$. With this procedure, we obtain the estimate $m^*\approx 0.07 m_e$ which is almost an order of magnitude smaller than the literature value such that we have come to the conclusion that a smaller effective mass cannot be the sole explanation of the observed mode spacing but rather the effective constriction is smaller than the lithographically defined geometry.
All these findings motivates more detailed studies, which go beyond the
scope of this work. Here, we highlight an interesting and promising approach to fabricate doping profiles in MoS$_2$ devices 
on length scales of tenth to hundreds of nanometer allowing to study quantum phenomena in this and related TMDC materials. 



In summary, we present a technique to fabricate mesoscopic devices based on MoS$_2$ encapsulated by hBN using an entirely dry transfer process, contacting the MoS$_2$ using few-layer graphene and adjusting the charge carrier concentration using photodoping. By transferring only a single FLG flake to fabricate a large number of contacts to the MoS$_2$, we reduced the complexity of the required van-der-Waals heterostructure significantly. Similar to remote doping in 2DEGs, we are able to adjust the doping level in MoS$_2$, while preserving a ``high'' charge carrier mobility, resulting in metallic transport at low temperatures. By using metal shadow masks combined with photodoping, we demonstrate a way to produce complex doping profiles on a scale close to the diffraction limit, allowing to observe size quantization effects in MoS$_2$. In low temperature transport measurements, we reproducibly observe signatures of quantized conductance in MoS$_2$ constrictions. Even though gaining precise control over the exact confinement potential proves still to be difficult, our simulations suggest that the shadow masking technique can prove useful to generate smooth and long-lasting doping profiles in van-der-Waals heterostructures. This technique could pave the way towards the fabrication of other mesoscopic devices in TMDCs, such as for example Aharonov-Bohm rings or antidot lattices, which are difficult to realize using electrostatic gating.\\



    \begin{figure}[h!]\centering
    \includegraphics[draft=false,keepaspectratio=true,clip,%
                       width=1\linewidth]%
                       {Fig5.eps}
    \caption[fig04]{(a) A P\"oschel-Teller potential well of the form $V\propto 1/\cosh^2$ with the depth of the potential $V_0$ and the full width at half maximum $w$. (b) Extracted energy extend $\Delta_n$ of the n-th diamond in Figure~4c. A straight line is fitted to the data between $n\,=\,3$ and $10$ yielding $\Delta_n\,=\,(-0.27\pm0.12)\,\text{meV}\cdot n+(5.22\pm0.96)$\,meV. (c) Simulation of the intensity distribution in the constriction region defined by a shadow mask assuming vertical illumination of the sample. (d) Intensity profile of a cross-section through the bottom hBN for 250~nm spaced shadow masks. The full-width-at-half-maximum yields a width of 220~nm. }
    \label{fig05}
    \end{figure}

\textbf{Acknowledgement}\\
The authors thank F.~Haupt for fruitful discussions and 
S.~Kuhlen for help with the figures. Support by the Helmholtz-Nanoelectronic-Facility (HNF), the DFG (SPP-1459), the ERC (GA-Nr. 280140), and the EU project Graphene Flagship (contract no. NECT-ICT-696656) are gratefully acknowledged. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan and JSPS KAKENHI Grant Numbers JP26248061, JP15K21722 and JP25106006.



 \begin{thebibliography}{}
\bibitem{Novo16}
K. S. Novoselov, A. Mishchenko, A. Carvalho, A. H. Castro Neto, 2D materials and van der Waals heterostructures. Science \textbf{2016}, 353, 6298.
  
\bibitem{Geim13}
A. K. Geim and I. V. Grigorieva, Van der Waals heterostructures. Nature \textbf{2013}, 499, 419-425.
 
\bibitem{Rad10}
B. Radisavljevic, A.Radenovic, J.Brivio, V. Giacometti and A. Kis, Single-layer MoS$_2$ transistors. Nat. Nanotechnol. \textbf{2011}, 6 , 147-150.

\bibitem{Herr13}
B. W. H. Baugher, H. O. H. Churchill, Y. Yang and P. Jarillo-Herrero, Intrinsic Electronic Transport Properties of High-Quality Monolayer and Bilayer MoS2. Nano Lett. \textbf{2013}, 13, 4212-4216.


