Low bias negative differential resistance in WSe₂/MoS₂ Planar Superlattice Diodes

Abstract

The development of nanoscale diodes with controllable charge transport and negative differential resistance (NDR) characteristics remains a key challenge in future electronics. In planar superlattices based on two-dimensional materials such as monolayer transition metal dichalcogenides (ML-TMDCs), precise structural parameter control for optimized device performance has not yet to be fully explored. In this study, using an extended transfer matrix method and the Landauer–Büttiker formalism, we analyze the carrier transport behavior and NDR features of a planar superlattice diode composed of WSe₂ and MoS₂ monolayers, which offers a promising platform for next-generation nanoelectronic devices. Our results indicate that the observed NDR behavior is primarily driven by resonant tunneling phenomena and strongly depends on structural parameters such as layer widths, unitcell size, and applied bias voltage. Furthermore, increasing the number of unitcells modifies the carrier transmission probability due to quantum interference effects, which in turn influences the current–voltage response. The ability to engineer the band structure through precise control of layer dimensions opens new opportunities for tunable nanoelectronic applications.

Introduction

Among the most significant 2D materials are monolayer transition metal dichalcogenides (ML-TMDCs) such as ${\text{MoSe}}_{2}$, ${\text{MoS}}_{2}$, ${\text{WS}}_{2}$, and ${\text{WSe}}_{2}$. Compared to other 2D materials, particularly graphene, these materials exhibit superior performance in various applications due to their distinctive physical properties. ML-TMDCs have a direct bandgap ranging from 1 to 2 eV, making them suitable for electronics and optoelectronics applications, such as nanometer-scale diodes and field-effect transistors (FETs), with high efficiency and lower energy consumption^{1,2,3,4,5,6,7,8}. These features are essential for the next generation electronic and optical devices, while graphene as very promising 2D material lacks a natural bandgap, which imposes limitations on various applications. Additionally, ML-TMDCs, due to their direct bandgap, simultaneously exhibit electronic and optical characteristics, enabling them to interact with light and facilitate electron transport simultaneously⁹. In contrast, graphene, despite its high electron mobility, has weak optical properties. An important feature of these materials is their spintronic properties, which are crucial for future quantum and computational technologies^10,11, and they can more easily integrate into semiconductor-based technologies such as CMOS¹², whereas graphene requires further modifications. Therefore, graphene, due to its high electron mobility and exceptional mechanical properties, is an attractive material, but ML-TMDCs offer broader applications and better compatibility with semiconductor technologies in various electronic and optoelectronic applications. For example, in reference¹³, an asymmetric van der Waals heterostructure composed of multilayer ${\text{MoS}}_{2}$ and ${\text{MoTe}}_{2}$ was fabricated. Due to its asymmetric structure, the device operates predominantly through a tunneling mechanism under negative bias, while under positive bias the current is governed by over-barrier thermionic emission and interlayer recombination. Source–drain current measurements demonstrated an on/off current ratio of approximately $6 \times 10^{8}$ and a rectification ratio up to $10^{8}$, indicating a highly efficient and controllable diode behavior in this device. The main objective of this work was to design and construct an asymmetric heterostructure based on TMDCs that can integrate multiple functionalities -including diode, transistor, photodetector, and programmable rectifier- within a single device.

Among quantum effects, resonant tunneling is one of the most important phenomena in superlattices that aims to address the energy consumption challenge in ultra-dense integrated circuit (IC) technologies. The interfaces in superlattices are locations where unique phenomena occur, enabling the enhancement of material potentials for various applications. Therefore, these structures are recognized as promising materials for high-performance nanoelectronics and nanophotonics devices, such as cascade lasers, photodetectors, tunable filters, FETs, memory devices, and resonant tunneling diodes (RTDs)^{14,15,16,17,18}. Unlike vertical heterostructures, which are formed by stacking layers on top of each other, planar superlattice (PSL) results from the side-by-side arrangement of ML-TMDCs on a common horizontal plane. The electronic properties, such as band structure, conductivity, and density of states, depend on geometric parameters of the structure, such as the type of materials in the superlattice, the layer size, and the carrier incidence angle. In planar superlattice made of ML-TMDCs (PSL-TMDCs), electrons can easily move from one region to another due to quantum confinement and defect-free structure, resulting in high transfer rates. Additionally, in structures with very narrow widths (less than 2 nm), the quantum tunneling effect is enhanced, leading to better carrier transport. Ultimately, due to their high electrical conductivity and nanoscale structure, these planar superlattices are ideal candidates for fast and low-power diodes and transistors. Achieving precise growth of lateral heterostructures with sharp atomic junctions and sub-nanometer width layers remains a key research challenge in this field. In addition to significant work done on the synthesis of lateral heterostructures^{19,20,21,22,23,24}, In 2017, a group of researchers successfully created sub-nanometer channels of ${\text{ML}} - {\text{MoS}}_{2}$ within the ${\text{WSe}}_{2}$ sheet using a unique method (Fig. 1a, b), resulting in the formation of PSL with sharp, defect-free junctions²⁵. Following this work, another group achieved similar structures using different ML-TMDCs²⁶. As a result, PSL-TMDCs, from an experimental perspective, are ideal for next-generation nanoelectronics, optoelectronics, and sensor devices due to their similar crystal structures, excellent lattice matching, high structural stability, and sub-nanometer width layers. Specifically, these PSLs are the most suitable option for diode design due to their ability to form precise bonds, bandgap engineering, and high sensitivity and efficiency.

