Microscopic Quantum Transport Processes of Out‐of‐Plane Charge Flow in 2D Semiconductors Analyzed by a Fowler–Nordheim Tunneling Probe

Weak interlayer couplings at 2D van der Waals (vdW) interfaces fundamentally distinguish out‐of‐plane charge flow, the information carrier in vdW‐assembled vertical electronic and optical devices, from the in‐plane band transport processes. Here, the out‐of‐plane charge transport behavior in 2D vdW semiconducting transition metal dichalcogenides (SCTMD) is reported. The measurements demonstrate that, in the high electric field regime, especially at low temperatures, either electron or hole carrier Fowler–Nordheim (FN) tunneling becomes the dominant quantum transport process in ultrathin SCTMDs, down to monolayers. For few‐layer SCTMDs, sequential layer‐by‐layer FN tunneling is observed to dominate the charge flow, thus serving as a material characterization probe for addressing the Fermi level positions and the layer numbers of the SCTMD films. Furthermore, it is shown that the physical confinement of the electron or hole carrier wave packets inside the sub‐nm thick semiconducting layers reduces the vertical quantum tunneling probability, leading to an enhanced effective mass of tunneling carriers.


Introduction
Van der Waals (vdW) vertical heterostructures with assorted 2D materials have drawn much attention for upcoming lowdimensional electronic and optical device applications thanks to unlimited combinatorial choices for assembling heterostacks and engineering diverse functionalities. [1][2][3][4][5] Atomically clean vdW interfaces with negligible chemical and electrostatic interlayer bonding make expectations of novel highperformance devices realistically achievable. [1,6] In particular, 2D materials with a sizable energy gap of 1 eV or more; for example, MoS 2 , WSe 2 , and hBN, have been implemented as high-quality tunnel barriers or as efficient charge-blocking layers in various device applications such as resonant tunnel diodes, [7,8] transistor, [9][10][11] logic devices, [4,12] memories, [10,13,14] light-emitting diodes, [15,16] and photodetectors. [5,17,18] Weak interlayer couplings at 2D van der Waals (vdW) interfaces fundamentally distinguish out-of-plane charge flow, the information carrier in vdWassembled vertical electronic and optical devices, from the in-plane band transport processes. Here, the out-of-plane charge transport behavior in 2D vdW semiconducting transition metal dichalcogenides (SCTMD) is reported. The measurements demonstrate that, in the high electric field regime, especially at low temperatures, either electron or hole carrier Fowler-Nordheim (FN) tunneling becomes the dominant quantum transport process in ultrathin SCTMDs, down to monolayers. For few-layer SCTMDs, sequential layer-by-layer FN tunneling is observed to dominate the charge flow, thus serving as a material characterization probe for addressing the Fermi level positions and the layer numbers of the SCTMD films. Furthermore, it is shown that the physical confinement of the electron or hole carrier wave packets inside the sub-nm thick semiconducting layers reduces the vertical quantum tunneling probability, leading to an enhanced effective mass of tunneling carriers.
