Electronic Barriers Behavioral Analysis of a Schottky Diode Structure Featuring Two-Dimensional MoS₂

Abstract

This research presents a comprehensive study of a Schottky diode fabricated using a gold wafer and a bilayer molybdenum disulfide (${\mathrm{MoS}}_2$) film. Through detailed simulations, we investigated the electric field distribution, potential profile, carrier concentration, and current–voltage characteristics of the device. Our findings confirm the successful formation of a Schottky barrier at the Au/${\mathrm{MoS}}_2$ interface, characterized by a distinct nonlinear I–V relationship. Comparative analysis revealed that the Au/${\mathrm{MoS}}_2$ diode significantly outperforms a traditional W/Si structure in terms of rectification performance. The Au/${\mathrm{MoS}}_2$ diode exhibited a current density of 1.84 × ${10}^{-9}$ A/${\mathrm{cm}}^2$, substantially lower than the 3.62 × ${10}^{-5}$ A/${\mathrm{cm}}^2$ in the W/Si diode. Furthermore, the simulated I–V curves of the Au/${\mathrm{MoS}}_2$ diode closely resembled the ideal diode curve, with a Pearson correlation coefficient of approximately 0.9991, indicating an ideality factor near 1. A key factor contributing to the superior rectification performance of the Au/${\mathrm{MoS}}_2$ diode is its higher Schottky barrier height of 0.9 eV compared to the 0.67 eV of W/Si. This increased barrier height is evident in the band diagram analysis, which further elucidates the underlying physics of Schottky barrier formation in the Au/${\mathrm{MoS}}_2$ junction. This research provides insights into the electronic properties of Schottky contacts based on two-dimensional ${\mathrm{MoS}}_2$, particularly the relationship between electronic barriers, system dimensions, and current flow. The demonstration of high-ideality-factor Au/${\mathrm{MoS}}_2$ diodes contributes to the design and optimization of future electronic and optoelectronic devices based on 2D materials. These findings have implications for advancements in semiconductor technology, potentially enabling the development of smaller, more efficient, and flexible devices.

1. Introduction

The semiconductor industry has catalyzed countless technological advancements that have transformed our society, enabling everything from mobile phones and automobiles to airplanes, satellites, and video games [1]. These tiny electronic components have enhanced our quality of life, connecting people across the globe [2]. To continuously improve human well-being, the industry is designing, fabricating, and characterizing increasingly complex electronic devices [3]. While traditionally based on silicon, these devices now incorporate other materials like semiconductors, conductors, and insulators [4]. Typically, the structure of these devices consists of multiple layers of metals, semiconductors, and insulators arranged in a specific order [5].

Silicon-based devices, however, have reached a roadblock as a consequence of Moore’s Law [6]. Initially proposed in 1965 by Gordon Moore, the law predicted that the number of transistors per unit area would double yearly [7]. This projection was later revised to a doubling every two years [8]. However, the miniaturization of silicon transistors is nearing its physical limit, restricted by atomic sizes and the heat generated within the system [9]. Despite these challenges, the semiconductor industry aims to deliver approximately one trillion transistors in a single device by 2030 [10]. To overcome these obstacles, low-dimensional electronic systems have emerged as a promising solution [11].

Low-dimensional systems exhibit unique properties associated with electron confinement in fewer than three spatial dimensions (0D, 1D, and 2D) [12]. This quantum confinement dramatically alters the behavior of electrons, leading to quantum phenomena not observed in bulk materials [13]. In these systems, electrons are restricted to move in a plane (2D), along a line (1D), or even at a single point (0D), significantly modifying the material’s electronic and optical properties compared to its bulk counterpart [14].

A common method to create low-dimensional systems is through heterostructures, where extremely thin layers of different semiconductor materials are stacked [15]. Transition metal dichalcogenides (TMDs) represent a family of materials with molecular layered structures, denoted by the general formula ${\mathrm{MX}}_2$, where M is a transition metal (Mo, W, etc.), and X is a chalcogen (S, Se, or Te) [16,17]. In these materials, M atoms are arranged in a plane surrounded by two planes of X atoms, held together by van der Waals forces in the transverse plane, allowing for electronic confinement in two dimensions [18]. These structures often crystallize in tetragonal (T), hexagonal (H), or octahedral unit cells, depending on the specific arrangement of the atomic layers [19].

