High Number of Transport Modes: A Requirement for Contact Resistance Reduction to Atomically Thin Semiconductors

Electrical contacts to atomically thin 2-D semiconductors are considered as the hindering aspect of electronic devices based on these materials. The high resistance of such contacts stems from their Schottky nature in contrast to the desired low-resistance Ohmic contacts. This issue of Schottky contacts is thus one of the major inhibitors to the integration of 2-D materials into mainstream technology. In this work, we explore contact resistance (<inline-formula> <tex-math notation="LaTeX">${R}_{\text {C}}$ </tex-math></inline-formula>) to atomically thin 2-D semiconductors in terms of the injected current through the Schottky barrier (SB) by using the Landauer-Büttiker formalism as well as experimental measurements and technology computer aided design (TCAD) simulations. We show that the SB height and width, which are determined by the metal–semiconductor interface and the number of charge carriers in the semiconductor channel, respectively, affect <inline-formula> <tex-math notation="LaTeX">${R}_{\text {C}}$ </tex-math></inline-formula> when it is relatively high (<inline-formula> <tex-math notation="LaTeX">${R}_{\text {C}} >$ </tex-math></inline-formula> 1 <inline-formula> <tex-math notation="LaTeX">$\text{k}\Omega \cdot \mu \text{m}$ </tex-math></inline-formula>). However, the number of transport modes for carrier injection is the limiting factor for aggressive <inline-formula> <tex-math notation="LaTeX">${R}_{\text {C}}$ </tex-math></inline-formula> lowering (<inline-formula> <tex-math notation="LaTeX">${R}_{\text {C}} < $ </tex-math></inline-formula> 1 <inline-formula> <tex-math notation="LaTeX">$\text{k}\Omega \cdot \mu \text{m}$ </tex-math></inline-formula>), even for near-zero SB height. Our results show that to reduce <inline-formula> <tex-math notation="LaTeX">${R}_{\text {C}}$ </tex-math></inline-formula> below <inline-formula> <tex-math notation="LaTeX">$100~\Omega \cdot \mu \text{m}$ </tex-math></inline-formula>, large number of transport modes are required, which can be accomplished through raising the number of channel carriers above 5. 1013 cm-2 by means of heavy doping or gating. Our conclusions offer insight for future contact engineering and can explain recently published state-of-the-art results.


I. INTRODUCTION
A S ATOMICALLY thin 2-D materials gain more attention due to their outstanding properties and extreme thinness, it becomes apparent that high contact resistance (R C ) is one of their major drawbacks [1], [2], [3]. Much effort has recently been invested to lower R C to 2-D semiconductors by increasing their charge carrier concentration (n) or by depinning the Fermi level at the metal/semiconductor interface [4], [5], [6], [7], [8], [9], [10], [11], [12]. However, the calculations of the intrinsic lower limit of contact resistivity to bulk semiconductors showed that the number of states available for carrier transport determines R C at the extreme end of scaling [13], [14]. Therefore, which of these approaches will reduce R C more dramatically remains an open question. Moreover, the main determining factor of R C and its relation to the aforementioned methods in the context of atomically thin 2-D semiconductors have not been thoroughly discussed.
Here, we address the questions of preferred method and limiting factor for aggressive R C reduction by utilizing a combination of three approaches. First, the usage of an analytical Schottky barrier field-effect transistor (SB-FET) model and application of various physics driven modifications. Second, the detailed experimental analysis of R C and the Schottky barrier (SB) height (φ B ) to monolayer (1L) MoS 2 . Third, the simulation of R C with Sentaurus technology computer aided design (TCAD) software [15], [16], [17]. First, we calculated the current injected through an SB between MoS 2 and metal contacts using the Landauer-Büttiker formalism with φ B and n acting as the model parameters [16], [18]. Previously, the model was only used for current calculations in the SB-FET OFF-state, and our adjustments enable its use in the ONstate as well, thus allowing for R C to be extracted at the desired transistor operation mode. Then, to complement the model modifications, we fabricated 1L MoS 2 transistors with Ni contacts in four-point probe (4PP) measurement structures and compared the experimentally extracted current and contact resistance to the model calculated values [17]. Finally, we validated the results by conducting TCAD simulations of 1L MoS 2 structures and extract R C by using the SB-FET model parameters as simulation inputs. We see that the SB width and height equivalently affect the current when the contact resistance is relatively high and crucially find that for extreme R C reduction, the critical limiting factor is the number of transport modes for injection and not the transmission coefficient through the barrier. Our results show that high number of transport modes, achievable by increasing n, are needed to reach desirably low R C even for low φ B , implying that high n is essential in lowering R C to atomically thin semiconductors.