vdW-assembled heterostructures have also been utilized as effective material metrology platforms to characterize lowdimensional material properties such as the fundamental electronic band structures and electron interactions with various quasiparticles like phonons and magnons. [19][20][21][22][23][24] The majority of device applications explored to date rely on low-energy direct quantum tunneling; namely, out-of-plane vertical charge flow around the Fermi levels of the vertical heterostacks. [19][20][21][22][23][24][25][26][27] It is well established that, as tunnel-electron energies increase at the tunnel junctions, high-energy quantum tunneling events such as Fowler-Nordheim (FN) tunneling or the thermally assisted Schottky emission mechanism play more dominant roles in the quantum charge flow. [24,28,29] FN tunneling, where charges tunnel into the conduction or valence band of tunnel barriers, is distinguished from low-energy direct tunneling that directly drives charge carriers to tunnel into the empty energy states of contact electrodes. FN tunneling has been widely implemented in various conventional electronic and optical device applications, e.g., flash memories, high-resolution electron microscopy, and field-emission displays, to name a few. [30][31][32][33] Moreover, since the energy distribution of the FN tunnel charge is much narrower than those of Schottky and thermionic emissions, FN tunneling has considerable potential as a material metrology probe. In particular, the interfaces of vdW-assembled 2D-2D planar junctions are ideal device platforms for validating the conceptual FN tunneling model that considers an atomically flat and clean metalinsulator interface. [34] Additionally, interesting questions arise about whether high-energy FN tunneling, which requires a transformation of the shape of the tunnel barrier, could be realized with single-atomic-layer insulators. Even with these intriguing questions and opportunities, however, experimental realizations of "high-energy" FN tunneling and device applications have been minimal for 2D vdW heterostructures to date. [32,33,35,36] In this report, we demonstrate that FN tunnel driven out-of-plane vertical charge flows are realized in 2D-2D vdW vertical tunnel junctions, which are assembled with ultrathin tunnel barriers down to monolayers of semiconducting transition metal dichalcogenides (SCTMDs). At low temperature, the high electric field-driven FN tunneling displays a layer-by-layer quantum tunneling process that identifies the Fermi level location of the 2D SCTMDs with respect to the conduction or the valence band of each layer. Confinement of electron or hole wave packets in the 2D-2D planar tunnel junctions reduces the vertical charge flow significantly, which relates to a greatly enhanced out-of-plane effective charge carrier mass in the sub-nm thick tunnel insulators. Figure 1a,b respectively show an optical microscope image and a simplified device schematic of the 2D-2D vdW heterostructures (details of Experimental Section in the Supporting Information). Thin graphite flakes (≥ 5 nm) with clean vdW contacts to the SCTMDs are essential to preserve the intrinsic electrical properties of the tunnel barriers, even down to monolayer thickness. [21][22][23] Thanks to negligible charge transfer between the graphite electrodes and the SCTMDs, [20,21] the tunnel barrier at the graphite-SCTMD interface, quantified by the Schottky barrier height (Φ SB ), is determined by the fundamental SCTMD material properties, namely the relative www.advelectronicmat.de positions of the conduction (E C ) or valence (E V ) band edge with respect to the Fermi level (E F ) of the electrode: Φ SB = E C -E F for n-type and Φ SB = E F -E V for p-type SCTMD films. Figure 1b also illustrates a series of tunneling processes as a sample bias voltage (V b ) sequentially drives the charge flow in the regimes of direct tunneling (dashed black lines), FN tunneling (solid blue line), Schottky emission (SE, solid orange lines), and thermionic emission (TE, solid red lines). [24,28,29] Due to the distinctive path of each tunneling process, V b -dependent tunnel events reveal specific properties of the tunnel junctions. For instance, high-energy tunneling such as FN, SE, and TE is sensitive to the electronic structure of the tunnel barrier material, whereas low-energy direct tunneling is subject to the barrier's mid-gap states between E C and E V . Since FN tunneling occurs only near the E F of the source electrode at low temperature, moreover, the electron energy distributions corresponding to FN tunneling become very narrow. Consequently, as illustrated in Figure 1b, FN tunnel driven charges are confined to an energy window around the E C (or E V ) edge, underlining the potential for a practical material metrology probe utilizing high-energy FN tunneling.

