Schottky barriers, emission regimes and contact resistances in 2H-1T’ MoS₂ lateral metal-semiconductor junctions from first-principles

The electronic structure calculations were performed within the DFT formalism as implemented in the Siesta code [21]. The exchange correlation functional was included in the Generalized Gradient Approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) parametrization [22]. Although for 2D materials the GGA gaps often match experimental optical gaps despite the well known DFT underestimation of band gaps, this is because the experimental gaps are subject to strong excitonic effects that reduce the single particle gaps [23]. Since band alignments and carrier injection are expected to depend on the single particle band structure, the limitation of plain LDA/GGA still stands. This might affect the overall value of the Schottky barrier heights (SBHs) we provide in this work, but relative differences in SBHs should be more accurate [24].

We use norm-conserving pseudopotentials [25] to describe the core electrons and a double-$\zeta$ polarized basis set for the valence electrons. The Brillouin zone was sampled using a grid of 9x16x1 k-points for the 6-atom orthogonal unit cell of the 2H phase, and its equivalent for supercell calculations. The real space grid cutoff was set to 250 Ry.

Ionic positions were relaxed using a force tolerance of 0.03 eV/Å for the Siesta calculation and 0.05 eV/Å for TranSIESTA calculations. In the case of structures with interfaces, we also performed a cell optimization until the strains were lower than 0.1 GPa (1 kbar). The system studied consisted of a single layer MoS₂ heterostructure, where the semiconducting phase of MoS₂ (2H) is contacted laterally to the 1T’ metallic phase. We considered two different orientations for the 2H-1T’ interface, armchair (ac) and zigzag (zz). Finally, in order to avoid interactions between images in the non-periodic direction, a vacuum of 20 Å was included.

The large signal contact resistance of an interface can be obtained from the ratio between the voltage drop across the junction and the current flowing through it. Using NEGF it is possible to calculate the current that flows through a device when a potential is applied at the semi-infinte contacts. In these calculations, however, the applied voltage ($V_{tot}$) is distributed between the interface present in the scattering region, ($V_{S_{A}-S_{B}}$), and the interface between the semi-infinite contacts with the device ($V_{L-S_{A}}$ and $V_{S_{B}-R}$), as seen in Figure 3. Therefore, the total applied voltage can be expressed as:

from where

Here $R_{tot}$ represents the total series resistance, and can be obtained from the I–V curves of the whole device. On the other hand, $R_{L-S_{A}}$ and $R_{S_{B}-R}$ are the resistances between the contacts and the electrodes, which are obtained from I–V reference curves of pristine devices, all of the same phase. Since pristine devices include two electrode-contact junctions, the value extracted for each reference calculation must be divided by two to consider the contribution of a simple electrode-contact interface:

where $V_{S_{A}-S_{A}}$ and $V_{S_{B}-S_{B}}$ are the potentials applied in the pristine devices to obtain the same current value as in the whole device.

Finally, $R_{S_{A}-S_{B}}$, which is the resistance corresponding to the junction between the two phases, can be calculated from the other terms by replacing eq. (7) into eq. (6) and reordering:

A similar analysis can be performed to obtain the small signal contact resistance, but in this case, instead of taking the ratio between voltage and current at each point, we take the derivative of the voltage with respect to the current to compute the resistance. Carrying this process out we have

where derivatives are evaluated at the bias that provides a current $I_{0}$, corresponding to the bias $V_{0}$ of interest for the whole device.

To understand the transport properties of 2H-1T’ junctions, we calculated the specific conductance (i.e. conductance per unit of channel width) using the Landauer formula [28]:

where $T(E)$ are the transmission coefficients averaged over all the ${\bf k}_{\parallel}$, obtained from the NEGF calculations, and $a_{t}$ is the width of the computational cell along the direction perpendicular to transport.

In the supplementary information (SI) we performed a detailed analysis of the transport properties across zigzag and armchair interfaces. As the results predict better conductance for armchair structures, due to the asymmetric transport in the 1T’ phase, we focus our studies on ac structures.

In Figure 4 we show the specific conductance as a function of the energy of the incoming carrier, for armchair 2H-1T’ and 2H-2H interfaces with positive and negative doping at different concentrations: intermediate (5$\times$10¹² cm^-2) and high (5$\times$10¹³ cm^-2). Comparing the transmission for the 2H-1T’ device to the 2H’-2H’ reference device, which provides the maximum attainable transmission, a notable reduction of the conductance is observed, mainly for the structures with intermediate doping concentration. The results also show that the injection of holes is slightly favored due to the higher number of transmission channels in the valence band.