\bibitem{Rus08}
S. Russo, J. B. Oostinga, D. Wehenkel, H. B. Heersche, S.S. Sobhani, L. M. Vandersypen, A. F. Morpurgo, Observation of Aharonov-Bohm conductance oscillations in a graphene ring. Phys. Rev. B \textbf{2008}, 77, 085413.


\bibitem{Ter16}
B. Terr\'es, L. A. Chizhova, F. Libisch, J. Peiro, D. J\"orger, S. Engels, A. Girschik, K. Watanabe, T. Taniguchi, S. V. Rotkin, J. Burgd\"orfer and C. Stampfer, Size quantization of Dirac fermions in graphene constrictions. Nat. Commun. \textbf{2016}, 7, 11528.


\bibitem{Elz04}
J. M. Elzerman, R. Hanson, L.W. Van Beveren, B. Witkamp, L.M.K. Vandersypen and L.P. Kouwenhoven, Single-shot read-out of an individual electron spin in a quantum dot. Nature \textbf{2004}, 430, 431-435.

\bibitem{Wee88}
B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel and C. T. Foxon, Quantized conductance of point contacts in a two dimensional electron gas. Phys. Rev. Lett. \textbf{1988}, 60, 848.

\bibitem{Wep13}
I. van Weperen, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and  L. P. Kouwenhoven, Quantized conductance in an InSb nanowire. Nano Lett. \textbf{2013}, 13, 387-391.

\bibitem{Joh92}
A. T. Johnson, L. P. Kouwenhoven, W. De Jong, N. C. Van der Vaart, C. J. P. M. Harmans, and C. T. Foxon, Zero-dimensional states and single electron charging in quantum dots. Phys. Rev. Lett. \textbf{1992}, 69, 1592-1596.

\bibitem{Ryc07}
A. Rycerz, J. Tworzydlo and C. W. J. Beenakker, Valley filter and valley valve in graphene. Nat. Phys. \textbf{2007}, 3, 172-175.

\bibitem{Tom11}
N. Tombros, A. Veligura, J. Junesch, M. H. D. Guimaraes, I. J. Vera Marun, H.  T.  Jonkman and B. J. van Wees, Quantized conductance of a suspended graphene
nanoconstriction. Nat. Phys. \textbf{2011}, 7, 697-700.


\bibitem{Bis14}
D. Bischoff, F. Libisch, J. Burgd\"orfer, T. Ihn, and K. Ensslin, Characterizing wave functions in graphene nanodevices: Electronic transport through ultrashort graphene constrictions on a boron nitride substrate. Phys. Rev. B \textbf{2014}, 90, 115405.

\bibitem{Gos12}
A. M. Goossens, S. C. M. Driessen, T. A. Baart, K. Watanabe, T. Taniguchi, and L. M. K. Vandersypen, Gate-Defined Confinement in Bilayer Graphene-Hexagonal Boron Nitride Hybrid Devices. Nano Lett. \textbf{2012}, 12, 4656-4660.

\bibitem{All12}
M. T. Allen, J. Martin, and A. Yacoby, Gate-defined quantum confinement in suspended bilayer graphene. Nat. Commun. \textbf{2012}, 3, 934.

\bibitem{Son15}
X.-X. Song, D. Liu, V. Mosallanejad, J. You, T.-Y. Han, D.-T. Chen, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo and G.-P. Guo, A gate defined quantum dot on the two-dimensional transition metal dichalcogenide semiconductor WSe$_2$. Nanoscale \textbf{2015}, 7, 16867-16873.

\bibitem{Ke16}
K. Wang, T. Taniguchi, K. Watanabe and P. Kim, Engineering Quantum Confinement in Semiconducting van der Waals Heterostructure. arXiv:1610.02929, (2016).

\bibitem{Gold1998}
D. Goldhaber-Gordon, Hadas Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav and M. A. Kastner, Kondo effect in a single-electron transistor. Nature \textbf{1998}, 391, 156.

\bibitem{Petta05}
J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby,  M. D. Lukin, C. M. Marcus, M. P. Hanson and A. C. Gossard Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science \textbf{2005}, 309, 2180.

\bibitem{Ju14}
L. Ju, J. Velasco Jr., E. Huang, S. Kahn, C. Nosiglia, H.-Z. Tsai, W. Yang, T. Taniguchi, K. Watanabe, Y. Zhang, G. Zhang, M. Crommie, A. Zettl and F. Wang, Photoinduced Doping in Heterostructures of Graphene and Boron Nitride. Nat. Nanotechnol. \textbf{2014}, 9, 348-352.