Fig. 1

Structural, electronic, and device modeling of MoS₂ channels embedded in WSe₂ monolayers. (a) Atomic-resolution ADF-STEM image showing periodic dislocations at the WSe₂/MoS₂ interface, responsible for the formation of sub-nanometer MoS₂ channels²⁵; (b) DFT-calculated band structure of the WSe₂–MoS₂–WSe₂ lateral heterojunction, revealing a direct bandgap of ${\text{E}}_{\text{g}}\sim 1.66\text{ eV}$²⁴; (c) Schematic of the PSLD composed of alternating ML–WSe₂ and ML–MoS₂ under an external bias; (d) Band structure of the unbiased superlattice showing the miniband gap (E_g = 1.462 eV).

Besides, it seems that the phenomenon of Negative Differential Resistance (NDR) can be another intriguing feature of PSLs that is directly related to resonant tunneling^{27,28,29,30,31}. NDR occurs when an increase in voltage leads to a decrease in current. On the current–voltage (I-V) curve, this phenomenon is observed as a descending region where the slope of the curve is negative. In superlattice-based devices, the current initially increases at low voltages due to quantum tunneling. However, as the voltage further increases, the energy states in quantum wells no longer remain in resonance, reducing the tunneling probability and consequently causing a decrease in current. This process results in a region where current decreases with increasing voltage, referred to as the NDR region. The position and size of the NDR region depend on the number and dimensions of the layers. The NDR phenomenon can be utilized in designing oscillators, amplifiers, and high-speed memory devices^32,33,34. Studies on vertical heterostructure of black phosphorus and hexagonal boron nitride (h-BN) demonstrate distinct NDR behaviors due to their different band structure characteristics and carrier transport properties. For instance, black phosphorus, owing to its moderate bandgap and high carrier mobility, can exhibit NDR with varying mPVR values; however, its operating voltages are generally higher. For example, in Ref.³⁵, which investigates controllable carrier tunneling in a van der Waals Graphene/Black Phosphorus heterostructure, the reported current is in the order of $10^{ - 5} {\text{A}}$ and no clear NDR behavior is observed in the I–V curve for lower voltages $\left( {{\text{Less than}}\;{ }1.5{\text{ V}}} \right)$. However, at higher biases $({\text{more than }}\;1.5{\text{ V)}}$, a current peak with an mPVR of approximately 2.6 is measured. Similarly, in Ref.³¹, where the transport characteristics of a Boron Nitride/Graphene heterostructure are studied as a function of the structure’s geometric parameters, the peak current for ${\text{V}}_{{\text{g}}} = 0$ is reported to be in the range of ${\text{I}}_{{{\text{peak}}}} = 0.05{-}0.16\left( {2{\text{e}}^{2} /{\text{h}}} \right)$, and the corresponding mPVR values span from 1.06 to 4.2 depending on the geometric configuration. In lateral structures, the NDR region can be more stable and precise compared to vertical structures, as the geometric control and material growth in a two-dimensional plane are more feasible. Thus, resonant tunneling and the NDR phenomenon in PSLs arise from the quantum and dimensional confinement features of these structures. Ultimately, PSLs become attractive options for designing high-speed, low-power nanoelectronics device. Therefore, PSL-TMDCs based diodes (PSLD-TMDCs) are expected to demonstrate optimized performance, smaller size, and improved efficiency. Numerous scientific and experimental studies demonstrate that lateral heterostructures based on two-dimensional materials, particularly TMDCs, are ideal candidates for designing advanced diodes^9,36,37. These studies, while confirming the applicability of these structures, demonstrate their unique potential in enhancing performance diode and reducing the limitations of traditional technologies.

Based on the above-mentioned discussions, in this paper, we will model and analyze the I-V characteristics of a diode based on PSLs using the developed transfer matrix method and the Landauer-Büttiker formalism. The results obtained from this modeling will be presented, and the engineering and control of the transport characteristics of this device will be discussed by geometric parameters of the structure. The results of this study highlight the potential of planar superlattices in the development of low-power, high-efficiency diodes and transistors. Future research can investigate the impact of external perturbations, such as temperature variations and substrate effects, to further optimize the performance of these devices for real-world applications. Additionally, integrating these structures with other two-dimensional heterostructures could lead to novel functionalities in optoelectronic and spintronic applications.