Molybdenum disulfide (${\mathrm{MoS}}_2$) possesses highly tunable electronic and optical properties, making it an exceptional candidate for enhancing the performance of Schottky diodes [20]. One of the most significant properties is its thickness-dependent bandgap, which transitions from an indirect bandgap of 1.2 eV in bulk form to a direct bandgap of 1.8 eV in monolayers, thus allowing control over optical absorption and electronic transition rates [21,22]. This tunability is essential in Schottky diodes like the Au/${\mathrm{MoS}}_2$ junction, as it enhances electron mobility and reduces recombination, resulting in more efficient charge carrier separation [23]. Additionally, ${\mathrm{MoS}}_2$ exhibits excellent electrical conductivity with a carrier mobility that can reach up to 500 cm²/Vs under optimal conditions, depending on layer thickness and external factors such as strain or electrostatic doping [3]. These properties contribute to lowering the current density in the Au/${\mathrm{MoS}}_2$ diode to as low as 1.84 × ${10}^{-9}$ A/${\mathrm{cm}}^2$ compared to traditional W/Si diodes as well as improving the Schottky barrier height (0.9 eV in Au/${\mathrm{MoS}}_2$) for enhanced rectification behavior [24].

Moreover, ${\mathrm{MoS}}_2$’s strong light–matter interaction and high absorption coefficient in the visible and near-infrared spectra are critical in optimizing the photocurrent generation in such diodes [17,25]. The ability to modulate the Schottky barrier height dynamically, depending on applied voltage and environmental conditions such as temperature, further allows for highly responsive electronic behavior, other authors have observed thermal interferences in grain boundaries or interfaces in layered semiconducting structures [26]. In addition to the boundaries, crystal defects such as vacancies interfere with thermal conductivity [27]. The tunable photoconductivity and electronic band structure also enable the optimization of optical responsivity and quantum efficiency, which are essential for low-leakage and high-sensitivity photodetection [2,17,28]. Overall, the versatility of ${\mathrm{MoS}}_2$’s electronic and optical properties directly contributes to the observed high-performance characteristics of the proposed Au/${\mathrm{MoS}}_2$ Schottky diode, particularly its ideality factor and current–voltage profile [29], making ${\mathrm{MoS}}_2$-based Schottky diodes suitable for high-frequency and high-temperature applications [30].

Schottky barrier junctions are indispensable in contemporary semiconductor technology. In this work, we explore the Schottky barrier formed at the ${\mathrm{MoS}}_2$–Au junction using the well-established Si–W junction as a reference, which is commonly employed in the semiconductor industry [31]. Our objective is to gain a deeper understanding of the Schottky barrier behavior in a low-dimensional diode constructed with ${\mathrm{MoS}}_2$ and provide insights for future developments of low-energy, low-dimensional electronic devices [3,11,32].

2. Methodology

Schottky barriers occur at the interface between a metal and a semiconductor or between two highly doped semiconductors. To form a Schottky diode in this work, we propose using gold as the metal contact with a work function of 5.1 eV and molybdenum disulfide with a work function of 4.2 eV. This difference in energy levels allows for the formation of a potential barrier known as a Schottky barrier when gold and ${\mathrm{MoS}}_2$ are joined.

In this phenomenon, the I–V curve exhibits nonlinear behavior, meaning that the current flowing through the contact depends on the polarity and magnitude of the applied voltage. In this work, the contact will be directly polarized (Figure 1) by applying a voltage to the metal contact and sweeping the voltage from 0.001 V to 0.25 V with the aim of modeling the I–V curve behavior. This is of great importance because, due to the rectifying nature of the diode, not only is energy consumption necessary but so is the characteristic shape that allows for rectification in the initial stage of the curve.

In order to investigate the nonlinear relationship between the current flowing through the M–S junction and the applied voltage, we proceed to perform a finite element analysis using a commercial software known as COMSOL Multiphysics 6.1. For this, we model a ${\mathrm{MoS}}_2$ wafer with a thickness of 2 atomic layers and an Au coating of 6 nm. The numerical analysis features a nanostructure with the following dimensions: 9.25 nm, 1 nm, and 1 nm for the x, y, and z axes, respectively. ${\mathrm{MoS}}_2$ has approximately a 1.34 nm thickness, while the Au coat has a thickness of 6 nm.