\bibitem{Neu16}
C. Neumann, L. Rizzi, S. Reichardt, B. Terres, T. Khodkov, K. Watanabe, T. Taniguchi, B. Beschoten, and C. Stampfer, Spatial control of laser-induced doping profiles in graphene on hexagonal boron nitride. ACS Appl. Mater. Interfaces \textbf{2016}, 8, 9377-9383.

\bibitem{Wan13}
 L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, One-Dimensional Electrical Contact to a Two-Dimensional Material. Science \textbf{2013}, 342, 614-617.
 
 \bibitem{Bansz15}
 L. Banszerus, M. Schmitz, S. Engels, J. Dauber, M. Oellers, F. Haupt, K. Watanabe, T. Taniguchi, B. Beschoten and C. Stampfer, Ultrahigh-mobility graphene devices from chemical vapor deposition on reusable copper. ‎Sci. Adv \textbf{2015}, 1, e1500222.	

\bibitem{Yu14}
L. Yu, Y.-H. Lee, X. Ling, E. J. G. Santos, Y. C. Shin, Y. Lin, M. Dubey, E. Kaxiras, J. Kong, H. Wang, and T. Palacios, Graphene/MoS$_2$ hybrid technology for large-scale two-dimensional electronics. Nano Lett. \textbf{2014}, 14, 3055-3063.

\bibitem{Roy14}
T. Roy, M. Tosun, J. S. Kang, A. B. Sachid, S. B. Desai, M. Hettick, C. C. Hu, and A. Javey, Field-effect transistors built from all two-dimensional material components. ACS Nano \textbf{2014}, 8, 6256-6264.

\bibitem{Cui15}
X. Cui, G.-H. Lee, Y. D. Kim, G. Arefe, P. Y. Huang, C.-H. Lee, D. A. Chenet, X. Zhang, L. Wang, F. Ye, F. Pizzocchero, B. S. Jessen, K. Watanabe, T. Taniguchi, D. A. Muller, T. Low, P. Kim	and J. Hone, Multi-terminal transport measurements of Mo  using a van der Waals heterostructure device platform. Nat. Nanotechnol. \textbf{2015}, 10, 534-540.

\bibitem{Kap14}
 R. Kappera, D. Voiry, S. E. Yalcin,B. Branch, G. Gupta, A. D. Mohite and M. Chhowalla, Phase-engineered low-resistance contacts for ultrathin MoS$_2$ transistors.
    Nat. Mater. \textbf{2014}, 13, 1128-1134.


\bibitem{Dressel09}
L.M. Malard, M.A. Pimenta, G. Dresselhaus, M.S. Dresselhaus, Raman spectroscopy in graphene. Phys. Rep. \textbf{2009}, 473, 51-87.

\bibitem{Gor11}
R. Gorbachev, I. Riaz, R. R. Nair, R. J. L. Britnell, B. D. Belle, E. W. Hill, K. S. Novoselov, A. K. Geim, P. Blake, Hunting for Monolayer Boron Nitride: Optical and Raman Signatures, Small \textbf{2011}, 7, 465-468.

\bibitem{Ton13}
P. Tonndorf, R. Schmidt, P. B\"ottger, X. Zhang, J. B\"orner, A. Liebig, M. Albrecht, C. Kloc, O. Gordan, D. R. T. Zahn, S. Michaelis de Vasconcellos, and R. Bratschitsch, Photoluminescence emission and Raman response of monolayer MoS$_2$, MoSe$_2$, and WSe$_2$. Opt. Express \textbf{2013}, 21, 4908-4916.

\bibitem{Sah13}
H. Sahin, S. Tongay, S. Horzum, W. Fan, J. Zhou, J. Li, J. Wu, and F. M. Peeters, Anomalous Raman spectra and thickness-dependent electronic properties of WSe$_2$. Phys. Rev. B \textbf{2013}, 87, 165409.

\bibitem{Rad13}
B. Radisavljevic and A. Kis, Mobility engineering and a metal-insulator transition in monolayer MoS$_2$. Nat. Mater. \textbf{2013}, 12, 815-820.


\bibitem{Bau13}
B. W. H. Baugher, H. O. H. Churchill, Y. Yang, and P. Jarillo-Herrero, Intrinsic Electronic Transport Properties of High-Quality Monolayer and Bilayer MoS$_2$. Nano Lett. \textbf{2013}, 13, 4212-4216.