Model and method

In this section, we will discuss on the model structure and the Hamiltonian with its solutions. In Fig. 1c, the model diode based on PSL (PSLD) formed by the lateral stacking of alternating layers of ${\text{ML}} - {\text{WSe}}_{2}$ and ${\text{ML}} - {\text{MoS}}_{2}$ with very thin thickness (on the order of a few angstroms). This arrangement leads to a planar heterostructure, in especial case a planar superlattice, where the electronic and optical properties of the structure can be controlled and engineered based on the geometric parameters. The length of the structure along the growth direction of the PSL is given by $L = W_{A} + N\Lambda$, where the unit cell size of the PSL is defined as $\Lambda = W_{A} + W_{B}$. ${\text{W}}_{{\text{A}}}$ and ${\text{W}}_{{\text{B}}}$ represent the width of the ML-WSe₂ and ML-MoS₂ layers, respectively, and are defined as ${\text{W}}_{{\text{A}}} = {\text{N}}_{{\text{A}}} {\text{a}}_{{\text{A}}}$ and ${\text{W}}_{{\text{B}}} = {\text{N}}_{{\text{B}}} {\text{a}}_{{\text{B}}}$, where ${\text{a}}_{{\text{A}}} \left( {{\text{a}}_{{\text{B}}} } \right)$ is the width and ${\text{N}}_{{\text{A}}} \left( {{\text{N}}_{{\text{B}}} } \right)$ is the number of hexagons forming the monolayer of A (B). In this structure, a sapphire substrate is used, which acts as an insulating base and provides mechanical stability to the structure. Due to the relatively large energy gap of ML-TMDCs, the effect of the substrate on the band structure of these materials is minimal²⁵, and the substrate effects will be neglected in this work. Due to the different energy gaps of layers A and B, small potential barriers form at the interface between these two materials. These barriers limit the movement of carriers and control the current in the structure. The extremely thin layer thickness allows the wavefunction to pass through these potential barriers, a phenomenon known as quantum tunneling, which enables the generation of current even in the presence of potential barriers. By varying the applied external voltage ${\text{V}}_{{\text{b}}}$, the tunneling probability and the amount of current passing through the structure are adjusted such that, with an increase in potential in forward bias, we expect higher current due to the reduction in barriers. These devices, due to the use of tunneling phenomena, can be applied in low-power consumption transistors, and additionally, due to the sensitivity of two-dimensional materials to environmental changes, they can be developed for optical and gas sensors as well as for using in photodetectors.

Figure 1c illustrates of PSLD-TMDCs under an applied external potential. The left electrode (cathode) and right electrode (anode) serve as the two main terminals of the circuit, positioned on either side of the structure and maintaining lateral connectivity. The bias voltage is applied to the system through these electrodes, causing a tilt in the band structure and generating current within the diode. Considering the structure arrangement and the bandgap of each layer, they form a Type-B superlattice. One of the fundamental aspects of any novel condensed matter structure is the study and characterization of its electronic properties. Among these, the band structure is a critical feature that must be analyzed for any system. Accordingly, Fig. 1d focuses on the band structure of the PSL-TMDCs composed of ${\text{ML}} - {\text{WSe}}_{2}$ and ${\text{ML}} - {\text{MoS}}_{2}$, with a unit cell of ${\Lambda } = 5{\text{a}}_{{\text{A}}} + 5{\text{a}}_{{\text{B}}}$, with the spin–orbit coupling (SOC) effects. The electronic characteristics of this structure discussed in detail in reference³⁸. The maximum of the valence band is located at an energy of ${\text{E}}_{{\text{v}}} = - 0.6469{\text{ eV}}$, and the minimum of the conduction band is at ${\text{E}}_{{\text{c}}} = + 0.8151{\text{ eV}}$, resulting in a fundamental energy gap of ${\text{E}}_{{\text{g}}} = 1.4620{\text{ eV}}$. Higher-order minibands within both the valence and conduction bands are also observed.

As mentioned earlier, we assumed that ML-TMDCs are periodically arranged within a single layer, and near the Dirac points, the electronic properties of these materials are described by a two-dimensional pseudo-Dirac Hamiltonian^39,40,41, we will use this equation to describe the transport behavior of PSLD-TMDCs. Metal electrodes on the left and right sides apply a linear electrostatic potential across the PSLD-TMDCs as ${\text{V}}_{{\text{b}}} = {\text{V}}_{{\text{R}}} - {\text{V}}_{{\text{L}}}$. W is the width of the structure, and we assume it is larger compared to L, which is the size of the superlattice in the propagation direction. The applied electrostatic potential ${\text{U}}\left( {\text{r}} \right)$ is considered uniform along the y-axis and varies along the x-axis. The x-component of this potential, which is related to ${\text{V}}_{{\text{b}}}$, is defined as ${\text{U}}\left( {\text{x}} \right) = - {\text{eV}}_{{\text{b}}} {\text{x}}_{{\text{m}}} /{\text{L}}$, where this relation represents the magnitude of the applied potential at the spatial position ${\text{x}}_{{\text{m}}}$. Given that the potential variations across each layer are very small $\left( {{\Delta U}_{{\text{n}}} \approx 1 - 2{\text{ meV}}} \right)$, the potential changes along the PSL will be considered step-like as:

$${\text{U}}_{{\text{n}}} \left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}c} {\frac{{ - {\text{eV}}_{{\text{b}}} }}{{\text{L}}}\left[ {{\text{w}}_{{\text{A}}} + \left( {\frac{{{\text{n}}_{{\text{A}}} - 1}}{2}} \right){\Lambda }} \right] } & {{\text{n}}_{{\text{A}}} = 1, 3, \ldots , N} & {\left[ {W{\text{Se}}_{2} Layers } \right]} \\ {\frac{{ - {\text{eV}}_{{\text{b}}} }}{{\text{L}}}\left[ {\left( {\frac{{{\text{n}}_{{\text{B}}} }}{2}} \right){\Lambda }} \right]} & {{\text{n}}_{{\text{B}}} = 2, 4, \ldots , N} & { \left[ {Mo{\text{S}}_{2} Layers } \right]} \\ \end{array} } \right.$$