As previously discussed, the diode’s rectifying property, stemming from its nonlinear current–voltage characteristics, is crucial for the rectification process. For reference, we will model an ideal silicon diode with a tungsten contact and apply the same voltage sweep in the same configuration as we apply to our ${\mathrm{MoS}}_2$/Au contact. The W/Si structure has a thickness of 254 $\mathsf{\mu }$m and a diameter of 5080 $\mathsf{\mu }$m. The silicon wafer is also 254 $\mathsf{\mu }$m thick, while the tungsten contact is 1 $\mathsf{\mu }$m.

The resulting signal will help us compare the shape of the I–V curve. The comparison will be performed using a Pearson correlation of the current density versus the applied voltage. Additionally, we present band diagrams of the materials and their behavior when brought into contact. This allows us to compare the electronic structure Schottky barrier of both devices and gain a better understanding of the proposed device.

3. Results and Discussion

3.1. Schottky Barrier Simulation

We successfully modeled a Schottky diode comprising a gold wafer and a bilayer ${\mathrm{MoS}}_2$ film. Figure 2 illustrates the electric field and potential distributions within the device when the anode voltage is swept. Theoretically, given $q{\psi }_m>q{\psi }_{SC}$ in Au/${\mathrm{MoS}}_2$, the semiconductor exhibits n-type behavior. Thus, the semiconductor acts as the cathode, and the metal serves as the anode.

As expected, due to the differing work functions of Au and ${\mathrm{MoS}}_2$, a Schottky barrier forms at their interface. The semiconductor’s higher work function ($q{\psi }_{sc}$) compared to the metal’s ($q{\psi }_m$) leads to electron transfer from the metal to the semiconductor, as shown in Figure 2. This transfer induces an electric field directed from the semiconductor toward the metal. The electric potential in this junction results in upward band bending, indicative of the barrier formation, which plays a critical role in diode performance [24,32].

In the context of the Au/${\mathrm{MoS}}_2$ Schottky diode, carrier concentration plays a pivotal role in shaping the diode’s electronic properties and overall performance [2]. A higher carrier concentration in the ${\mathrm{MoS}}_2$ semiconductor reduces the Schottky barrier height (SBH) due to enhanced Fermi level pinning at the metal–semiconductor interface. This effect facilitates easier electron injection from the metal into the semiconductor under forward bias, improving current conduction and reducing the series resistance of the diode [24]. Furthermore, controlling the carrier concentration enhances the diode’s rectification ratio, allowing it to switch between high forward current and low reverse leakage, crucial for high-frequency applications [33,34].

Figure 3 shows the full carrier behavior as well as the concentration throughout the system when the voltage is swept from 0.001 V to 0.25 V in 0.001 V steps. At a low applied voltage of 0.001 V (Figure 3a), the ${\mathrm{MoS}}_2$/Au junction exhibits a depletion region, characterized by a relatively low hole concentration near the interface. As the applied voltage increases to 0.25 V (Figure 3b), a hole accumulation region forms due to the injection of holes from the Au contact. This transition from depletion to accumulation indicates the diode’s rectification behavior [3,34].

Similarly, at 0.001 V (Figure 3c), the ${\mathrm{MoS}}_2$ layer accumulates electrons. However, as the applied voltage increases to 0.25 V (Figure 3d), the electron concentration decreases near the Au contact due to hole injection and subsequent recombination. This depletion of electrons further reinforces the diode’s rectifying characteristics, a behavior essential for high-speed and low-leakage applications [3,35].

The simulation effectively demonstrates the formation and distribution of electric potential and field within the ${\mathrm{MoS}}_2$/Au Schottky diode. The abrupt change in potential at the metal–semiconductor interface confirms the presence of a Schottky barrier, a fundamental characteristic of such junctions. The electric field lines, originating from the higher-potential region (Au) and terminating in the lower-potential region (${\mathrm{MoS}}_2$), are consistent with the expected direction of current flow under forward bias conditions [2,22,30]. The control over carrier concentration in ${\mathrm{MoS}}_2$ not only optimizes the rectification ratio but also enhances the photoconductivity and response time, particularly in optoelectronic applications [23,36].