II. CURRENT MODELING OF SB-FET
The calculation of the current and contact resistance using the SB-FET model relies on the Landauer-Büttiker formalism, which accounts for two modes of current injection, namely, thermionic emission over the SB and field emission (tunneling) through the SB [16], [18]. The injected electron current is expressed as where I is the current, q is the electron charge, h is Planck's constant, M(E) is the number of transport modes for conduction in the semiconductor channel, T (E) is the transmission coefficient through the barrier, f (E) is the Fermi-Dirac distribution, and V C is the voltage drop across the contacts. Note that because our purpose is to understand which aspect of the SB limits the reduction of R C , we assume that the current is contact-resistance-limited and therefore do not include carrier scattering inside the channel in our current model. Furthermore, we only address the case of long contacts that are not limited by the contact transfer length. However, for aggressively scaled contacts, more considerations should be applied to accommodate the further limiting factor. We choose to model the current as a result of the contact voltage drop, which we experimentally determine and separate later on. The total current is thus the sum of the charge carrier transmission across all the available transport modes for conduction at energies above the conduction band edge for a given V C .
The above expression is valid when the current is primarily governed by electron injection into the conduction band but can readily be adjusted for hole injection into the valence band. The resulting current for E < φ B (i.e., tunneling through the barrier) is dominated by field emission and thermionic field emission, and for E > φ B , the resulting current is dominated by thermionic emission. We calculate M(E) by counting the number of modes along the transport direction in the first Brillouin zone, as previously shown by Suryavanshi et al. [19]. The subtraction of the Fermi-Dirac distributions ensures that the number of electrons participating in the conduction is determined exclusively by V C [16].
The transmission coefficient is normally calculated by assuming a fully depleted conducting channel and a triangularshaped SB with a base width equal to the characteristic geometrical screening length (λ geo ) [15], [16], [20]. The length is defined by [21] where ε 2D is the 2-D channel dielectric constant, ε i is the gate insulator dielectric constant, t i is the gate insulator thickness, and t 2D is the 2-D channel thickness, resulting in a constant λ geo for a given FET. This extraction process is therefore valid for SB-FETs operating in the OFF-state where these assumptions hold. When the SB-FET operates in the ON-state however, high carrier density is induced by the gate inside the channel, and it is therefore no longer fully depleted. To overcome the limitations imposed by the above assumptions, we adjusted the transmission coefficient calculation based on previous work that detail the expansion of the SB into the channel of FETs based on 2-D materials (2-D-FETs) [22], [23]. First, we adjust the SB width to accurately reflect the electrostatic effect resulting from the carrier concentration in the channel induced by the back-gate voltage. As discussed previously by Kang et al. [22], the band-bending due to the SB must decay across the 2-D channel when the dielectric screening length is longer than the channel thickness, as shown in Fig. 1. The barrier penetration length inside the 2-D channel (λ 2D ) is determined by solving Poisson's equation and given by where ε 0 is the vacuum permittivity, V GS is the back-gate voltage, V T is the FET threshold voltage, and G is the band movement factor that determines the ratio between the backgate voltage and the band-bending [24]. The 1-D solution of Poisson's equation is best suited for 1L channels, while thicker multilayered channels may require a more detailed 2-D solution to calculate λ 2D , like bulk semiconductors. However, the main point of the manuscript still holds as the limiting factor of R C will remain as discussed by Baraskar et al. [13] and Maassen et al. [14]. For simplicity, we assume that the applied voltage V C distributes equally between the source and drain contacts, and therefore, only half of it drops on each one, resulting in the 0.5 coefficient in the above expression.
Note that this is likely not the case due to the inherent different nature of current injecting and collecting contacts and the contact-to-contact variability in a fabricated device [23]. The band movement factor is given by and is derived from the solution of a series capacitance network [24]. The channel charge carrier concentration during the ON-state operation of the FET (i.e., for V GS > V T ) induced by the back-gate voltage is given by where C i is the back-gate insulator capacitance. We divide by the 2-D channel thickness to transition between the surface charge concentration and the volume charge concentration. The barrier width is expectedly found to be gate-dependent at these conditions. Second, we modify the transmission coefficient calculation to consider the entire current path and account for the varying λ 2D .