(1)

Considering the Dirac-like equation for ML-TMDCs and the electrostatic potential described in Eq. (1), and also incorporating the Sturm–Liouville theorem due to the Hermiticity⁴², the Hamiltonian for the PSL will be expressed as follows:

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H}_{t} = - i\hbar \left[ {\sqrt {v_{F} \left( x \right)} \sigma_{x} \partial_{x} \sqrt {v_{F} \left( x \right)} + v_{F} \left( x \right)\sigma_{y} \partial_{y} } \right] + \frac{\Delta \left( x \right)}{2}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma }_{z} - V\left( x \right)\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma }_{z} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{1} }}{2} + U\left( x \right)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{I}$$

(2)

Here, the Fermi velocity is defined as ${\text{v}}_{{\text{F}}} = {\raise0.7ex\hbox{${{\text{at}}}$} \!\mathord{\left/ {\vphantom {{{\text{at}}} \hbar }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\hbar $}}$, where a is the lattice constant for ML-TMDCs, t is the effective hopping integral, and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma }$ represents the Pauli matrices. Additionally, the definition $V_{soc} = \lambda \tau s$ has been used, where $2{\uplambda }$ is the spin–orbit splitting ${\text{(L}}\cdot{\text{S)}}$, s is the spin index $\left( {{\text{s}} = \pm 1} \right)$, ${\uptau }$ is the index corresponding to the valleys ${\text{K}}$ and $K^{\prime}\left( {\tau = \pm 1} \right)$, and ${\Delta }$ represents the energy gap in ML-TMDCs. This Hamiltonian is solved using the extended or developed transfer matrix method, by considering the periodicity along the x-axis, with the wavefunction described as a plane wave in this direction. The wavefunctions for the sublattices p and q of the ML-TMDC materials are written as ${\uppsi }_{{{\text{p}},{\text{q}}}} \left( {{\text{x}},{\text{y}}} \right) = {\uppsi }_{{{\text{p}},{\text{q}}}} \left( {\text{x}} \right){\text{e}}^{{{\text{ik}}_{{\text{y}}} {\text{y}}}}$. Finally, after lengthy calculations, the reflection coefficient ${\text{r}}\left( {{\text{E}},{\text{V}}_{{\text{b}}} ,{\text{k}}_{{\text{y}}} } \right)$, and transmission coefficient ${\text{t}}\left( {{\text{E}},{\text{V}}_{{\text{b}}} ,{\text{k}}_{{\text{y}}} } \right)$ for PSLD-TMDCs under an external potential are obtained as³⁸:

$$\left\{ \begin{gathered} r\left( {E,V_{b} ,k_{y} } \right) = \frac{{X_{22} P_{0} e^{{ \pm i\theta_{0} }} - X_{11} P_{e} e^{{ \pm i\theta_{e} }} - X_{12} P_{0} P_{e} e^{{ \pm i(\theta_{e} + \theta_{0} )}} + X_{21} }}{{X_{22} P_{0} e^{{ \mp i\theta_{0} }} + X_{11} P_{e} e^{{ \pm i\theta_{e} }} - X_{12} P_{0} P_{e} e^{{ \pm i(\theta_{e} - \theta_{0} )}} - X_{21} }} \hfill \\ t\left( {E,V_{b} ,k_{y} } \right) = \frac{{2P_{0} \cos \theta_{0} }}{{X_{22} P_{0} e^{{ \mp i\theta_{0} }} + X_{11} P_{e} e^{{ \pm i\theta_{e} }} - X_{12} P_{0} P_{e} e^{{ \pm i(\theta_{e} - \theta_{0} )}} - X_{21} }} \hfill \\ \end{gathered} \right.$$

(3)

The transmission probability dependent on the indices ${\text{s}}_{{\text{z}}}$ and ${\uptau }$, with the energy and the incidence angle of the carriers as ${\text{E}}$ and ${\uptheta }_{0}$, reads²⁹:

$$T_{\tau }^{{s_{z} }} \left( {E,V_{b} } \right) = \int\limits_{{ - {\raise0.7ex\hbox{$\pi $} \!\mathord{\left/ {\vphantom {\pi 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}^{{{\raise0.7ex\hbox{$\pi $} \!\mathord{\left/ {\vphantom {\pi 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} {\left| {t\left( {E,V_{b} ,k_{y} } \right)} \right|^{2} \cos \alpha d\alpha }$$

(4)

In PSLD-TMDCs under a bias potential ${\text{V}}_{{\text{b}}}$, and when the valence band of the superlattice is fully occupied and the conduction band is empty, we can obtain the current by inserting ${\text{T}}_{{\uptau }}^{{{\text{s}}_{{\text{z}}} }}$ into the Landauer-Buttiker formula^29,43,44, which will be expressed as follows:

$$I_{SD} = I_{0} \int\limits_{{ - \mu_{S} }}^{{ + \mu_{D} }} {dE\left| E \right|T_{\tau }^{{s_{z} }} \left( {E,V_{b} } \right)}$$

(5)

The constant factor ${\text{I}}_{0} = {\raise0.7ex\hbox{${2{\text{geW}}}$} \!\mathord{\left/ {\vphantom {{2{\text{geW}}} {{\text{v}}_{{\text{F}}} {\text{h}}^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{\text{v}}_{{\text{F}}} {\text{h}}^{2} }$}}$ is given, and considering the spin and valley degeneracy, the degeneracy factor for this structure is ${\text{g}} = 4$.