\bibitem{Sch14}
H. Schmitt, S. Wang, L. Chu, M. Toh, R. Kumar, W. Zhao, A. H. Castro Neto, J.Martin, S. Adam, B. \"Ozyilmaz, and G. Eda, Transport Properties of Monolayer MoS$_2$ Grown by Chemical Vapor Deposition. Nano Lett. \textbf{2014}, 14, 1909-1913.

\bibitem{Won15}
 D. Wong, J. Velasco Jr,	L. Ju, J. Lee, S. Kahn, H.-Z. Tsai, C. Germany, T. Taniguchi, K. Watanabe, A. Zettl, F. Wang and M. F. Crommie,	Characterization and manipulation of individual defects in insulating hexagonal boron nitride using scanning tunnelling microscopy. Nat. Nanotechnol. \textbf{2015}, 10, 949-953.


\bibitem{Ding1978}
R. Dingle, H. L. St\"ormer, A. C. Gossard, and W. Wiegmann, Electron mobilities in modulation-doped semiconductor heterojunction superlattices. Appl. Phys. Lett. \textbf{1978}, 33, 665.


\bibitem{Ros13} H. Rostami, A. G. Moghaddam, and R. Asgari, Effective lattice Hamiltonian for monolayer MoS$_2$: Tailoring electronic structure with perpendicular electric and magnetic fields. Phys. Rev. B \textbf{88}, 085440  (2013).

\bibitem{Kor14}
A. Kormanyos, V. Zolyomi, N. D. Drummond, and G. Burkard, Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides. Phys. Rev. X \textbf{4}, 011034, (2014).


\bibitem{Bie05}
M. J. Biercuk, N. Mason, J. Martin, A. Yacoby, and C. M. Marcus, Anomalous Conductance Quantization in Carbon Nanotubes. Phys. Rev. Lett. \textbf{2005}, 94, 026801.
 

\bibitem{Mos05}
J. Moser, T. Zibold, D. Schuh, M. Bichler, F. Ertl, G. Abstreiter, M. Grayson, S. Roddaro, and V. Pellegrini, Aluminum arsenide cleaved-edge overgrown quantum wires. Appl. Phys. Lett. \textbf{2005}, 87, 052101.

\bibitem{Scap06}
G. Scappucci, L. Di Gaspare, E. Giovine, A. Notargiacomo, R. Leoni, and F. Evangelisti, Conductance quantization in etched Si/SiGe quantum point contacts. Phys. Rev. B \textbf{2006}, 74, 035321.

\bibitem{Hon15}
  J. Hong, Z. Hu, M. Probert, K. Li, D. Lv, X. Yang, L. Gu, N. Mao, Q. Feng, L. Xie, J. Zhang, D. Wu, Z. Zhang, C. Jin, W. Ji, X. Zhang, J. Yuan, and Z. Zhang, Exploring atomic defects in molybdenum disulphide monolayers. Nature Comm. \textbf{6}, 6293 (2015).

\bibitem{Lan} L. D. Landau and E. M. Lifshitz,
  {\em Quantum Mechanics\/}, prob. 5 in \S 23 of vol. 3 of {\em Course of Theoretical Physics\/}
 (Pergamon Press, London, 1958).


\bibitem{Note1}
The simulations were performed using COMSOL Multiphysics and assumes vertical illumination of the sample.

\bibitem{Capp13}
E. Cappelluti, R. Roldan, J. A. Silva-Guillen, P. Ordejon and F. Guinea. Tight-binding model and direct-gap/indirect-gap transition in single-layer and multilayer MoS$_2$. Phys. Rev. B \textbf{2013}, 88, 075409.