3.2. I–V Curve Analysis

Schottky diodes inherently display a nonlinear I–V relationship, where the current flow is dependent on both the polarity and the magnitude of the applied voltage. In this study, the diode was forward biased by applying a voltage to the metal contact and sweeping it from 0.001 V to 0.25 V. This voltage sweep is crucial for modeling the I–V behavior and understanding the rectifying nature of the diode, particularly in the initial stages of the curve where rectification occurs.

To further explore the nonlinear relationship between current and applied voltage, a finite element analysis was conducted using COMSOL Multiphysics 6.1. By applying a voltage sweep ranging from 0.0001 V to 0.25 V to the forward-biased diode model, the resulting current density was analyzed and plotted against the applied voltage. This analysis provides valuable insights into the behavior of the Schottky diodes, highlighting the distinct advantages of the Au/${\mathrm{MoS}}_2$ diode in rectification applications.

To comprehensively evaluate the performance of our proposed diode compared to a classical Schottky diode, we analyzed the current density versus applied voltage. The results, depicted in Figure 4, demonstrate that the current density through the classical W/Si nanostructure emerges at 3.62 × ${10}^{-5}$ A/${\mathrm{cm}}^2$. In contrast, the Au/${\mathrm{MoS}}_2$ junction exhibits a significantly lower current density of 1.84 × ${10}^{-9}$ A/${\mathrm{cm}}^2$. This comparison highlights that our proposed diode features a substantially lower on/off ratio than the classical Si-based structure. However, the rectifying behavior is expressed in its nonlinear behavior. Therefore, we analyzed the reciprocal relationship between the J/V curve of the ideal Schottky contact (W/Si) and the proposed structure to determine if both curves follow a Schottky-like shape. Additionally, in the simulated Au/${\mathrm{MoS}}_2$ structure, the lattice temperature was 298 K.

3.3. Comparison with Ideal Diode

To describe the relationship between the simulated I–V curves and the ideal diode curve, a Pearson correlation analysis was conducted. Given that each curve consists of N data points, the Pearson correlation coefficient is defined as

$$\rho (A,B)={\displaystyle \frac{1}{N-1}}\sum _{i=1}^N\left({\displaystyle \frac{A_i-{\mu }_A}{{\sigma }_A}}\right)\left({\displaystyle \frac{B_i-{\mu }_B}{{\sigma }_B}}\right)$$

(1)

where ${\mu }_A$ and ${\sigma }_A$ are the mean and standard deviation of A, respectively, and ${\mu }_B$ and ${\sigma }_B$ are the mean and standard deviation of B. The correlation coefficient matrix for each pairwise variable combination is defined by

$$\left({\displaystyle \genfrac{}{}{0pt}{}{\rho (A,A)\rho (A,B)}{\rho (B,A)\rho (B,B)}}\right)$$

(2)

As A and B are correlated to themselves, the diagonal entries are just 1. Then, the resultant matrix is

$$\begin{array}{cc}1.0000& 0.9991\\ 0.9991& 1.0000\end{array}$$

As a result of the analysis, it can be seen that the correlation matrix exhibits a ≈99.91% similarity to an ideal diode. Hence, it can be concluded that the proposed model has an ideality factor close to 1.

3.4. Band Diagram Analysis

The band diagrams of the Au/${\mathrm{MoS}}_2$ and W/Si junctions provided insights into the underlying physics of Schottky barrier formation.

To obtain a comprehensive energy levels understanding, we used a classical schematic band diagram. The components in here include ${\mathrm{E}}_0$, vacuum energy; ${\mathrm{E}}_v$, the maximum energy of the valence band; ${\mathrm{E}}_c$, minimum energy of the conduction band; ${\mathrm{E}}_F$, Fermi level; q${\psi }_m$, the metal work function; q${\psi }_{sc}$, semiconductor work function; ${\chi }_m$, metal affinity; and ${\chi }_{sc}$, semiconductor affinity.

Figure 5 shows the proposed materials in isolation. It is aligned according to the vacuum level (${\mathrm{E}}_F$). The Au work function (q${\psi }_m$) differs from that of ${\mathrm{MoS}}_2$ (q${\psi }_{SC}$), as shown in Figure 5a. In a similar way, Figure 5b depicts it for W and Si. This indicates that a mismatch exists in the Fermi level, and since q${\psi }_m$≠q${\psi }_{SC}$ in both cases, an electronic barrier will be formed in both devices.