Results and discussions

It is worth mentioning that throughout this report, all studies will be carried out for valley $\text{K}$ and the spin-up state (${\text{S}}_{\text{up}}$), by considering SOC. Additionally, the unitcell size and the length of the PSL structure are defined as ${\Lambda } = 5{\text{a}}_{{\text{A}}} + 5{\text{a}}_{{\text{B}}}$ and ${\text{L}} = {\text{W}}_{{\text{A}}} + 82{\Lambda }$, respectively, unless explicitly stated. The study of the physics related to transmission probability in diodes based on PSL is carried out due to its scientific and practical importance. This investigation helps to understand electron behavior in multilayer structures and control current in nanoscale devices. In the ${\text{ML}} - {\text{WSe}}_{2}$ and ${\text{ML}} - {\text{MoS}}_{2}$ semiconductor materials, the band structure is such that electron transfer does not occur easily unless the electrons possess sufficient energy to overcome the bandgap. The PSL structure comprising these two semiconductors leads to the formation of new bands and different energy states, such characteristics can be controlled and engineered through the structural parameters of the superlattice, and consequently, the electron transport behavior can be modified. Therefore, analyzing the carrier transmission probability in this structure provides information about the location and size of the bandgap and the density of states distribution. At the nanoscale, one of the critical phenomena in carrier transport is tunneling. The transport characteristics in nanoscale PSLs can also be utilized for precise control of electronic states in the quantum scale. These features are of great significance in quantum computing, as the ability to control electron transfer and create specific quantum states can lead to designing qubits with tailored characteristics.

In Fig. 2a, the behavior of transmission probability spectrum for different voltages in the PSLD-TMDCs structure is examined. ${\text{V}}_{SD} = 0,{ }50,{ }100,{ }200,500{\text{ mV}}$, represented by red, yellow, green, blue, and black, respectively. According to this figure, we observe that the band structure, as well as the position and size of the bandgap, play a significant role in the transport behavior of the structure, Therefore, the transmission probability at the energy gap is completely zero. Based on the curves in Fig. 2b , c, we observe that as the voltage increases, the valence band is affected more than the conduction band. Specifically, with a voltage increase of ${\Delta V}_{{\text{b}}} = 500{\text{ mV}}$, the valence band edge (conduction band edge) is shifted by ${\Delta E}_{{\text{v}}} = 582{\text{ meV}}$ $\left( {{\Delta E}_{{\text{c}}} = 27{\text{ meV}}} \right)$ toward lower energies (redshift), Consequently, the bandgap increases as $\Delta E_{g} = 555{\text{ meV}}$ approximately proportional to the applied voltage. It is evident in Fig. 2d that despite the bandgap increase with the applied potential $\left( {{\text{V}}_{b} } \right)$, the higher-order minibands overlap, causing the higher bandgaps disappears that leads to an increase in transmission probability of carriers at higher applied potentials.

Fig. 2

(a) Transmission probability behavior of the PSLD-TMDCs under different external voltages, (b) conduction band of (a), (c) valence band of (a), (d) transmission behavior as a function of external voltage and carrier energy. Such that the unitcell and length of the superlattice are equal to $\Lambda =5{a}_{A}+5{a}_{B}$ and $L={W}_{A}+82\Lambda$, respectively.