 \end{thebibliography}

\end{document}


1612.01118.extracted_bib
{Novo16 K. S. Novoselov, A. Mishchenko, A. Carvalho, A. H. Castro Neto, 2D materials and van der Waals heterostructures. Science 2016, 353, 6298.
1612.01118.extracted_bib
{Geim13 A. K. Geim and I. V. Grigorieva, Van der Waals heterostructures. Nature 2013, 499, 419-425.
1612.01118.extracted_bib
{Rad10 B. Radisavljevic, A.Radenovic, J.Brivio, V. Giacometti and A. Kis, Single-layer MoS$_2$ transistors. Nat. Nanotechnol. 2011, 6 , 147-150.
1612.01118.extracted_bib
{Herr13 B. W. H. Baugher, H. O. H. Churchill, Y. Yang and P. Jarillo-Herrero, Intrinsic Electronic Transport Properties of High-Quality Monolayer and Bilayer MoS2. Nano Lett. 2013, 13, 4212-4216.
1612.01118.extracted_bib
{Rus08 S. Russo, J. B. Oostinga, D. Wehenkel, H. B. Heersche, S.S. Sobhani, L. M. Vandersypen, A. F. Morpurgo, Observation of Aharonov-Bohm conductance oscillations in a graphene ring. Phys. Rev. B 2008, 77, 085413.
1612.01118.extracted_bib
{Ter16 B. Terr\'es, L. A. Chizhova, F. Libisch, J. Peiro, D. J\"orger, S. Engels, A. Girschik, K. Watanabe, T. Taniguchi, S. V. Rotkin, J. Burgd\"orfer and C. Stampfer, Size quantization of Dirac fermions in graphene constrictions. Nat. Commun. 2016, 7, 11528.
1612.01118.extracted_bib
{Elz04 J. M. Elzerman, R. Hanson, L.W. Van Beveren, B. Witkamp, L.M.K. Vandersypen and L.P. Kouwenhoven, Single-shot read-out of an individual electron spin in a quantum dot. Nature 2004, 430, 431-435.
1612.01118.extracted_bib
{Wee88 B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel and C. T. Foxon, Quantized conductance of point contacts in a two dimensional electron gas. Phys. Rev. Lett. 1988, 60, 848.
1612.01118.extracted_bib
{Wep13 I. van Weperen, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Quantized conductance in an InSb nanowire. Nano Lett. 2013, 13, 387-391.
1612.01118.extracted_bib
{Joh92 A. T. Johnson, L. P. Kouwenhoven, W. De Jong, N. C. Van der Vaart, C. J. P. M. Harmans, and C. T. Foxon, Zero-dimensional states and single electron charging in quantum dots. Phys. Rev. Lett. 1992, 69, 1592-1596.
1612.01118.extracted_bib
{Ryc07 A. Rycerz, J. Tworzydlo and C. W. J. Beenakker, Valley filter and valley valve in graphene. Nat. Phys. 2007, 3, 172-175.
1612.01118.extracted_bib
{Tom11 N. Tombros, A. Veligura, J. Junesch, M. H. D. Guimaraes, I. J. Vera Marun, H. T. Jonkman and B. J. van Wees, Quantized conductance of a suspended graphene nanoconstriction. Nat. Phys. 2011, 7, 697-700.
1612.01118.extracted_bib
{Bis14 D. Bischoff, F. Libisch, J. Burgd\"orfer, T. Ihn, and K. Ensslin, Characterizing wave functions in graphene nanodevices: Electronic transport through ultrashort graphene constrictions on a boron nitride substrate. Phys. Rev. B 2014, 90, 115405.
1612.01118.extracted_bib
{Gos12 A. M. Goossens, S. C. M. Driessen, T. A. Baart, K. Watanabe, T. Taniguchi, and L. M. K. Vandersypen, Gate-Defined Confinement in Bilayer Graphene-Hexagonal Boron Nitride Hybrid Devices. Nano Lett. 2012, 12, 4656-4660.
1612.01118.extracted_bib
{All12 M. T. Allen, J. Martin, and A. Yacoby, Gate-defined quantum confinement in suspended bilayer graphene. Nat. Commun. 2012, 3, 934.
1612.01118.extracted_bib
{Son15 X.-X. Song, D. Liu, V. Mosallanejad, J. You, T.-Y. Han, D.-T. Chen, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo and G.-P. Guo, A gate defined quantum dot on the two-dimensional transition metal dichalcogenide semiconductor WSe$_2$. Nanoscale 2015, 7, 16867-16873.
