Phonon-assisted carrier transport through a lattice-mismatched interface

MoS2 typically exhibits unconventional layer-thickness-dependent electronic properties. It also exhibits layer-dependent band structures including indirect-to-direct band transitions, owing to which the electronic and carrier transport properties of a lattice-mismatched, conducting, two-dimensional junction are distinct with the naturally stepwise junction behaving as a 1D junction. We found distinguishable effects at the interface of vertically stacked MoS2. The results revealed that misorientationally stacked layers exhibited significantly low junction resistance and independent energy bandgaps without bending owing to their effectively decoupled behavior. Further, phonon-assisted carriers dominantly affected the lattice-mismatched interface owing to its low junction resistance, as determined via low-temperature measurement. Our results could facilitate the realization of high-performance MoS2 transistors with small contact resistances caused by lattice mismatching. The unusual effects observed when ultrathin semiconductors are stacked into multi-layers may make it easier to wire these materials into high-performance transistors. Molybdenum disulfide (MoS2), a substance that forms graphene-like crystals, has electronic properties that depend on its thickness. Seong Chan Jun from Yonsei University in Seoul, Korea, and colleagues have developed a MoS2 device that combines thickness variations with intentionally misaligned crystal lattices. The team fabricated MoS2 transistors that contain either a naturally occurring crystal “step” or one produced by transferring a thicker MoS2 flake onto a thinner layer. Electrical tests revealed that devices with artificial steps could be connected into circuits with much lower resistance than typical for 2D semiconductors. Energy band measurements suggest that vibrations at the mismatched thin and thick MoS2 interfaces improve the flow of electrons into external metal electrodes. We showed the distinctive unconventional junction effect of MoS2 junctions: a lattice mismatched MoS2. It is unique to observe the difference originated from the atomic interrelation at the interface. The results revealed the dominant scattering source at the conventional naturally stepwise junction, while the misorientationally stacked layer exhibited effectively decoupled behavior and a significantly smaller junction resistance via phonon assist carrier. Therefore, our finding in this paper clearly shows the different mechanisms in carrier transport at both junction interface of MoS2.

To utilize 2D TMDs as a homojunction device, a difference in thickness is necessary. However, homojunctions using 2D TMDs have a perfectly aligned stepwise junction, which is distinct from a randomly stacked homojunction. Thus far, there have not been systematic studies on the electronic properties of these two types of junctions. We fabricated field effect transistor (FET) devices with a single MoS 2 flake that consisted of different numbers of layers ranging from thin (less than three layers) to thick (more than four layers). Such devices are termed "naturally stepwise junction" (NSJ) devices. Subsequently, we compared the performance of the NSJ devices with that of a lattice-mismatched junction (LMJ) device. Although both type-I and type-II junctions have been reported recently for the NSJ structure 28,29 , our experimental and calculational results are consistent with a type-I junction. In this study, surface-potential measurements using the scanning Kelvin probe microscopy (SKPM) technique allowed us to determine the work function of MoS 2 for different layer thicknesses. In addition, we could deduce the junction type from the conduction-band minimum (CBM) and valence band maximum (VBM) obtained from density functional theory (DFT). For the LMJ structure, the potential barrier was very small, and layer decoupling at the interface was observed. Furthermore, the band structures of LMJ and NSJ as well as temperature-dependent current values were investigated. Figure 1a shows a schematic of the fabricated device. The MoS 2 devices had a thickness step in the middle, which was formed via two different approaches. For the ohmic contact, graphene electrodes were adopted and deposited Pd/Cr/Au (30 nm/5 nm/20 nm) as metal pads (Fig. S2). Metal electrodes were patterned by e-beam lithography (TESCAN VEGA3). For the NSJ, we used a MoS 2 flake including naturally adjacent atomic steps with a perfectly aligned atomic lattice (Fig. 1b). The LMJ-MoS 2 devices were produced by transferring a thicker MoS 2 flake onto a thinner one using a transferring device (MSTech Probe Station Series MST 5500B, Fig. S1).

