Multiple Control of Few-layer Janus MoSSe Systems

In this computational work based on density functional theory we study the electronic and electron transport properties of asymmetric multi-layer MoSSe junctions, known as Janus junctions. Focusing on 4-layer systems, we investigate the influence of electric field, electrostatic doping, strain, and interlayer stacking on the electronic structure. We discover that a metal to semiconductor transition can be induced by an out-of-plane electric field. The critical electric field for such a transition can be reduced by in-plane biaxial compressive strain. Due to an intrinsic electric field, a 4-layer MoSSe can rectify out-of-plane electric current. The rectifying ratio reaches 34.1 in a model junction Zr/4-layer MoSSe/Zr. This ratio can be further enhanced by increasing the number of MoSSe layers. In addition, we show a drastic sudden vertical compression of 4-layer MoSSe due to in-plane biaxial tensile strain, indicating a second phase transition. Furthermore, an odd-even effect on electron transmission at the Fermi energy for Zr/$n$-layer MoSSe/Zr junctions with $n=1, \, 2,\, 3, \,\dots,\, 10$ is observed. These findings reveal the richness of physics in this asymmetric system and strongly suggest that the properties of 4-layer MoSSe are highly tunable, thus providing a guide to future experiments relating materials research and nanoelectronics.


I. INTRODUCTION
Two-dimensional (2D) transition metal dichalcogenides (TMDs) have potential applications in electronics/optoelectronics [1,2], due to the presence of a direct band gap and sufficiently high mobility. The computed band gaps of TMDs vary by ∼ 2 eV, depending on the chemical composition [3], which provides rich opportunities for different applications.
Since the electron affinity of S is lower than that of Se, there is electron accumulation on the S side of monolayer MoSSe. Consequently, monolayer MoSSe possesses an intrinsic out-of-plane electric dipole, pointing from S to Se. Multilayer MoSSe can have a potential buildup in the out-of-plane direction if the electric dipoles of individual layers are aligned in the same direction. In this case, the band gap of multilayer MoSSe decreases with the number of MoSSe layers n, until it closes at n = 4 [17]. Since it is the intrinsic electric field that causes the band gap closing in 4-layer MoSSe, we wondered whether the band gap can reopen upon the application of a compensating external electric field. It is not a surprise, as we will show in the results section, that a metal to semiconductor transition in 4-layer MoSSe can be induced by an external out-of-plane electric field. However, the critical electric field that is required to induce the metal to semiconductor transition in 4-layer MoSSe is much smaller than that for a semiconductor to metal transition in bilayer MoS 2 [18]. In the context of a field effect transistor, an external electric field can be applied via a dual gate configuration, whereas a single gate configuration induces electrostatic doping. We will also show how the electronic structure of 4-layer MoSSe is affected by electrostatic doping.
Furthermore, the internal electric field within 4-layer MoSSe may allow rectification of an out-of-plane electric current, which sets another research goal of this study.
Guo and Dong showed that the band gap of monolayer MoSSe can be significantly tuned by in-plane biaxial strain. [19] Strain engineering of the electronic structure of heterogeneous bilayers MoSSe/WX 2 (X=S, Se) [20] and MoSSe/WSSe [21] have also been reported. These discoveries motivate us to investigate the influence of in-plane biaxial strain on the critical electric field for the metal to semiconductor transition. In addition, we will also examine the influence of out-of-plane pressure on the critical electric field.
The rest of the paper is organized as follows. In Section II, we present the computational methods and simulation parameters. We then show our computational results in Section III, which is further divided into subsections concerning the atomic structure, external electric field, electrostatic doping, strain, and rectifying effect respectively. Finally, conclusions are given in Section IV.

II. METHOD
Our calculations are based on density functional theory (DFT) [22,23] as implemented in the SIESTA package [24]. The effective screening medium (ESM) method is used to simulate the effects of out-of-plane electric field and electrostatic doping [25,26]. The DFT + NEGF method [27][28][29] is used to simulate electron transport properties of a Zr/4-layer MoSSe/Zr junction. We calculate the electron transmission and the electric current via the Caroli formula [30] and the Landauer formula [31,32] respectively.
We apply a double-ζ polarized (DZP) basis set [24] to expand the Kohn-Sham orbitals and the electron density. A mesh cutoff of 150 Ry is set to sample real space. A 15 × 15 Monkhorst-Pack k-point mesh [33] is chosen to sample the 2D reciprocal space for ionic relaxations. For self-consistent calculations, the k-point mesh is increased to 21 × 21 to guarantee convergence. The k-point mesh is further increased to 101 × 101 for calculating electron transmission. We adopt norm-conserving pseudo-potentials as generated by the Troullier-Martins scheme [34] and the Perdew-Burke-Ernzerhof (PBE) exchange correlation energy functional [35]. The Van der Waals interaction is taken into account via the DFT-D2 method [36]. The atomic structure of 4-layer MoSSe is fully optimized in all our calculations except for those under finite bias, since the atomic force under non-equilibrium conditions may not be reliable. The numerical tolerances for the density matrix, the Hamiltonian matrix, and the atomic force are no larger than 1 × 10 −4 , 2 × 10 −3 eV, and 0.02 eV/Å respectively.