In this study, the current–voltage behavior of PSLD-TMDCs for low voltages in the range of $- 0.5{\text{ V}} < {\text{V}}_{{\text{b}}} < 0.5{\text{ V}}$, is investigated. These types of diodes, which are based on 2D materials, are expected to exhibit different behavior compared to conventional diodes. In Fig. 3a, the current behavior is shown over a wider voltage range, where several significant features can be observed for a forward bias voltage (0 to 500 mV), which comes from various physical effects in the diode structure. As observed, for a voltage change of $\Delta {\text{V}}_{{\text{b}}} = 500{\text{ mV}}$, the current change is approximately $\Delta {\text{I}} = 0.5{\text{ mA}}$, whereas in graphene superlattices, these changes have been reported in the microampere range^43,44. Therefore, the current in the proposed structure is nearly two orders of magnitude larger than in graphene-based structures. The general behavior of the curve in this range indicates an increase in current with increasing voltage, due to the reduction in the potential barrier and the facilitation of carrier transport via various mechanisms such as tunneling and carrier injection. In this structure, at low voltages, transport mainly occurs through quantum tunneling, and as the voltage increases, injected carriers contribute to the increase in current. Another significant factor in the current increase is the unique properties of the 2D materials ${\text{ML}} - {\text{WSe}}_{2}$ and ${\text{ML}} - {\text{MoS}}_{2}$ in the design of this structure. One of the key characteristics of ML-TMDCs is the high electrical conductivity of these materials, and their quantum properties, such as high carrier mobility and strong spin–orbit coupling, can also influence the behavior of this curve. Another important consideration in PSLs composed of ${\text{ML}} - {\text{WSe}}_{2}$ and ${\text{ML}} - {\text{MoS}}_{2}$ is that the layer thickness is in the nanometer range. Therefore, in the study of the band structure of this superlattice, mini-bands are formed. These mini-bands can be controlled and engineered based on the structural parameters of the superlattice. Additionally, with carrier injection through voltage bias, these mini-bands will overlap. Mini-bands overlap leads to several important physical phenomena, including resonant tunneling, which is used in resonant tunneling diodes (RTDs) or quantum lasers. Furthermore, when carriers move from one mini-band to another, a sudden abrupt change in current occurs, which causes the emergence of a negative differential resistance (NDR) region, Formation of several small peaks, or oscillatory current behavior with details are depicted in Fig. 3b. As seen in this figure, in certain parts of the current curve, we observe that as the applied voltage increases, the current starts to decrease after reaching its peak, indicating the presence of a negative differential resistance region. In nonlinear electronic devices, the current–voltage curve does not follow Ohm’s law, and in fact, NDR is a unique property that typically appears in such devices. From figure, three regions with negative differential resistance are indicated by red loops at ${\text{V}}_{{\text{b}}} = 37{\text{ mV}}$, ${\text{V}}_{{\text{b}}} = 54.5{\text{ mV}}$, and ${\text{V}}_{{\text{b}}} = 64.5{\text{ mV}}$. Generally, an electronic device with an NDR characteristic has the ability to convert the applied DC power into AC power and also amplify the applied AC signal⁴⁵. This physical characteristic is typically used in electronic oscillators, amplifiers, as well as in memory and switching circuits^46,47,48. The criterion used to measure the effectiveness of the NDR phenomenon in devices is the maximum peak-to-valley ratio (mPVR) in the I-V curve, electronic device designers aim to achieve the largest mPVR because in these conditions the device exhibits better NDR feature. This parameter represents the sensitivity of current to voltage at different points and is often used to assess switching performance and diode response in the circuits. In the diagram, for the three specified regions, the mPVR values are 1.61, 1.30, and 1.37, respectively. In reference⁴⁹, NDR was observed in black phosphorus (BP)-based FETs, attributed to the ambipolar behavior of BP and strong gate–channel coupling. The study showed that both the peak voltage and PVR can be effectively tuned by the gate voltage, with source–drain currents reaching the milliampere range, comparable to the structure investigated here. In our work, however, NDR and PVR are instead engineered through the geometrical parameters of the device, offering new opportunities for high-speed and multivalued electronics based on two-dimensional materials.

Fig. 3

(a) The I-V curve of the PSLD-TMDCs, where the unitcell of the structure is considered as $\Lambda =5{\text{a}}_{\text{A}}+5{\text{a}}_{\text{B}}$, and the length of the structure is taken as $\text{L}={\text{W}}_{\text{A}}+82\Lambda$, (b) Part of the zoom of Figure (a).

In the voltage range of ${\Delta V}_{{\text{b}}} = 0{\text{ to}} - 500{\text{ mV}}$, the I-V curve exhibits interesting features related to the specific PSLD-TMDCs structure and the governing physical phenomena. In the range ${\Delta V}_{{\text{b}}} = 0{\text{ to}} - 100{\text{ mV}}$, the current gradually increases. This behavior is caused by resonant tunneling and energy level alignment phenomena, and around ${\text{V}}_{{\text{b}}} = - 150{\text{ mV}}$, the current curve begins to decrease, which could be due to changes in quantum state alignment or reduced tunneling efficiency. In the following we will study the control and engineering of the transport characteristics of model diode, such as current value, mPVR position and size, based on the geometric parameters of PSL-TMDCs.

One of the most important structural parameters of superlattices that can affect electronic and transport characteristics is the number of unitcells in the superlattice, or in other words, the length of the structure. In Fig. 4, the red, yellow, green, blue, and red curves correspond to the number of unitcells $N = 5,{ }25,{ }50,{ }75,{ }100$ $\left( {L = 17.91,{ }82.94,{ }164.23,{ }245.52,{ }326.81{ }nm} \right)$, respectively. According to a physical principle, the band structure is a fundamental characteristic of each system, and in superlattice structures, with the increase in N is expected to result in no change in the band structure. However, in this structure, since there is an external bias voltage, this can affect each layer. Consequently, because all the unitcells experience different potentials, therefore, by changing N, we will observe changes in the physical characteristics of the device, particularly in carrier transport. As shown in Fig. 4b–d, we observe that with an increase in the length of the structure, the edges of the valence and conduction bands do not show any shifts. Additionally, in some regions of the energy range, we notice that with the increase in N, the transmission probability decreases, which can be attributed to quantum interference and increased scattering resulting from the increased number of interfaces. In Fig. 4a, the transmission is plotted as a function of external potential and N at energy ${\text{E}} = + 0.83{\text{ eV}}$ within the conduction band. It is observed that with the increase in applied voltage and N at low voltages, the carrier transmission probability increases.