We also performed NEGF calculations at finite bias, showing the resulting current vs. voltage (I–V) characteristics in Figure 5. For the high doping structures, we observe a slightly asymmetry at forward and reverse biases, specially in the $n$-doped structure, but overall there is an ohmic behavior in the I–V curve. For intermediate doping concentrations, we observe an exponential increase of the current at forward bias, indicative of a Schottky regime, and a poor rectifying behavior, also typical of Schottky contacts. In this regime, the transport mechanism can be (a) thermionic emission (TE) over the Schottky barrier (SB), (b) field emission (FE) with electrons around the Fermi level tunneling through the SB, or (c) thermionic-field emission (TFE), where the tunneling electrons contributing to the current are quite above the semiconductor Fermi level, but still below the top of the SB [29].

In order to determine which of these mechanisms dominates in 2H-1T’ junctions, we performed a temperature study of the forward I–V characteristics. The results are shown in Figure 6. Plots a) and c) show the forward bias I–V characteristic for n and p-doped structures, respectively, at different temperatures, while plots b) and d) show the energy $E_{0}$ extracted from a fit of the I–V curves to $J\propto\exp(qV/E_{0})$. When FE or TFE dominate, we have $E_{0}=E_{00}\coth(E_{00}/kT)$ [29], where $E_{00}$ is an energy related to how fast the transmission coefficient through the barrier increases as the forward bias is increased. If $kT\ll E_{00}$, then FE will dominate, while when $kT\sim E_{00}$ TFE is the main mechanism [30]. On the other hand, if $kT\gg E_{00}$ then TE dominates, and we will have $E_{0}\sim\eta kT$, where $\eta$ is an ideality factor accounting for the variation of the SBH with the applied bias.

All these considerations can be summarized into the following prescription, which can also be applied to experimental measurements, allowing the determination of the dominating transport mechanism: for a range of temperatures $T$, fit the I–V curves to $J\propto\exp[qV/E_{0}(T)]$; then fit $E_{0}(T)$ to

and finally compare $E_{00}$ to $kT$ to determine whether FE, TFE or TE dominates, obtaining the ideality factor $\eta$ as a by-product. Performing this analysis prior to an activation-energy study should be required in order to ensure that TE dominates transport across the junction, since that is the regime assumed in the activation-energy study. Otherwise, a too strong dependence of the SBH with the metal-semiconductor bias, even leading to unphysical negative values for the SBH [31]. Further information within the tunneling regime can be obtained with the analysis by Mouafo et al. [32].

We have fitted the $n$-doped structure in Figure 6.b) to Equation (11), obtaining $E_{00}=60.72$ meV and $\eta=1.33$. So, for this case we are in the FE regime, with a small temperature assistance. The $p$-doped case is more complicated. We see in Figure 6.c) how, specially at low temperatures, the curves present two regions with separate temperature dependence; at small bias the $T$ dependence is weak, becoming stronger at bias $\gtrsim 0.25$ V. Figure 6.d) shows $E_{0}$ extracted from a fit in the forward bias [0.3, 0.4] V range, where we see that, at low $T$, we have some contribution of TFE current, overwhelmed by TE current at medium and large temperatures (the fitted parameters are $E_{00}=16.25$ and $\eta=2.04$).

It is desireable to find $E_{00}$ by other means in order to check the consistency of the treatment and have further validation for the claimed transport regime. In the case of a 3D structure and a parabolic barrier, $E_{00}$ can be easily evaluated from the metal-semiconductor (MS) junction parameters, finding that $E_{00}=q\sqrt{N_{D}\hbar^{2}/4\varepsilon_{s}m^{\ast}}$ [29], where $N_{D}$ is the 3D dopant concentration, $\varepsilon_{s}$ is the semiconductor dielectric constant and $m^{\ast}$ is its effctive mass, under the parabolic dispersion assumption. For the 2D MS junction, the barrier profile is no longer parabolic [33], difficulting the obtaining of an analytical expression. However, $E_{00}$ can be numerically estimated noting that the transmission coefficient for carriers coming at the Fermi level $E_{F}$ may be written as [34]:

where $V_{B}$ is the Schottky barrier height in the semiconductor side and $V$ is the applied bias. We have carried out this approach, plotting the transmission coefficient as a function of the applied bias in Figure 7 for the $n,p=5\times 10^{12}$ cm^-2 doping at an incoming electron energy of $E_{F}+\delta E$ eV¹¹1The incoming energy with respect to the Fermi level must be increased (decreased) for electrons (holes) to ensure that the incoming carriers have allowed energies. and fitting to Equation (12), for several $\delta E$. The obtained slopes have been extrapolated to $\delta E=0$ with a cosh function, and from there we have obtained for the $n=5\times 10^{12}$ cm^-2 case $E_{00}=61.0$ meV, in excellent agreement with the value obtained from the temperature analysis, thus corroborating the assignation to the FE regime. For the $p=5\times 10^{12}$ cm^-2 case we have obtained $E_{00}=15.7$ meV, which, being quite smaller than $kT$ at room temperature, corroborates the assignment to the TE regime.

In experiments, Schottky barrier heights ($\Phi_{B}$) are normally obtained through the activation-energy method, which uses the thermionic emission equations to extract $\Phi_{B}$ from the $\ln(I/T^{\alpha})$ vs. $1/T$ plot. While this method can be applied to the intermediate $p$-doped case at high bias, see Figure LABEL:fig:lnTvsT in the SI, where we find an activation energy of 0.28 eV; it is not suitable for the intermediate $n$-doped and low bias $p$-doped cases because the doping concentrations we considered are high enough to observe transport governed by FE or TFE. Therefore, we studied the effect of the gate bias on the Schottky barriers from local density of states (LDOS) plots. Although penetration of the metallic states into the semiconductor gap renders the direct determination of the SBH difficult, we can use the macroscopic average of the Hartree portential, which traces quite closely the conduction and valence band profile [35], to extract the $n$-doped and $p$-doped barrier according to:

where $V_{bulk}$ is the average Hartree potential at the semiconductor side far from the influence of the interface, $V_{i}$ is the average Hartree potential at the interface, and $E_{CBE}$ and $E_{VBE}$ are the conduction band and valence band edges, respectively, extracted from the LDOS. Finally, in order to take into account the case of an applied bias across the junction, we define $E_{F}^{1T^{\prime}}$ and $E_{F}^{2H}$ as the respective Fermi levels.

The LDOS plot for the undoped armchair 2H-1T’ interface at zero bias is shown in Figure 8. From the plot, it can be seen that $\Phi_{B}$ for the hole injection is lower than for electrons, the values being 0.60 eV and 0.97 eV, respectively. This result is in good agreement with other theoretical values reported in literature, where the height of the $p$ and $n$-barriers were found to be 0.71 eV and 0.96 eV, respectively [19].

The LDOS plots for $n$-doped and $p$-doped armchair (zigzag) structures at zero bias are shown in Figure 9 (Figure LABEL:fig:ldos-zz-all, in SI). The barrier heights ($\Phi_{B}$) and the depletion layer width ($W$) extracted from Figures 8 and 9 (Figure LABEL:fig:ldos-zz-all) are presented in Table 1 (Table LABEL:table:barriers-zz). Of course, the depletion layer width is reduced as the doping level increases, leading to the ohmic-like behavior of the contact seen in Figure 5. We also observe that the Schottky barrier decreases when the electrostatic doping increases, and it slightly varies for ac and zz interfaces. This evidence indicates that Schottky-Mott Rule does not apply in these highly doped 2H-1T’ semiconductor-metal juctions, as expected for non-ideal systems [36]. A similar behavior was also observed in DFT studies of a Ag/Si 3D junction, where increase of the semiconductor doping level led to a reduction of the SBH [35] by similar amounts. Note that this dependence of the SBH on the doping level is qualitatively different from that reported in graphene-silicon contacts [37], because there the variation in electrostatic doping was applied to the graphene “metallic” component of the junction, thus changing the metal workfunction. Additionally, our results point to the possibility that the 2H-1T’ junction is free from Fermi level pinning, a possibility already hinted at by Katagiri et al. [38], since the Fermi level position at the interface spans most of the semiconductor gap region (cf. the two 5$\times 10^{13}$ cm^-3 plots in Figure 9 for the two extreme cases). This is opposite to junctions with 3D metals, where pinning of the Fermi level was found [39, 40, 41]. This might open up the possibility of ambipolar injection into MoS₂, similarly to what has been observed in MoTe₂ under weak Fermi level pinning conditions [42].