Fowler-Nordheim Tunneling in Monolayer SCTMDs
A collection of I-V b characteristic curves at T = 6.0 K is displayed in Figure 1c for vertical tunnel junctions with monolayer MoS 2 , MoSe 2 , WS 2 , and WSe 2 tunnel barriers. The I-V b characteristics exhibit a clear nonlinearity as a function of V b (Figure 1c inset), indicating that the transport of charge carriers through the monolayer SCTMD is governed by quantum mechanical tunneling processes. At low temperatures where thermally activated Schottky emission becomes effectively suppressed, a series of quantum transport processes of direct tunneling, FN tunneling, and channel-limited ohmic conduction sequentially dominate the quantum transport through the junctions. As illustrated in Figure 1b, FN tunneling initiates when the lowered (raised) Fermi level of the biased electrode (the right one in Figure 1b) aligns with the valence (conduction) band edge of the tunnel barrier at −eV b (eV b ), accompanied by a transformation of the tunnel barrier shape from trapezoidal to triangular. The FN tunnel driven current can be expressed as where A, h, d, m e , and m* respectively indicate the tunnel junction area, Planck's constant, tunnel barrier width, free electron mass, and effective mass of the tunneling charge carrier. [28,37] The linear progressions in the FN tunnel relation of ln(I/V b 2 ) versus 1/V b at high V b , as marked with dotted lines in Figure 1d indicate that the vertical charge flows are primarily driven by FN tunneling, even in the devices with a d ≈ 0.6 nm thick monolayer SCTMD. However, when comparing different monolayers, MoSe 2 and WSe 2 clearly show a linear behavior when 1/V b is less than ca. 1 V −1 . Note that the voltages at which FN tunneling starts to develop are lower for the S-based SCTMDs than for their Se-based counterparts, indicating that Φ SB is lower in the devices with WS 2 and MoS 2 than in those with MoSe 2 and WSe 2 . Our previous report showed that sulfur atom vacancies and the resultant mid-gap states, likely leading to the Φ SB reductions in the MoS 2 and WS 2 devices, are conspicuous in the S-based monolayer SCTMD films. [20] The lowered tunnel barrier heights and the substantial leakage current through the mid-gap states allow a large amount of low-energy charge carriers to flow, even before high-energy FN tunneling becomes effective. In comparison, WSe 2 and MoSe 2 films are less prone to form selenium atom vacancies, so that the high-energy FN tunnel features can clearly appear in the tunnel junctions with monolayer WSe 2 and MoSe 2 where there is a larger Φ SB and a reduced leakage current.

Fowler-Nordheim Tunneling as a Material Characterization Probe
High-energy FN tunnel characteristic measurements can address a couple of critical 2D SCTMD material properties. To accurately assess V FN , we replot the data as the numerical deriv- to an effective FN tunnel junction resistance at a specific V b . Figure 2 displays a series of FN plot derivative curves for graphite-WSe 2graphite junctions with mono-, bi-, and tri-layer WSe 2 . The FN plot derivative curves, shown with solid blue lines in Figure 2, are obtained by numerically averaging as many as 61 individual plots (gray dots in Figure 2), some of which represent data with different gate voltages applied to the Si/SiO 2 /hBN back gate. Note that the V b at which the FN tunnel junction resistance reaches its first local maximum (first minimum on the derivative plot), indicated with vertical orange arrows in Figure 2, consistently decreases as the WSe 2 layer number increases.