For most contacts to 2-D semiconductors, when charge carriers are injected from the metal into the semiconductor channel, they must pass through a van der Waals (vdW) gap prior to the SB, as shown in Fig. 1(a) [22], [25]. A band diagram representing the energy barriers along the current path is shown in Fig. 1(b), note that λ 2D depends on the back-gate voltage, as shown schematically in blue. The transmission of electrons through the vdW gap (T vdW ) is calculated using the Wentzel-Kramer-Brillouin (WKB) approximation and is given by the rectangular potential barrier tunneling probability where L vdW is the vdW gap width, m e is the electron effective mass, and χ 2D is the 2-D material electron affinity, which acts as the vdW gap height. We used a 3-Å-wide gap for our calculation to coincide with previous work [19] resulting in T vdW = 1%-2%; however, various MoS 2 -metal bond lengths have previously been reported [22]. By adjusting for the correct vdW gap width, the model can be generalized for any atomically thin semiconductor-metal contact. The transmission coefficient through the SB (T SB ) is also calculated using the WKB approximation and is then given by The transmission coefficient through the barriers is therefore and is used for the current calculation shown in (1). Any spatial variations affecting the carrier injection, such as SB height inhomogeneity [26], [27], can be accounted for by implementing a variance function into the relevant variable [e.g., φ B (x)] in the transmission coefficient and barrier penetration length calculations.
We conduct further analysis to complement our modeling efforts by experimentally obtaining φ B and n and using them as the key parameters for comparison with our model results. Note that n affects not only λ 2D but also the number of transport modes through the change in the conduction band edge location. As n increases, the conduction band edge approaches the 2-D semiconductor Fermi level and becomes lower relative to the metal Fermi level. As presented by Suryavanshi et al. [19], the number of transport modes monotonically increases for energy states above the conduction band edge. Therefore, increasing n allows the metal to inject electrons into relatively higher energy states with respect to the semiconductor conduction band edge, which are associated with larger number of transport modes.
To test our approach experimentally we patterned 4PP test structures by e-beam lithography and used e-gun evaporated Ni as contacts to MoS 2 grown by chemical vapor deposition (CVD) on SiO 2 (90 nm)/Si (p++) substrates, where the Si serves as a back-gate, as illustrated in Fig. 2(a) [28]. As was previously reported, Ni can form chemical reaction free contacts to MoS 2 , therefore exhibiting a vdW gap [29], [30], [31]. An atomic force microscopy image of the device structure is presented in Fig. 2(b), and more details on the device fabrication are reported elsewhere [23]. I D -V GS measurements, presented in Fig. 2(c), show typical gate control with I on /I off ratio > 10 8 and 20-µA/µm current at V GS = 40 V and V DS = 1 V for a 3-µm-long device. The measurements were performed at various temperatures ranging from 125 to 300 K in vacuum ambient condition of 3·10 −5 mbar. The subthreshold current increases with the temperature due to the higher contribution of the thermionic current that is suppressed at lower temperatures. Fig. 2(d) and (e) depicts the determination of the SB height by first extracting the slopes of ln(I D /T 1.5 ) versus 1000/T for various gate biases, then choosing the barrier height at flat band conditions from the effective barrier height versus V GS plot as the point at which the relation diverges from linearity [15]. The SB height is found to be φ B ≈ 150 meV.
The calculated and measured current versus charge carrier concentration is presented in Fig. 3(a). The physical parameters used for calculations are ε 2D = 4, ε i = 3.9, m e = 0.45m 0 , χ 2D = 4.05 eV, and L vdW = 3 Å. We used 4PP measurements to determine and separate the effective contact voltage drop (V DS,eff ) as previously mentioned by subtracting the channel region voltage drop (V Ch ) from the applied voltage V DS [17]. V Ch is given by where V 2−3 is the voltage drop between contacts 2 and 3, L Ch is the total channel length, and L 2−3 is distance between contacts 2 and 3. The voltage used for the current modeling calculation is then adjusted accordingly. Although the main assumption of our contact resistance model is that the current is contact-resistance-limited, it does not necessarily hold for the fabricated 4PP devices. In fact, the fabricated channels are 3 µm long and the device currents are channel-resistancelimited. However, by accounting for the effective voltage drop on the contacts, we can compare the modeled and experimentally measured currents using the aforementioned effective voltage drop adjustment. The experimentally extracted and modeled R C versus n is presented in Fig. 3(b), showing the well-known back-gate voltage dependency of R C [32]. Our modeled I and R C both exhibit good agreement with the measurements, affirming the model modifications we made. Some mismatch is observed which can be attributed to our assumptions that the voltage drop distributes equally between the source and drain contacts and to the process variability of the MoS 2 growth and contact metal deposition. Lastly, we compare our modeled R C with values obtained from Sentaurus TCAD simulations and evaluate the relative effect of the SB height and width on R C using both methods. The barrier height is adjusted with the variation of φ B , and the barrier width is adjusted with the variation of n (and λ by consequence). Besides varying n, we account for the V GS effect on R C by inserting the band bending induced by the backgate voltage into the transmission coefficient calculation. The structure used for the TCAD simulations is similar to previous publications [33], [34], [35]. Briefly, a single layer of MoS 2 with 0.6 nm thickness is defined on top of a SiO 2 /Si substrate to reflect the fabricated 1L MoS 2 devices. A 3-Å-wide vdW gap between the MoS 2 layer and the metal contacts is modeled by a vacuum layer (ε = 1) in the designated area. The simulation then solves Poisson's equation coupled with drift-diffusion equations, and the boundary conditions we set act as the applied gate and drain voltages.