Fig. 4

(a) Carrier transmission probability as a function of ${\text{V}}_{\text{b}}$ and N, (b, c, d) Carrier transmission probability as a function of the number of superlattice unitcells, (e, f) I-V characteristics of the PSLD-TMDCs with an applied voltage of ${\text{V}}_{\text{b}}=100\text{ meV}$ and $\Lambda =5{\text{a}}_{\text{A}}+5{\text{a}}_{\text{B}}$ for different numbers of unitcells.

In the I-V curves shown in Fig. 4e, the increase in current with the increase in N is clearly visible. This increase in current can be attributed to several physical reasons. As mentioned before, with an increase in voltage, the potential barriers are decreased or lowered (see Fig. 1), allowing more carriers to transfer that consequently increases the current. However, increasing the number of unitcells provides more possible pathways for carrier transport, leading to enhanced resonant tunneling probability, which can increase the total current, even if the transmission probability in each individual path decreases. Moreover, in two-dimensional materials such as ${\text{ML}} - {\text{WSe}}_{2}$ and ${\text{ML}} - {\text{MoS}}_{2}$, the density of states is highly dependent on the dimensions and the number of layers of the planar superlattice, and this can lead to an increase in the number of accessible states, which in turn may contribute to an increase in the total current. Finally, as previously mentioned, with the increase in the number of periods in the structure and the increase in scattering sources, the transmission probability decreases. However, we observe that this scattering does not necessarily result in a decrease in total current, because we will have an increase in the number of carriers with the increase in the length of the structure. Therefore, the observed behavior is the result of a complex tradeoff between various physical phenomena.

In this section, the transport behavior of the diode is analyzed based on different ratios of the layer widths that make up the unitcell. As mentioned earlier, the unitcell size of the superlattice is defined as $\Lambda = N_{A} a_{A} + N_{B} a_{B}$. In this analysis, the total number of hexagons forming each unitcell is kept constant at ${\text{N}}_{{\text{A}}} + {\text{N}}_{{\text{B}}} = 10$, while only ${\text{N}}_{\text{A}(\text{B})}$ changes. For example, when the unitcell is set to $\Lambda =9{\text{a}}_{\text{A}}+1{\text{a}}_{\text{B}}$ ($\uplambda ={\text{N}}_{\text{A}}:{\text{N}}_{\text{B}}=9:1$), it means that the hexagon number forming layer A ($\text{ML}-{\text{WSe}}_{2}$) is nine, while the width of layer B ($\text{ML}-{\text{MoS}}_{2}$) comprises just one hexagon. According to Fig. 5a, when λ changes, the conduction band edge shifts slightly for both ${\text{V}}_{\text{b}}=0$ and ${\text{V}}_{\text{b}}=100\text{ meV}$, but the conduction band edge significantly shifts towards lower energies (red-shift), resulting in a larger bandgap for the structure. This increase in the bandgap can be attributed to two physical factors. Firstly, when the width of layer A is greater than that of layer B, knowing that the hexagonal size of $\text{ML}-{\text{MoS}}_{2}$ (${\text{a}}_{\text{B}}=3.310\text{ \AA }$) is smaller compared to $\text{ML}-{\text{WSe}}_{2}$ (${\text{a}}_{\text{A}}=3.193\text{ \AA }$), the unitcell size will be decreased (as shown on the upper x-axis of the diagram). Therefore, according to a fundamental physical principle with the reduction of Λ, the bandgap is increased. The second reason for the bandgap increase is that when the width of layer B relative to layer A increases, $\text{ML}-{\text{MoS}}_{2}$ becomes the dominant material in the unitcell, and since the bandgap of this layer $({\Delta }_{{\text{MoS}}_{2}}=1.66\text{ eV}$) is larger than that of $\text{ML}-{\text{WSe}}_{2}$ (${\Delta }_{{\text{WSe}}_{2}}=1.60\text{ eV}$), that increases the bandgap of the superlattice . In Fig. 5b, the plot representing the transmission probability of carriers as a function of energy for different λ values as $\uplambda =9:1, 7:3, 4:6\text{ and }2:8$ corresponding to the red, green, blue, and black curves, respectively. Figure 5c, d present magnified views of the conduction and valence bands, respectively. Based on these figures, we observe the changes in the band edges in these curves, indicating that carrier transmission probability is strongly dependent on the band structure of the superlattice. Additionally, the intensity and position of the transmission peaks are affected by different λ values. For instance, in Fig. 5d, within the valence band energy range, increasing the ${\text{MoS}}_{2}$ layer contribution shifts the mini-band edge towards lower energies. However, the transmission probability for different energies exhibits varying behaviors. Specifically, at the energy $\text{E }= -0.8\text{ eV}$, the blue curve experiences the greatest reduction in transmission, while at $\text{E }= -1.0\text{ eV}$, the green curve shows the greatest decrease. Generally, for the red curve, since the diode experiences the smallest bandgap, the highest transmission probability for various energies is observed.

Fig. 5

(a) Band structure of PSL-TMDCs versus variation in layer width (${\text{N}}_{\text{A}}:{\text{N}}_{\text{B}}$); (b) Carrier transmission probability at ${\text{V}}_{\text{b}}=100\text{ meV}$ and ${\text{N}}_{\text{A}}+{\text{N}}_{\text{B}}=10$; (c, d) Conduction and valence band of the transmission spectrum in panel (b), respectively; (e, f) I–V characteristics of the PSLD-TMDC with varying layer widths.