1612.01118.extracted_bib
{Ke16 K. Wang, T. Taniguchi, K. Watanabe and P. Kim, Engineering Quantum Confinement in Semiconducting van der Waals Heterostructure. arXiv:1610.02929, (2016).
1612.01118.extracted_bib
{Gold1998 D. Goldhaber-Gordon, Hadas Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav and M. A. Kastner, Kondo effect in a single-electron transistor. Nature 1998, 391, 156.
1612.01118.extracted_bib
{Petta05 J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson and A. C. Gossard Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 2005, 309, 2180.
1612.01118.extracted_bib
{Ju14 L. Ju, J. Velasco Jr., E. Huang, S. Kahn, C. Nosiglia, H.-Z. Tsai, W. Yang, T. Taniguchi, K. Watanabe, Y. Zhang, G. Zhang, M. Crommie, A. Zettl and F. Wang, Photoinduced Doping in Heterostructures of Graphene and Boron Nitride. Nat. Nanotechnol. 2014, 9, 348-352.
1612.01118.extracted_bib
{Neu16 C. Neumann, L. Rizzi, S. Reichardt, B. Terres, T. Khodkov, K. Watanabe, T. Taniguchi, B. Beschoten, and C. Stampfer, Spatial control of laser-induced doping profiles in graphene on hexagonal boron nitride. ACS Appl. Mater. Interfaces 2016, 8, 9377-9383.
1612.01118.extracted_bib
{Wan13 L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, One-Dimensional Electrical Contact to a Two-Dimensional Material. Science 2013, 342, 614-617.
1612.01118.extracted_bib
{Bansz15 L. Banszerus, M. Schmitz, S. Engels, J. Dauber, M. Oellers, F. Haupt, K. Watanabe, T. Taniguchi, B. Beschoten and C. Stampfer, Ultrahigh-mobility graphene devices from chemical vapor deposition on reusable copper. ‎Sci. Adv 2015, 1, e1500222.
1612.01118.extracted_bib
{Yu14 L. Yu, Y.-H. Lee, X. Ling, E. J. G. Santos, Y. C. Shin, Y. Lin, M. Dubey, E. Kaxiras, J. Kong, H. Wang, and T. Palacios, Graphene/MoS$_2$ hybrid technology for large-scale two-dimensional electronics. Nano Lett. 2014, 14, 3055-3063.
1612.01118.extracted_bib
{Roy14 T. Roy, M. Tosun, J. S. Kang, A. B. Sachid, S. B. Desai, M. Hettick, C. C. Hu, and A. Javey, Field-effect transistors built from all two-dimensional material components. ACS Nano 2014, 8, 6256-6264.
1612.01118.extracted_bib
{Cui15 X. Cui, G.-H. Lee, Y. D. Kim, G. Arefe, P. Y. Huang, C.-H. Lee, D. A. Chenet, X. Zhang, L. Wang, F. Ye, F. Pizzocchero, B. S. Jessen, K. Watanabe, T. Taniguchi, D. A. Muller, T. Low, P. Kim and J. Hone, Multi-terminal transport measurements of Mo using a van der Waals heterostructure device platform. Nat. Nanotechnol. 2015, 10, 534-540.
1612.01118.extracted_bib
{Kap14 R. Kappera, D. Voiry, S. E. Yalcin,B. Branch, G. Gupta, A. D. Mohite and M. Chhowalla, Phase-engineered low-resistance contacts for ultrathin MoS$_2$ transistors. Nat. Mater. 2014, 13, 1128-1134.
1612.01118.extracted_bib
{Dressel09 L.M. Malard, M.A. Pimenta, G. Dresselhaus, M.S. Dresselhaus, Raman spectroscopy in graphene. Phys. Rep. 2009, 473, 51-87.
1612.01118.extracted_bib
{Gor11 R. Gorbachev, I. Riaz, R. R. Nair, R. J. L. Britnell, B. D. Belle, E. W. Hill, K. S. Novoselov, A. K. Geim, P. Blake, Hunting for Monolayer Boron Nitride: Optical and Raman Signatures, Small 2011, 7, 465-468.
1612.01118.extracted_bib
{Ton13 P. Tonndorf, R. Schmidt, P. B\"ottger, X. Zhang, J. B\"orner, A. Liebig, M. Albrecht, C. Kloc, O. Gordan, D. R. T. Zahn, S. Michaelis de Vasconcellos, and R. Bratschitsch, Photoluminescence emission and Raman response of monolayer MoS$_2$, MoSe$_2$, and WSe$_2$. Opt. Express 2013, 21, 4908-4916.