Characterization of junction devices
To compare transport characteristics between NSJ and LMJ, both types of devices were experimentally measured and theoretically analyzed via surface potential measurement, I-V measurement with temperature variation, Yfunction analysis, and DFT analysis. The surface potential value was obtained using SKPM, and the work function translated from the value was obtained through Eq. (1) in chapter 2 of the supporting information. CBM and VBM were obtained by applying DFT, which is the theoretical calculation method (see chapter 4 of the supporting information). For the geometry optimization calculations, the Brillouin zone (BZ) was sampled in the Monkhorst-Pack scheme where the convergence of energy as a function of the number of k-points was tested. The Yfunction method was used to obtain the contact resistance. The data obtained from the I-V results were calculated using Eqs. (2)(3)(4)(5) from the supporting information for strong inversion cases. Additionally, using Eq. (6) from the supporting information to plot Arrhenius, we deduced the junction barrier height. In this work, each I-V data was measured in a low-temperature chamber and controller (LakeShore 336 Temperature controller) since we had to know the change in temperature.

Results and discussion
Layer dependent homojunction of MoS 2 Figure 1d shows the output curves of a representative NSJ-MoS 2 device consisting of two and five layers clearly indicating the quasi-current-rectifying effect and the ntype response to the back-gate voltage. Such asymmetrical current-voltage behavior implies a built-in electric field at the junction, which can arise from the thicknessdependent electron affinity and bandgap of MoS 2 12,26,27,30-33 . The LMJ-MoS 2 device, fabricated by transferring a six-layer (6L) MoS 2 onto a 2L flake yielding a large-area 2D interface, as shown in Fig. 1c, and the corresponding output curves are presented in Fig. 1e. Contrary to the NSJ-MoS 2 device, these output curves are linear and symmetric indicating that the misorientationally stacked device had ohmic-like contacts at the vdW interface (rectification ratio for both types can be seen in Fig. S3).

Naturally stepwise junction (NSJ)
To understand the band alignment of the layerdependent MoS 2 junction, we evaluated the layerdependent work function using SKPM (Fig. S4).  34 . The CBM decreases as the number of layers increases, while the VBM decreases. The CBM of thin MoS 2 is higher than that of thick MoS 2 and the VBM is lower. Therefore, when two MoS 2 that have different thicknesses meet, a type-I junction is formed.
The distinctive difference between the NSJ and LMJ is the lattice alignment. For the LMJ, the lattice structures of the two layers are misoriented with respect to each other. Conversely, in the NSJ, each layer of MoS 2 is wellstacked to form smoothly oriented lattice structures. We performed DFT calculations to elucidate the different electronic properties of these two cases (Figs. S10-11). For the NSJ, the interface is considered 1D and the effect of the imperfectly terminated edge of the thicker side can be very significant. Figure 2c shows the calculated local density of states (LDOS) for the 2L-6L NSJ. Here, the LDOS away from the interface recovers the expected bulk band edges for 2L and 6L MoS 2 . However, at the interface, we observed a reduced LDOS at the band edges. This reduced LDOS, albeit only occurring across a few atomic layers, can be a significant source of carrier scattering. By considering the CBM and the work function of each side of the MoS 2 , the band alignment is determined for the NSJ, as shown in Fig. 2b. The current rectifying behavior is mainly due to the existence of band bending and a potential barrier at the interface. We observed a higher current amplitude under negative voltage bias on the thicker layer. Although MoS 2 homojunction certainly forms a type I junction, the CBM difference between layers can be much smaller than that of the work function, especially in the case that a monolayer MoS 2 is not involved. Therefore, as described in Fig. 2b, the potential barrier for electrons can be minimized when it comes from a thicker layer to a thinner layer.