To understand the effect of the applied voltage on the band alignment of the 2H-1T’ interfaces, we also represented the LDOS of the ac structures under finite bias, as can be seen in Figures 10 and 11 for $n$-doped and $p$-doped structures, respectively. The $\Phi_{B}$ calculated according to Eqs. (13) and (14) are presented in Table 1 for different biases.

In all cases we observe $\Phi_{B}$ slightly reduces (increases) when the 2H-1T’ junction is reverse (forward) biased, again by amounts similar to what was observed in the Ag/Si junction [35], where this variation was attributed to the effect of image forces [30]. For the case of intermediate n(p)-doping concentrations, 5$\times$10¹² cm^-2, when the 2H side is forward biased with respect to the 1T’ phase, V$>$0 (V$<$0) in Figures 10 and 11, the 2H band edge raises (lowers) and the effective barrier is reduced, increasing the current. When the junction is reverse biased the band bending increases, and so does the depletion layer; as result, only a small leakage current flows. On the other hand, when the transport is ohmic (high doping concentrations 5$\times$10¹³ cm^-2), the band bending is almost imperceptible and the current flows independently of the bias polarization in both types of doping.

We calculated the large signal contact resistance ($R_{c}$), relevant for digital applications, of the 2H-1T’ interfaces ($R_{2H-1T^{\prime}}$) at 300 K using Eq. (8) in section 2.2:

where the voltages $V_{2H-1T^{\prime}}$, $V_{2H-2H}$ and $V_{1T^{\prime}-1T^{\prime}}$ have been extracted from the I–V curves shown in Figure LABEL:fig:iv-ref-all, at the same current value $I$. The resulting contact resistances for different electrostatic dopings as a function of $V_{MS/SM}$ are shown in Figure 12. We also show a comparison of the large- and small-signal contact resistances, in Figure LABEL:fig:iv-ref-all, finding a qualitatively similar behavior.

As expected from the I–V curves in section 3.1, positive doping yields lower contact resistance than negative doping by a factor of $\sim\!\!10$ ($\sim\!\!4$) in the intermediate (high) doping case. We also observe that in the intermediate-doping/Schottky case the contact resistance has an exponential dependence with $V_{MS}$, very clear at forward bias. Having established in Sec. 3.1 that the junction operates in the FE (TE) regime for the $n$ ($p$) case, the exponential decrease of $R_{c}$ at forward bias is due to the narrowing and lowering of the barrier seen from the semiconducting side ($V_{bi}$, see Figure 9). In the reverse bias, the lowering of $R_{2H-1T^{\prime}}$ is due to the narrowing of the tunnel barrier for a fixed energy of the carrier coming from the metal side. In the high-doping/ohmic case, a linear drop in $R_{2H-1T^{\prime}}$ is observed at forward bias. This is due to the (linearly) increased transmission as bias is raised.

Houssa et al. also performed calculations of contact resistances for 2H-1T—as opposed to 1T’—lateral heterojunctions but without including electrostatic doping. They obtained $R_{c}$ values on the order of 40 and 30 k${\rm\Omega\!\cdot\!\mu m}$ for armchair and zigzag interfaces, respectively, using the transfer length method [18]. These values are much lower than our results for intermediate doping; however, the differences may be related to the procedure of computation of the $R_{c}$. They used the transfer length method, with semiconductor lengths up to 4.8 nm. As we can see in Figure 2, the depletion width for the undoped case is $>8$ nm, meaning that they treated the fully depleted case, with some amount of indirect doping from the metal contacts.

We also compared our results to experimental measurements of MoS₂-based FETs with metal-semiconductor junctions. Nourbakhsh et al. reported $R_{c}$ values of 1 k${\rm\Omega\!\cdot\!\mu m}$ at gate voltages of 3.5 V, i.e. for sufficiently high negative charge induced in the channel [43]. This result is in agreement with our calculations for negative 5$\times$10¹³ cm^-2 doping concentration at forward bias.

On the other hand, Kappera et al. reported $R_{c}$ values of 0.24 k${\rm\Omega\!\cdot\!\mu m}$ at zero gate voltage [16]. This extreme low value, obtained without inducing any charge in the channel, has been explained by considering the functionalization of 1T’ phase with chemical dopants (H, Li, or H₂O), present during the local transformation of semiconducting 2H phase into metallic 1T’ phase [18].