III. RESULTS
We present our results in five parts. In Section III A, we first display our atomic structure of 4-layer MoSSe. Next, in Section III B we show a metal to semiconductor transition in 4-layer MoSSe induced by an out-of-plane external electric field. Third is Section III C, on the effects of electrostatic doping due to a single back gate. Fourth, we examine how the critical electric field for the metal to semiconductor transition is affected by in-plane biaxial strain in Section III D. Last, we demonstrate that 4-layer MoSSe can rectify out-of-plane electric current in Section III E. stacking, which is energetically more favorable than on-top site interlayer stacking [37,38].
There are two types of hollow site interlayer stacking, as shown on the right in Fig. 1.
Type 1 hollow site interlayer stacking (h1 ) is slightly higher in energy than type 2 hollow site interlayer stacking (h2 ) [37,38]. In this study, we focus on the 4-layer MoSSe with a stacking sequence of h1 -h2 -h1 . Nevertheless, we also verify our major findings for h1 -h1 -h1 stacking 4-layer MoSSe and h2 -h2 -h2 stacking 4-layer MoSSe.  . The sulfur side faces the top gate. Compared with the h1 -h2 -h1 stacking, the h1 -h1 -h1 (h2 -h2 -h2 ) stacking has a larger (smaller) critical field in magnitude. In order to understand such a difference, we analyse the internal potential buildup between adjacent MoSSe layers. In practice, we take the planar average of the electrostatic potential over the 2D plane parallel to the Janus MoSSe.
Denoting the plane averaged electrostatic potential at the vertical position of a Se/S atom as V Se/S , we find that the potential buildup V Se − V S at the three van der Waals gaps of the 4-layer MoSSe with h1 -h2 -h1 stacking are 1.39 eV (h1 ), 1.22 eV (h2 ), and 1.44 eV (h1 ) respectively. As such, the potential buildup at a van der Waals gap between two MoSSe layers with h1 stacking is larger than that for h2 stacking by 0.17/0.22 eV. A larger internal electric field requires a larger external electric field to compensate and to open a band gap around the Fermi level. Therefore, the critical electric field for the h1 -h1 -h1 (h2 -h2 -h2 ) stacking is larger (smaller) than that for the h1 -h2 -h1 stacking.

C. Electrostatic doping
In the previous section, we have shown that a metal to semiconductor transition can be induced by an out-of-plane electric field when the system is charge neutral. In this section, we examine the effects of electrostatic doping (charging) due to a back gate. the results in the previous section. Fig. 4b (Fig. 4c)   The red squares in Fig. 6 show the calculated electric current density J in the Zr/4-layer MoSS/Zr junction as a function of bias voltage. The bias voltage V b is defined by we see that η(0.02 V) is as small as 1. Particularly, T 10 (E) is more than 10 orders of magnitude smaller than T 1 (E) within this energy range. In contrast, T n (E) is always above 10 −2 (and smaller than 1) regardless of n for E ∈ [0.5 : 0.8] eV. Thus, we infer from the zero bias transmission function that thicker MoSSe should exhibit a higher rectifying ratio. We verify this idea using a Zr/5-layer MoSSe/Zr junction and obtain the higher value η = 85.6 at V b = 5.5 V.
The electron transmission at Fermi energy decays differently with the number of MoSSe layers n depending on the parity of n. This odd-even effect originates from the paritydependent potential buildup at the contact between the n-layer MoSSe and the right Zr electrode. Further analysis of this phenomenon is presented in Appendix C.

IV. CONCLUSION
We have performed a detailed study of the electronic and electron transport properties of n-layer MoSSe (n = 1−10) with focus on n = 4 using first principles based density functional theory. Several significant findings obtained are as follows: 1) We show that a metal to semiconductor transition can be induced by an out-of-plane electric field; 2) Sensitive to the stacking order, the critical electric fields for the 4-layer MoSSe with h1 -h1 -h1 , h1 -h2 -h1 , and h2 -h2 -h2 stackings are, respectively, −0.252, −0.187, and −0.145 V/Å; 3) The critical electric field for the h1 -h1 -h1 stacking is highest (in magnitude) because the internal potential buildup with a h1 stacking is larger than that with a h1 stacking. We can reduce        Fig. C1 shows the electron transmission at the Fermi energy T n (E F ) for Zr/n-layer MoSSe/Zr junctions under zero bias with n = 1, 2, 3, . . . , 10. For simplicity, we will write T n (E F ) as T n in the rest of this section. As seen from the figure, T n decays differently for even n and odd n. When n ≥ 5, T n can be well fit by a function of the form T n = λe −γn , T o n ≈ 9.44 × 10 −2 e −2.39 for odd n and T e n ≈ 3.75 × 10 −1 e −2.43n for even n. The fitted curves are also shown in Fig. C1. Interpolating, we see that T o n < T e n at the same number of layers n, or the same width of an effective tunneling barrier. As such, we infer that the height of the effective tunneling barrier in odd junctions is higher than in even junctions. There are two possible reasons for such a difference in the height of the tunneling barrier.
The first one lies in the n-layer MoSSe itself. Recall that the potential buildup at each van der Waals gap is stacking-dependent and that the potential buildup at an h1 stacking is larger. Given that the number of h1 stackings is greater (lesser) than the number of h2 stackings by one in even (odd) junctions, we expect that even junctions have a higher effective barrier. However, this is contradictory to the fact that T o n < T e n . The second possible reason pertains to the contact between the n-layer MoSSe and the Zr electrodes. The contact between the left Zr electrode and the n-layer MoSSe is the same for all the junctions, while the contact on the right side is different for even and odd junctions, as depicted in Figs. C2a and C2b. As a consequence, there is a parity-dependent potential difference between the adjacent Zr and S atomic layers as shown in Fig. C2c. For n ≥ 5, the odd junctions possess an ∼ 0.25 eV higher potential difference V Zr − V S than the even junctions. It seems that the potential buildup at the right contact competes with the potential buildup at van der Waals gaps within a multilayer MoSSe, with the former being dominant. Therefore, the effective tunneling barrier of odd junctions is higher than that of even junctions. It is worth mentioning that the number of electrons at the right contact also exhibits an oscillatory behavior for n ≥ 5, as shown in Fig. C3.