Lattice-mismatched junction (LMJ) Figure 3a, b show the distribution of the wave function in real space along the layers of a perfectly aligned, uniform 6L MoS 2 structure and a misaligned 2L/4L structure, respectively. The results show that the wave functions are well separated between the misaligned 2L and 4L MoS 2 in the latter structure, while the perfectly aligned 6L MoS 2 has a strongly mixed wave function. Thus, when the lattice structures of randomly stacked MoS 2 layers are mismatched, each stacked layer is electronically decoupled and should be regarded as two separate materials. Consequently, the lattice-mismatched homojunction interface is 2D. Figure 3c presents a surface-potential image of a MoS 2 flake obtained by SKPM. The MoS 2 flake contains various regions with different thicknesses. In Fig. 3c, we observe a clear transition in the surface-potential values as the Fig. 1 Images of a FET device with thickness-dependent MoS 2 isotype heterojunction with graphene electrodes and Pd metal pads. Our surfacepotential measurement suggests that work-function value increases with thickness of MoS 2 layer. Schematics for b NSJ device and c LMJ device indicating the presence of junction resistance in addition to each contact resistance for layers 1 and 2. Current-voltage relationship at various back gate voltages for d NSJ and e LMJ number of layers is increased from two to seven. As the thickness is increased, the MoS 2 nanosheet tends to have a lower surface potential and higher work function. This result strongly supports the physical picture of interlayer decoupling owing to lattice-mismatches between the overlapped MoS 2 flakes. To systematically analyze these layer-dependent work-function differences and the layer decoupling effect, we extracted the junction resistance of each NSJ and LMJ device.
For an LMJ, the thickness of the device is far smaller than the relevant carrier screening length. Figure 3g shows a rotated schematic of an LMJ device and an energy band diagram for the LMJ case. Since the actual band structure is restricted in atomic thickness, the potential barrier between the two layers can be identified by the CBM difference, and it is much smaller than that of the NSJ case in most cases. Additionally, the LDOS for each atomic region  (Figs. S10-11). In addition, for an LMJ, the relative misorientation of each layer implies a mismatch in reciprocal space, as illustrated in Fig. 3h. Hence, the carrier transport across this 2D interface would require the assistance of a large-momenta supplied by optical phonons. To understand the effect of the lattice misorientation, explicit conductance simulations were performed 35 . The corresponding transmission coefficients are plotted in Fig. 3i. The result of this calculation is that the misoriented junction exhibits similar interlayer conductance with respect to bulk MoS 2 and even higher conductance in the valence band region and around CBM. The reasons for such unexpected behavior lie in a "charge compression" effect. In the bulk system, black curve in Fig. 3i, the repulsion of the electronic clouds of the S atoms pushes the valence band to higher energy, especially at the Γ point, producing a decrease in the indirect band gap 29,36,37 . Additionally, the rotational angle in the LMJ case may be another issue, but studies regarding twisted, misaligned TMD heterojunctions and graphene layers suggest an insensitivity to angle 38,39 . Basically, the twist angle controls the effective interlayer coupling and induces a van Hove singularity at some energy 39 .