1612.01118.extracted_bib
{Sah13 H. Sahin, S. Tongay, S. Horzum, W. Fan, J. Zhou, J. Li, J. Wu, and F. M. Peeters, Anomalous Raman spectra and thickness-dependent electronic properties of WSe$_2$. Phys. Rev. B 2013, 87, 165409.
1612.01118.extracted_bib
{Rad13 B. Radisavljevic and A. Kis, Mobility engineering and a metal-insulator transition in monolayer MoS$_2$. Nat. Mater. 2013, 12, 815-820.
1612.01118.extracted_bib
{Bau13 B. W. H. Baugher, H. O. H. Churchill, Y. Yang, and P. Jarillo-Herrero, Intrinsic Electronic Transport Properties of High-Quality Monolayer and Bilayer MoS$_2$. Nano Lett. 2013, 13, 4212-4216.
1612.01118.extracted_bib
{Sch14 H. Schmitt, S. Wang, L. Chu, M. Toh, R. Kumar, W. Zhao, A. H. Castro Neto, J.Martin, S. Adam, B. \"Ozyilmaz, and G. Eda, Transport Properties of Monolayer MoS$_2$ Grown by Chemical Vapor Deposition. Nano Lett. 2014, 14, 1909-1913.
1612.01118.extracted_bib
{Won15 D. Wong, J. Velasco Jr, L. Ju, J. Lee, S. Kahn, H.-Z. Tsai, C. Germany, T. Taniguchi, K. Watanabe, A. Zettl, F. Wang and M. F. Crommie, Characterization and manipulation of individual defects in insulating hexagonal boron nitride using scanning tunnelling microscopy. Nat. Nanotechnol. 2015, 10, 949-953.
1612.01118.extracted_bib
{Ding1978 R. Dingle, H. L. St\"ormer, A. C. Gossard, and W. Wiegmann, Electron mobilities in modulation-doped semiconductor heterojunction superlattices. Appl. Phys. Lett. 1978, 33, 665.
1612.01118.extracted_bib
{Ros13 H. Rostami, A. G. Moghaddam, and R. Asgari, Effective lattice Hamiltonian for monolayer MoS$_2$: Tailoring electronic structure with perpendicular electric and magnetic fields. Phys. Rev. B 88, 085440 (2013).
1612.01118.extracted_bib
{Kor14 A. Kormanyos, V. Zolyomi, N. D. Drummond, and G. Burkard, Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides. Phys. Rev. X 4, 011034, (2014).
1612.01118.extracted_bib
{Bie05 M. J. Biercuk, N. Mason, J. Martin, A. Yacoby, and C. M. Marcus, Anomalous Conductance Quantization in Carbon Nanotubes. Phys. Rev. Lett. 2005, 94, 026801.
1612.01118.extracted_bib
{Mos05 J. Moser, T. Zibold, D. Schuh, M. Bichler, F. Ertl, G. Abstreiter, M. Grayson, S. Roddaro, and V. Pellegrini, Aluminum arsenide cleaved-edge overgrown quantum wires. Appl. Phys. Lett. 2005, 87, 052101.
1612.01118.extracted_bib
{Scap06 G. Scappucci, L. Di Gaspare, E. Giovine, A. Notargiacomo, R. Leoni, and F. Evangelisti, Conductance quantization in etched Si/SiGe quantum point contacts. Phys. Rev. B 2006, 74, 035321.
1612.01118.extracted_bib
{Hon15 J. Hong, Z. Hu, M. Probert, K. Li, D. Lv, X. Yang, L. Gu, N. Mao, Q. Feng, L. Xie, J. Zhang, D. Wu, Z. Zhang, C. Jin, W. Ji, X. Zhang, J. Yuan, and Z. Zhang, Exploring atomic defects in molybdenum disulphide monolayers. Nature Comm. 6, 6293 (2015).
1612.01118.extracted_bib
{Lan L. D. Landau and E. M. Lifshitz, {\em Quantum Mechanics\/, prob. 5 in \S 23 of vol. 3 of {\em Course of Theoretical Physics\/ (Pergamon Press, London, 1958).
1612.01118.extracted_bib
{Note1 The simulations were performed using COMSOL Multiphysics and assumes vertical illumination of the sample.
1612.01118.extracted_bib
{Capp13 E. Cappelluti, R. Roldan, J. A. Silva-Guillen, P. Ordejon and F. Guinea. Tight-binding model and direct-gap/indirect-gap transition in single-layer and multilayer MoS$_2$. Phys. Rev. B 2013, 88, 075409.