In Fig. 5e, the I-V characteristic of this device is studied for different values of λ. According to this figure, we observe that for a forward bias voltage of ${\Delta \text{V}}_{\text{b}}=+500\text{ mV}$, the current increases by ${\Delta \text{I}}_{\text{bias}}=+0.53\text{ mA}$, and for a reverse bias voltage of ${\Delta \text{V}}_{\text{b}}=-500\text{ mV}$, the current changes by ${\Delta \text{I}}_{\text{bias}}=-0.13\text{ mA}$. Moreover, at lower voltages (${\Delta \text{V}}_{\text{b}}=\pm 130\text{ mV}$), the current changes are almost identical for different λ values, but at higher voltages, due to the smaller bandgap of the superlattice for the red curve and the relatively easier tunneling that occurs, it shows larger current variations. Figure 5f shows a small part of the I-V curve, where various oscillations are clearly visible due to different quantum tunneling phenomena in the superlattice. We also observe that with an increase in the contribution of the ${\text{MoS}}_{2}$ layer (decrease in the unitcell size), which leads to different carrier transmission probabilities, the oscillations and mPVR become larger, and the NDR phenomenon becomes more pronounced, resulting in larger peaks observed in the black curve. Therefore, we can conclude that when $\text{ML}-{\text{MoS}}_{2}$ is the dominant material within a unitcell, the I-V curve exhibits more nonlinear behavior and more pronounced NDR characteristics.

In such diodes, which are designed based on superlattice technology, one of the key structural parameters that can control their transport behavior is the width of the layers forming the superlattice. In this section, we will change the number of hexagons forming each layer $\left( {{\text{N}}_{{{\text{A}}\left( {\text{B}} \right)}} } \right)$, thereby altering the size of the superlattice unitcell, which in turn changes the length of the structure. In Fig. 4, the change in structure length was due to an increase in the number of unitcells, while the size of ${\Lambda }$ remained constant. However, in this section, the change is due to variations in the layer widths, or in the other words, changes in the size of ${\Lambda }$, while the number of unitcells remains constant (${\text{N}} = 50$). In Fig. 6a, we observe the band structure behavior and the bandgap size of the superlattice with respect to changes in layer widths. It is worth to note that these changes do not significantly affect the conduction band edge, but the valence band edge shifts towards higher energies (blue-shift), resulting in a reduced bandgap. Consequently, as the unitcell size increases from ${\Lambda }\left( {2:2} \right) = 1.3{\text{ nm}}$ to ${\Lambda }\left( {10:10} \right) = 6.5{\text{ nm}}$, the bandgap decreases by $\Delta {\text{E}}_{{\text{g}}} = 51{\text{ meV}}$.

Fig. 6

(a) Band structure of PSL-TMDCs versus variation in superlattice unitcell size ($\Lambda$); (b) Carrier transmission probability at ${\text{V}}_{\text{b}}=100\text{ meV}$ and $\text{N}=50$; (c, d) Conduction and valence band of the transmission spectrum in panel (b), respectively; (e, f) I–V characteristics of the PSLD-TMDCs with varying superlattice unitcell size.

Figure 6b shows the carrier transmission probability as a function of energy for different values of Λ, and Fig. 6c, d display a small part of the transmission curve within the conduction and valence band energy ranges, respectively. The changes in band structure lead to variations in the intensity and position of the peaks in these curves, so that in Fig. 6d, we observe that with the unitcell size increase, the conduction band edge shifts towards higher energies (peaks in the energy range of ${\text{E}} = - 0.72{\text{ eV}}$), while the transmission probability decreases, but for higher energies, an increase in transmission probability is noticeable in both the valence and conduction bands. In the I-V curve of Fig. 6e, one can see an increase in current with the increase in unitcell size in both forward and reverse bias voltage. The main reason for this increase is the reduced bandgap, which facilitates enhanced tunneling. Another contributing factor could be the increased carrier transmission probability across a wide range of energies. In Fig. 6f, which focuses on a small part of the current–voltage curve, we observe that as the size of Λ increases, the diode exhibits more nonlinear behavior. Notably, in the black and blue curves, larger peaks are observed, indicating better NDR characteristics. Therefore, these predictable effects are significantly more nonlinear and complex compared to conventional semiconductor diodes and can be utilized in designing diodes with specialized behaviors for nanoelectronic and nanophotonic applications.

Conclusion

In conclusion, our investigation confirms that planar superlattices based on ML-TMDCs exhibit strong negative differential resistance (NDR) characteristics at low bias, making them highly promising for next-generation nanoelectronic devices. The observed NDR behavior is primarily driven by resonant tunneling effects, which are strongly dependent on the structural parameters of the superlattice, such as layer width, unitcell size, and applied voltage. Through our theoretical modeling using the transfer matrix method and the Landauer-Büttiker formalism, we have analyzed the influence of geometric parameters on the transport properties of these diodes. Our findings indicate that increasing the number of unitcells leads to modifications in the transmission probability due to quantum interference effects, which in turn affects the overall current–voltage response. Furthermore, the ability to engineer the band structure through precise control of layer dimensions opens new opportunities for tunable nanoelectronic applications.