Junction resistance analysis by using Y-function method
To quantitatively evaluate the junction resistances at the interfaces, we employed the Y-function method. Figure 4a shows a representative transfer curve of a MoS 2 homojunction device. We choose the samples showing gate independent attenuation as "strong inversion" (Fig. S7). Because of the interfaces in both junction devices, the junction resistances should be considered as both effective mobility attenuation factors and the contact resistances. Figure 4b shows the extracted sum of the contact resistance and junction resistance as a function of the work function and CBM difference for both homojunction devices. Because the contact resistances are expected to have small values (less than a few kΩ) and do not differ regardless of the device structure and thickness, the difference in the resistance originates entirely from the junction resistance at the interface. The resistance of the NSJ increases exponentially with the surface-potential difference between the two MoS 2 layers. It is evident that the barrier resulting from the band bending in the NSJ case can be determined by the Fermi-level difference. This indicates that the junction resistance dominates the total resistance. It also justifies our assumption in the modification of the Y-function method. In the LMJ case, the barrier at the interface can be determined by the alignment of the CBM in equilibrium. The suggested values in Fig. 4b are given by the work function and CBM transition shown in Fig. 2a. The LMJ devices exhibit significantly smaller resistances than the NSJ devices in most cases and show a rough dependence on the energy difference, although the LMJ cases have many uncontrolled variables at the interface (Figs. S8-9). We measured the entire devices at room temperature, and the phonon-assist effect may have been significant. Cases with an energy difference smaller than 50 meV exhibit particularly small resistance values. Furthermore, at energies higher than 50 meV, the resistance is drastically increased to values comparable to the NSJ cases. The phonon-assisted charge-transfer mechanism may explain this behavior given the reported optical phonon energy of MoS 2 40 .
To further verify this conclusion, we fabricated both types of MoS 2 homojunction devices on the same MoS 2 Fig. 4 a The transfer curve of MoS 2 homojunction transistor (inset). The mobility attenuation factor, θ, with back gate bias. b Sum of contact resistance and junction resistance for various homojunction devices extracted using Y-function analysis. c Device image designed to separate junction resistance from contact resistance. Contact resistance was determined by measuring a pair of electrodes on the same MoS 2 layer (1a-1b, 2a-2b, and 3a-3b). d Extracted junction resistance for NSJ and LMJ cases flake, as shown in Fig. 4c. We used MoS 2 flakes with 2L (blue dotted line) and 10L (yellow dotted line) regions for the NSJ. Another MoS 2 flake of 10L was transferred onto the 2L region for forming the LMJ. The MoS 2 devices were measured by using 1a-1b, 2a-2b, and 3a-3b contacts where the contact resistances for the three devices were extracted via the Y-function method. The contact resistances for the three devices were significantly smaller than the total device resistance (<5 kΩ) (Fig. 4d). Both devices were measured by using 1a-2a and 2a-3a contact cases. By substituting the measured contact resistance from the total resistance, we separated the junction resistance for each device. The junction resistance of the LMJ device was significantly smaller than that of the NSJ device. For all NSJ cases and most LMJ cases, it is reasonable to assume that the contact resistance has a negligible contribution to the total device resistance. Thus, we conclude that the carrier transport mechanisms in the both MoS 2 homojunction cases differ significantly. While the NSJ case exhibits a strong potential-barrier dependence based on the work-function difference, the decoupled LMJ emphasizes the importance of phonons in the charge transfer across the misaligned lattices.

Junction barrier height characterization of homojunctions by temperature modulation
In intrinsic MoS 2 devices, the temperature affects the I-V characteristics in two dominant ways. The first is the thermionic effect where the number of charge carriers increases as the temperature rises from low to high. In contrast, phonon scattering, which interferes with the movement of electrons, increases as the temperature rises. In junction devices, there is another way for temperature to affect the I-V characteristics. After the temperature rises and the phonon scattering energy becomes higher than the barrier height of the junction, the charge carrier can overcome the junction barrier easily by the phononassist effect. Therefore, temperature-dependent electrical measurements were carried out to characterize the specific junction barrier height according to the lattice matching of the junction.