Figure 6 shows a schematic diagram of the metal and semiconductor once they are joined. The bands’ positions are profiled by the Fermi level (${\mathrm{E}}_F$). The alignment between the metal and the semiconductor Fermi level place the conduction band near to the junction ${\mathrm{E}}_F$. Therefore, the bands ${\mathrm{E}}_0$, ${\mathrm{E}}_F$, and ${\mathrm{E}}_V$ are bent downwards. At the interface in the W/Si and Au/${\mathrm{MoS}}_2$ structures, a Helmholtz double layer is formed. Consequently, a net electric field is established from the semiconductor to the metal in order to reach equilibrium. It causes the energy band edges in the semiconductor to bend upwards. It is associated to the accumulated charge and the electric field between the semiconductor and the metal surface.

The proposed structure (Figure 6a) and the reference diode (Figure 6b) show similar behavior. As we can verify, the metal’s work function is larger than that of the semiconductor, as shown in Figure 7. Therefore, the metal band bends downwards. The energy band alignment at the interface revealed a significant difference in barrier heights between the two junctions. The lower barrier height in the Au/${\mathrm{MoS}}_2$ junction contributed to the improved rectification performance observed in the I–V curves.

In order to calculate the Schottky barrier height (SBH), we considered the energy level distribution depicted in Figure 5. Here, we can observe that the metal work function is equal to the SBH plus the semiconductor electron affinity.

$$q{\psi }_m=q{\psi }_{Bn}+\chi$$

(3)

As we can see in Figure 6, it is possible obtain the SBH from the sum of the built-in potential and the energy between the ${\mathrm{E}}_c$ and ${\mathrm{E}}_f$.

$$q{\psi }_{Bn}=q{\psi }_{Bi}+(E_c-E_f)$$

(4)

$$q{\psi }_{Bn}=q{\psi }_{Bi}-KTIn\left({\displaystyle \frac{N_d}{N_c}}\right)$$

(5)

By solving Equations (3)–(5) and referencing data from [46,47,48,49], we determined that the SBH for the Au/${\mathrm{MoS}}_2$ contact is approximately 0.9 eV. This is significantly higher than the 0.67 eV SBH of W/Si [48]. Figure 7 visually illustrates this difference and compares the energy bands (${\mathrm{E}}_C$ and ${\mathrm{E}}_V$) for both junctions. The higher SBH of Au/${\mathrm{MoS}}_2$ means that electrons require less energy to overcome the barrier and enter the conduction band in W/Si. However, despite its lower barrier, W/Si demands a significantly higher energy input (10 times more) to facilitate current flow due to its larger size.

Diodes with higher SBHs (like Au/${\mathrm{MoS}}_2$) exhibit higher turn-on voltages, lower reverse leakage currents, higher operating temperatures, and lower switching speeds. Conversely, diodes with lower SBHs (like W/Si) have lower turn-on voltages, higher reverse leakage currents, lower operating temperatures, and higher switching speeds. These characteristics make high-barrier diodes well-suited for high-temperature, high-frequency applications requiring low leakage current, such as microwave and communication circuits. Low-barrier diodes, on the other hand, are ideal for low-power, low-frequency applications needing a low voltage drop, such as rectifiers and protection circuits.