First, we can examine the similarity between our modified model and the TCAD simulation by comparing the band structure used for the R C calculation. Fig. 4(a) and (b) shows Poisson's equation and the intricate numerical solutions for the band structure, respectively, for a 150-meV SB. The lateral decay of the bands is clearly shown by the large scale for the xaxis in comparison to the 1L thickness. The numerical solution provided by TCAD simulations is more exact but requires high computation power to achieve, and the simple Poisson's equation solution however is less accurate but is considerably easier to obtain. Therefore, we can use our calculation for the barrier penetration length, λ 2D , as a good estimate. Fig. 4(c) and (d) displays the calculated heat maps of R C as a function of φ B and n as extracted from our model and TCAD simulations, respectively. The markers represent reported stateof-the-art experimental data, and the dashed lines represent the 1000-, 500-, and 200-·µm contours [4], [6]. The general trends of R C are similar between our model and the TCAD simulation, again reiterating the power of our model to evaluate R C with a simple calculation as opposed to a robust numerical differential equation solver. We note that although a specific value for the vdW gap width (3 Å) is used for the MoS 2 -Ni contact, other values can otherwise be used to account for different atomically thin semiconductor-metal contact interfaces. Fig. 5 shows the model and TCAD simulation calculation of R C for various vdW gap widths, showing the generalization of both the model and simulation. Note that due to software limitations, the TCAD R C computation uses parabolic band structure and density of states (DOS), instead of the 1L MoS 2 DOS calculated by Suryavanshi et al. [19]. When calculating the R C heat map using our model with parabolic bands, mirroring the TCAD simulation, we observe similar trends; however, the absolute values of R C are marginally larger when parabolic bands are used.
In the midrange of R C (∼1 k ·µm), we observe from our model and TCAD results a dependency on both φ B and n as there is no clear factor limiting the current. However, to reach desirably low R C (<200 ·µm), high carrier concentrations (narrow SB and large number of transport modes) are required even for near-zero SB height, suggesting that high n is crucial in reducing R C to 2-D semiconductors [36]. Interestingly, when n is increased beyond 3·10 13 cm −2 , the R C dependency on φ B weakens as the transmission coefficient through the SB becomes close to unity (T vdW is independent on n). This stems from the strong dielectric screening that dramatically reduces the SB width, which greatly increases the tunneling probability and makes the SB height a lesser determining aspect of the transmission coefficient. We thus see that the number of transport modes limits the current at low or nearzero φ B . We note that this could also limit edge contacts,  which potentially shunt and reduce the vdW gap, as shown in the heat maps presented in Fig. 5 [22].

III. CONCLUSION
In conclusion, our work shows that although both the SB height and width generally determine R C , the increase of carrier concentration is more consequential in reducing R C to 500 ·µm and beyond. We found that for such low R C , the major current injection limiting factor is the number of transport modes (determined by n), while the transmission coefficient (and by extent φ B ) has a lesser effect. Our results suggest that increasing the carrier concentration (e.g., by strong gating or doping) is a prerequisite for aggressively lowering R C , for example, n > 5·10 13 cm −2 is needed to reduce R C below 100 ·µm even for near-zero φ B [5], [10], [37], [38]. These findings can help explain recent state-of-the-art results and provide a guideline for future contact engineering to atomically thin 2-D semiconductors.
ACKNOWLEDGMENT Fabrication was carried out at the Technion Micro-Nano Fabrication & Printing Unit (MNF&PU).