We associate V FN with the sample bias where the FN plot derivative reaches its first minimum and accordingly assign Φ SB of the junction as eV FN . As the V b increases beyond V FN , the tunnel junction resistance quickly switches from a higher to a lower resistance state, reflecting a direct transformation of the tunnel barrier from a trapezoid with a fixed d to a triangle with a shortening d. Since the FN tunnel junction resistance lowers its value as V b progressively increases, we can relate the first local FN tunnel plot derivative minimum to the point where FN tunneling starts to dominate. Similarly, the last minimum in the derivative curve is interpreted as the transition point from FN tunneling to channel-limited ohmic conduction. Thus, the region between the first and last minima represents the domain in which FN tunneling governs the charge transport. In Figure 2a, the monolayer WSe 2 junction displays two minima in the derivative curve, confirming the occurrence of FN tunneling in this area. The vertical orange arrows in Figure 2 indicate V FN for each device, and the corresponding Schottky barrier heights are 1.23 eV (Figure 2a), 1.03 eV (Figure 2b), and 0.93 eV (Figure 2c) for mono-, bi-, and tri-layer www.advelectronicmat.de WSe 2 junctions, respectively. We attribute the observed layernumber-dependent Φ SB reduction to the energy gap shrinkage in multilayer WSe 2 . [38] With the estimated Φ SB , the lower boundaries of the WSe 2 quasiparticle energy gaps (E G ) can be determined to be E G ≥ 2Φ SB . For example, the estimated energy gap of the monolayer WSe 2 , E G (1L-WSe 2 ) ≥ 2.4 eV, agrees well with previous reports, suggesting that the Fermi level of WSe 2 is located in the middle of the energy gap or marginally shifted toward the valence band edge. [20,38] The red dotted lines in Figure 2 indicate the FN tunnel plot derivative curves at T = 300 K for each tunnel junction. A single minimum in the FN plot derivative forms at a bias voltage that is slightly lower than V FN at T = 6.0 K, reflecting the narrowing energy gaps of the semiconducting films upon elevating the temperature. [20,39] Our approach to analyzing Φ SB based on the FN plot slopes can be further corroborated with the theoretical model proposed by K. H. Gundlach: the energy barriers in metal-insulator-metal tunnel junctions can be related to the peak positions in dln(I)/ dV b versus V b plots. [40,41] The solid green lines in Figure 2b,c display the dln(I)/dV b -V b relations for each WSe 2 tunnel junction, with the first peak aligned well with V FN . Figure 3a-d summarize the layer-number-dependent Φ SB of the WSe 2 , MoSe 2 , WS 2 , and MoS 2 tunnel devices, respectively. Diminishing Φ SB with increasing layer number is consistent across the type-VI SCTMDs. Note that Φ SB of the devices with MoS 2 films are consistently lower than the heights for other junctions, and also Φ SB for the monolayer WS 2 device is severely underestimated since FN tunneling was not fully developed for these measurements (Figure 1d).

Redefinition of Vertical Effective Mass in Ultrathin SCTMDs
Φ SB can be determined from the location of V FN in the FN tunnel resistance plots and d by measuring the SCTMD thickness with an atomic force microscope. Therefore, it is possible to extract the charge carrier effective masses in the SCTMD vertical junctions from the measured gradients. As displayed in Figure 4, the extracted carrier effective masses are found to be much larger than the free electron mass in the monolayer devices, m*(1L-WSe 2 )/m e ≅ 3. As the WSe 2 layer number increases, the effective mass decreases and converges to m*/m e ≅ 0.2-0.3. The convergence for high layer numbers to a value of ca. 0.2m e for WSe 2 is in reasonable agreement with earlier measurements for thick WSe 2 layers (> 10 nm). [36] WSe 2 at 300 K and MoSe 2 at 6 K show a similar dependency of effective carrier mass on layer number ( Figures S2 and S3, Supporting Information).