In LMJ, decoupling between two different MoS 2 occur causing band bending by Fermi level pinning at the junction to be drastically reduced compared to NSJ (Fig. 5a, b). To observe a reduced barrier height by decreasing Fermi level pinning, two probe based I-V were measured from 77.4 to 300 K. In the case of the NSJ Fig. 5 Energy-band diagrams for MoS 2 a NSJ, and b LMJ. c The junction barrier height for NSJ, LMJ, and intrinsic MoS 2 . The junction barrier height at the NSJ interface is higher than the junction barrier height at the LMJ interface. The drain current-voltage curves with various temperatures from 77.4 to 300 K for d NSJ device, e LMJ device, and f intrinsic MoS 2 device, (inset) drain current versus temperature at various drain-source biases device, drain current shows a monotonous tendency without large temperature dependent change below 250 K and shows rectifying behavior like a Schottky junction due to the barrier at the junction (Fig. 5d). However, above 250 K, the current level increases sharply and the rectifying ratio decreases. This is because the charge carrier was impeded by the junction barrier and the thermionic effect cannot appear under 250 K. However, above 250 K, the phonon-assist effect allows the charge carriers to sufficiently overcome the junction's barrier in NSJ. Conversely, in the LMJ, as the temperature increases, the drain current level initially increases before approximately 150 K and then decreases after approximately 200 K, which is similar to an intrinsic MoS 2 device (Fig. 5e). Since the barrier height of LMJ is relatively lower than that of NSJ, and the phonon-assist carrier appears before 77.4 K, the level of drain current of the LMJ device is higher even at low temperature, and it shows more ohmic I-V characteristics than the NSJ device (Fig. 5c). These results are related to the phonon-assisted carrier and show good agreement with Fig. 4b. When the temperature is further increased, the drain current of LMJ decreases since the mobility decreases due to phonon scattering. Since the junction barrier height of LMJ is relatively small, the tendency of the drain current shows similar results as an intrinsic MoS 2 device. Since an intrinsic MoS 2 device does not have a junction other than the graphene contact, the overall trend is similar to LMJ devices (Fig. 5f). However, LMJ have small junction barriers such that the drain current decreases more when the temperature decreases.
As a result of the Arrhenius plot (Fig. S12), we calculated the barrier height. According to a previous study, and because multilayered graphene/MoS 2 heterostructure has negative Schottky barriers 41 , the results indicate that the junction barrier height at LMJ is much lower than the junction barrier height at NSJ, and this is enough for electrons to overcome the barrier by phonon-assist. Moreover, the barrier height difference between both MoS 2 homojunctions is strong evidence that perfectly aligned band structure causes a stronger Fermi level pinning effect than a lattice-mismatched band structure.

Contact resistance enhancement by lattice-mismatched homojunction effect
A thicker MoS 2 exhibits smaller contact resistance for high-work function metals owing to the lower Schottky barriers within them 42 . By utilizing the small junction resistance of the LMJ, we demonstrated high-performance MoS 2 transistors with a reduced contact resistance leading to an enhanced two-terminal mobility. We fabricated two devices with different contact structures for comparison, as illustrated in Fig. 6a, b. When a drain bias was applied, thin MoS 2 generally had a high Schottky barrier for high-work function metals leading to a large contact resistance and a reduced two-terminal field-effect mobility (Fig. 6c). However, thicker MoS 2 had smaller Schottky-barrier heights owing to the lowered position of the conduction band. When thick MoS 2 was inserted, charge carriers could be more easily transported into the metal (Fig. 6d). The output curves in Fig. 6e, f indicate that the 2 L MoS 2 device with Pd contacts had a high Schottky barrier. However, the 2L MoS 2 device with the stacked contacts of 6L MoS 2 exhibited a dramatic improvement in the charge-carrier injection from the electrode to the channel.
In this study, we evaluated the junction decoupling effects of MoS 2 layers for a lattice mismatched junction structure. SKPM and a junction-resistance evaluation suggested that the work function of MoS 2 changed according to the number of layers and indicated the formation of a finite potential barrier at the layer-dependent MoS 2 junction. Subsequent analysis revealed that the junction resistance for a lattice-mismatched state was far smaller than that for a perfectly aligned state. The small junction resistance of the LMJ structure was utilized for contact resistance reduction. The junction effects revealed may be useful in the development of new strategies that can be widely employed to engineer the energy-band structure in the heterostructure on the basis of the combination of various TMDs.