4. Implications and Outlook

Molybdenum disulfide (${\mathrm{MoS}}_2$) has emerged as a highly versatile material for next-generation Schottky diodes due to its tunable electronic and optical properties. As described in [24], the transition of ${\mathrm{MoS}}_2$’s bandgap from an indirect gap of 1.2 eV in bulk to a direct gap in monolayers (1.8 eV) enables precise control of electron transport, which is crucial for enhancing charge mobility and reducing recombination losses. This tunability is particularly advantageous for Schottky diodes, where the integration of ${\mathrm{MoS}}_2$ with gold contacts, as shown in this work, significantly improves rectification performance and reduces inhomogeneities at the metal–semiconductor interface [5]. ${\mathrm{MoS}}_2$ also exhibits strong light–matter interaction and a high absorption coefficient, especially in the visible spectrum, making it an excellent candidate for high-performance optoelectronic devices, such as photodetectors and solar cells [26,32]. Additionally, multilayered ${\mathrm{MoS}}_2$, as highlighted in [14], allows for the modulation of interlayer coupling, which is key to improving carrier concentration and switching speeds in field-effect transistors (FETs). Furthermore, the ability to control defect densities, such as sulfur vacancies, has been shown to enhance the Schottky barrier height and overall diode performance, making ${\mathrm{MoS}}_2$ ideal for high-sensitivity applications [3]. Looking ahead, a promising area of research involves the use of solution-processed ${\mathrm{MoS}}_2$ for scalable Schottky diode manufacturing. Spin-coating techniques, such as those demonstrated in [50], offer precise control over the deposition of ${\mathrm{MoS}}_2$ films, which is vital for tuning the Schottky barrier height and improving overall diode performance. Additionally, vacancy engineering and doping strategies [9,11] can optimize the carrier concentration and barrier properties, making ${\mathrm{MoS}}_2$ ideal for low-power and high-sensitivity applications. Another exciting direction is the exploration of hybrid heterostructures that combine ${\mathrm{MoS}}_2$ with other transition metal dichalcogenides (TMDs) [51,52]. These van der Waals heterostructures offer significant improvements in charge mobility and reduced leakage currents, which are crucial for applications in flexible and wearable electronics. Moreover, heterostructures of ${\mathrm{MoS}}_2$ with other materials have demonstrated improved stability and performance in photodetection and catalytic applications [28,53]. These advancements suggest that ${\mathrm{MoS}}_2$ is well-suited for high-frequency, high-power applications where traditional materials face limitations [24]. ${\mathrm{MoS}}_2$’s tunable electrical properties and high-temperature stability make it particularly suited for high-power applications [24,30]. As research into low-dimensional electronic systems accelerates, ${\mathrm{MoS}}_2$ offers unique quantum properties that can significantly enhance device performance compared to bulk semiconductors. Quantum confinement in ${\mathrm{MoS}}_2$ leads to enhanced electron transport, making it ideal for high-speed, high-frequency applications [2]. This is why ${\mathrm{MoS}}_2$ could have a role in the future of advanced electronics, particularly in enabling the development of smaller, faster, and more energy-efficient electronic components and devices, such as transistors and photodetectors [9,17], which incorporate Schottky contacts. Future research into heterostructures, solution processing, and miniaturization will be critical in fully realizing the potential of ${\mathrm{MoS}}_2$ for next-generation technologies [2,4,9,24,32,50].

Our research endeavors to contribute to this expanding knowledge base by providing a more comprehensive understanding of the electronic properties in a Schottky contact based on bidimensional ${\mathrm{MoS}}_2$. Modeling a ${\mathrm{MoS}}_2$ wafer with electronic characteristics similar to bi-dimensional ${\mathrm{MoS}}_2$ such as electronic confinement in two planes, this work explores the complex interplay between electronic barriers, system dimensions, and their subsequent influence on current flow through ${\mathrm{MoS}}_2$-based Schottky diodes. These insights are crucial for future device design and optimization.

5. Conclusions

This research successfully modeled a Schottky diode composed of a gold wafer and a bilayer ${\mathrm{MoS}}_2$ film. The simulation results effectively demonstrate the formation and distribution of electric potential and field within the ${\mathrm{MoS}}_2$/Au Schottky diode. The abrupt change in potential at the metal–semiconductor interface confirms the presence of a Schottky barrier, a fundamental characteristic of such junctions. The electric field lines, originating from the higher-potential region (Au) and terminating in the lower-potential region (${\mathrm{MoS}}_2$), are consistent with the expected direction of current flow under forward bias conditions.

The I–V curve analysis revealed that the Au/${\mathrm{MoS}}_2$ diode exhibits a significantly lower current density compared to the classical W/Si structure, highlighting its superior rectification behavior. Furthermore, the Pearson correlation analysis demonstrated a high degree of similarity between the simulated I–V curves and the ideal diode curve, indicating an ideality factor close to 1. The band diagram analysis provided insights into the underlying physics of Schottky barrier formation in both the Au/${\mathrm{MoS}}_2$ and W/Si junctions. The higher Schottky barrier height in the Au/${\mathrm{MoS}}_2$ junction was identified as the key factor contributing to its improved rectification performance.