The increased charge carrier effective masses for low layer numbers reflect the unique quantum tunneling processes in vdW-coupled planar 2D-2D junctions with an ultrathin tunnel insulator. In 2D vdW materials, the energy-momentum relations along the z-axis cannot be explicitly defined because of the weak interlayer coupling. Thus, the "effective" masses extracted from the out-of-plane FN tunneling should differ from both the effective masses obtained for 3D bulk materials and the in-plane 2D carrier effective masses. Considering the energy-time uncertainty principle ΔEΔt ∼ ħ/2, we can argue that charge carriers tunneling through the 2D barrier momentarily stay in the tunnel junction for Δt ∼ ħ/(2ΔE), which is related to the resonant tunnel lifetime in a semiconductor tunnel junction where ΔE is the half-width half-maximum of the transmission coefficient. [42] During this transitory stay, charges travel by λ = vΔt, later expressed as λ = ħ/(2m*ΔE) 1/2 by combining the de Broglie wave p = m*v = ħ/λ and the uncertainty principle for Δt; namely, a lighter effective mass indicates that tunneling charges can transport deeper into the barriers. For the particular case that the thickness of an insulating layer d becomes thinner than λ, the penetration depth λ can naturally be replaced by d. Following this simple heuristic argument, we www.advelectronicmat.de can make two postulates: i) the bulk effective mass m* defines the critical thickness d c = ħ/(2m*ΔE) 1/2 at which the vertical quantum tunneling exhibits a crossover from bulk to 2D multiband transport, and ii) the "equivalent" charge carrier effective mass in an atomically thin SCTMD insulator reconfigures as m* 2D = ħ 2 /(2ΔEd 2 ). Figure 4 compares the experimentally extracted masses with the theoretically deduced "equivalent" effective mass variations. We consider ΔE for WSe 2 layers as 0.6 eV and the thickness of monolayer WSe 2 as 0.65 nm. ΔE is determined to be half of the voltage range, 0.8-2.0 V, where FN tunneling behavior is observed from the 2L, 3L, and 4L WSe 2 in Figure 2,5. With the bulk effective mass set as m* = 0.25m e , d c is estimated to be 1.79 nm, and thus the crossover between bulk and multiband 2D SCTMDs is expected to occur at three layers of WSe 2 . The effective mass variations derived from simulated I-V b characteristic responses incorporating microscopic out-of-plane quantum tunneling processes (open red circles in Figure 4) are well matched to the experimental data and the simple heuristic model calculation (more details in the Supporting Information).

Sequential Layer-by-layer Fowler-Nordheim Quantum Tunneling
We argue that a sequential layer-by-layer FN tunneling process regulates the high-energy out-of-plane charge flow, and also that the FN tunneling measurements presented here are sensitive enough to probe the microscopic quantum tunneling events in the 2D vdW semiconducting layers. We find that graphite-SCTMD-graphite vertical tunnel junctions, particularly the devices with WSe 2 and MoSe 2 , reveal fine spectral features in a layer-number-specific manner other than the leading derivative dip at V FN in the FN plot derivative curves. Figure 5a shows the FN plot derivative for a 4L-WSe 2 junction at T = 6.0 K. When V b increases beyond V FN (0.88 V), several additional features appear for the V b range up to 2 V: specifically, four additional distinct minima can be discerned in the FN plot derivative curve, indicated by the gray arrows in Figure 5a. Similar spectral features are also found in other WSe 2 devices, as shown in Figure 2.
The schematic illustrations in Figure 5b-f display the successive microscopic layer-by-layer tunneling process in p-type 4L-WSe 2 devices. In tunnel junctions with 3D bulk materials, charge carriers that have tunneled through the barrier need to transport through the tunnel insulator's conduction (or valence) band before draining out to a contact electrode. However, for ideal 2D SCTMD tunnel devices at low temperature, the highenergy FN tunnel driven charges successively access the individual layers of the SCTMD as the sample bias progressively increases. First, E F of the graphite source becomes aligned with E V of the bottom-most WSe 2 layer (i.e., next to the drain electrode) at V b = V FN (Figure 5c), initiating FN tunneling. As V b continuously increases and tunnel charges fill the bottom-most WSe 2 layer, E F of the graphite source remains fixed to E V of this layer. At the same time, the increasing electric field keeps tilting the triangular-shaped tunnel barrier and lowering the  www.advelectronicmat.de FN tunnel junction resistance. As the tunneling charges continue to enter the bottom-most WSe 2 layer, the energy cost for electrons to fill this individual WSe 2 layer is also built up. The augmented charging energies finally lead to an increase in the FN tunnel junction resistance. The switchover from a low to a high FN tunnel junction resistance develops a spectral peak (or bump) in the FN plot derivative curve. Next, the Fermi level of the second layer (from the bottom) is pulled down to the Fermi level of the graphite source once the first (or bottom-most) layer is filled, after which FN tunneling to the second WSe 2 layer starts (Figure 5d), producing a second minimum in the FN tunnel junction resistance plot. In the following V b range of 1.0 V ≤ V b ≤ 1.2 V, tunneling holes fill the second WSe 2 layer, and subsequent FN tunnel junction resistance variations lead to the second FN plot derivative bump. Accordingly, we can associate the third and fourth FN plot derivative peaks with the tunneling hole carriers filling the third and top-most WSe 2 layers. When the last WSe 2 layer is completely filled, the vertical charge flows are limited by the ohmic conduction of the graphite electrodes.

Conclusion
We investigate the microscopic quantum transport processes of out-of-plane vertical charge flows in atomically thin 2D vdW SCTMD layers, especially in the "high-energy" FN tunneling regime. The high electric field-driven vertical charge flows can be utilized as a reliable material characterization probe to quantitatively address the relative Fermi level locations of the SCTMDs and the layer numbers of WSe 2 and MoSe 2 . Detailed FN plot analyses indicate that out-of-plane charge flows are regulated by sequential tunneling, involving all layers of the SCTMD tunnel barriers successively. Moreover, electron and hole carriers tunneling across the 2D-2D planar junctions are held by momentary confinements in the ultrathin tunnel barriers, which materializes as enhanced out-of-plane charge carrier effective masses in the FN tunnel relations. The high-energy FN tunneling realized in our studies can provide a solid experimental foundation for understanding vertical charge flows in 2D semiconductors, and thus promote FN tunnel device applications with vdW-assembled vertical heterostructures.

Experimental Section
Device Fabrication: In vertical 2D vdW heterostructures, the preparation of atomically clean interfaces was critical for an accurate and reliable material characterization of the vertical junctions. At first, 60 nm to 100 nm thick highly crystalline hBN flakes were mechanically exfoliated on a 90 nm thick SiO 2 layer on Si substrate. Then a mechanically isolated graphite flake of thickness 5 nm or more was transferred to a prelocated hBN flake on the SiO 2 /Si substrate using a dry transfer method. Polymer stacks of PMMA (poly (methyl methacrylate))-PSS (polystyrene sulfonate) layers was utilized for such tasks and carefully adjust the thickness of each layer to enhance the optical contrast of the exfoliated ultrathin 2D layered materials. The PMMA film in warm (60 °C) acetone was removed and further anneal the samples at 350 °C for several hours in a mixture of Ar:H 2 = 9:1 to ensure residue-free graphite surfaces. Then by following a similar dry transfer protocol, SCTMD films were transferred on top of the pre-located Si/SiO 2 /hBN/graphite stack. In removing the polymer, the temperature to 250°C was reduced

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to avoid undesired defect state formation in the SCTMDs. In some devices, instead of using the PSS/PMMA polymer stacks, a Gel-Pak was utilized to exfoliate and transfer SCTMDs to form SCTMD-graphite vertical junctions. Lastly, a top graphite flake exfoliated on a PMMA-PSS polymer stack was transferred to complete the vertical graphite-SCTMD-graphite planar tunnel junctions. The tunnel devices through electron-beam lithography and metal lift-off (Ti ≈ 5 nm/Au ≈ 80 nm) was completed. High-purity (> 99.995%) SCTMD crystals of WSe 2 , MoSe 2 , WS 2 , and MoS 2 from HQ Graphene were purchased with no additional dopants added during the growth procedure.
I-Vb characteristic Measurements: The I-V b characteristic measurements were carried out by sweeping a bias voltage V b at a specific gate voltage V g using a DC voltage source (Yokogawa GS200). The current was converted into voltage using a low-noise voltage preamplifier (Digital Instruments DL1211) and then recorded using a digital multimeter (Agilent 34410A). The gate voltage was varied over a range of V g = −30 V to 30 V, with an increment of ΔV g = 1 V. The measurements were conducted at temperatures of 6.0 and 300 K, at the minimum pressures of 10 -7 Torr and 2 × 10 -3 Torr.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.