Efficient and Versatile Modeling of Mono- and Multi-Layer MoS₂ Field Effect Transistor

Abstract

Two-dimensional (2D) materials with intrinsic atomic-level thicknesses are strong candidates for the development of deeply scaled field-effect transistors (FETs) and novel device architectures. In particular, transition-metal dichalcogenides (TMDCs), of which molybdenum disulfide (MoS${}_2$) is the most widely studied, are especially attractive because of their non-zero bandgap, mechanical flexibility, and optical transparency. In this contribution, we present an efficient full-wave model of MoS${}_2$-FETs that is based on (1) defining the constitutive relations of the MoS${}_2$ active channel, and (2) simulating the 3D geometry. The former is achieved by using atomistic simulations of the material crystal structure, the latter is obtained by using the solver COMSOL Multiphysics. We show examples of FET simulations and compare, when possible, the theoretical results to the experimental from the literature. The comparison highlights a very good agreement.

1. Introduction

Mono-layer transition metal dicalchogenides are chemical compounds in which molecules are formed by one transition metal atom (Mo, W, Pt, etc.) and two atoms belonging to group 16 of the periodic table of elements (S, O, Pt). During the last decade, an increasing interest on the use of MoS${}_2$ has gradually emerged since this material exhibits several unprecedented properties, such as scalability [1], tunability [2], low noise figure [3], ambipolarity [3], non-zero bandgap, and, in the meantime, compatibility with the current complementary metal oxide semiconductor (CMOS) technology, as shown in literature [4]. MoS${}_2$ suits for a large plethora of applications in the nano-electronics area [5], ranging from field-effect transistors (MoS${}_2$-FETs) and gas sensors [6,7], to photo-detectors [8] and solar cells [9].

MoS${}_2$-FETs have been broadly studied by the literature, providing important and promising experimental data showing how these devices behave ([10,11,12]). However, from a design point of view, it is equally important to establish numerical methods that can predict the electrical properties of MoS${}_2$ based FETs. Cao et al. reported a model of FET specifically realized for monolayer TMDCs, considering interface traps, mobility degradation and inefficient doping effects [13]; in literature [14] it is possible to find a simulation study of a MoS${}_2$ FET for analog circuits; Zhang et al. illustrated another approach to model MoS${}_2$ FETs in [15], completed with a comparative study between CMOS FETs and MOS${}_2$-FETs. In 1992, Miller showed a FET modeled with a ferroelectric gate oxide, called ferroelectric-metal field-effect transistor (FEM-FET), by means of approximated methods [16], demonstrating how this device could be used as a non volatile memory unit.

In this work, we present an efficient and versatile model for the analysis and simulation of the FET, based on the following steps: (1) study of the material (MoS${}_2$) at the atomistic level, (2) derivation of constitutive relations and (3) their insertion in the full-wave solver (COMSOL) for the simulation at the continuum (device) level. It is remarkable to note that the set (2) is a key-development, as it introduces the possibility of simulating defects and particular contacts with the substrate. In the following, we firstly provide the theoretical foundations of the FET model; then, we describe the computational platform for the ab-initio (atomistic) simulations. Subsequently, we perform COMSOL simulations, present and compare some results with respect to data from the literature. As a further issue, we consider the use of hafnium-zirconium oxide (Hf${}_x$Zr${}_{1-x}$O${}_2$, $x=0.3$) as a substrate ferroelectric material, which exhibits high tunability and compatibility with the CMOS technology [17,18]. The last part provides conclusions of our work.

2. Materials and Methods

2.1. Theoretical Background

The benchmark models have been realized using the semiconductor physics module provided by COMSOL Multiphysics. This module implements Poisson’s equation, which links the potential (V) to the charge density ($\rho$), according to expression (1):

$$\nabla ·(-{ϵ}_0{ϵ}_r\nabla V)=\rho$$

(1)

where ${ϵ}_0$ and ${ϵ}_r$ are the vacuum and relative permittivities, respectively.

2.1.1. Semiconductor Material Model Interface

The semiconductor material model interface is used to implement the equations for semiconducting materials derived from the semi-classical model. The charge present in the channel is computed by Equation (2):

$$\rho +=q(p-n+N_d^{+}-N_a^{-})$$

(2)

where $q=-e$, being e the elementary electron charge, p and n are the carrier concentration and $N_d^{+}$ and $N_a^{-}$ are the donor and acceptor concentration, respectively, which correspond to the particle density in the ionized regions. Complete ionization is assumed.

Both electron (${\mathbf{J}}_n$) and hole (${\mathbf{J}}_p$) currents respect the conservation law according to Equation (3):

$$\nabla ·{\mathbf{J}}_n=0\nabla ·{\mathbf{J}}_p=0$$

(3)

Carrier currents are then computed according Equation (4a,b):

$${\mathbf{J}}_n=qn{\mu }_n\nabla E_c+{\mu }_n{k}_{B}T\nabla n+qn{D}_{n,th}\nabla lnT$$

(4a)

$${\mathbf{J}}_p=qp{\mu }_p\nabla E_c-{\mu }_p{k}_{B}T\nabla p-qp{D}_{p,th}\nabla lnT$$

(4b)

where ${\mu }_p$ and ${\mu }_n$ are holes and electrons mobility respectively, $D_{p,th}$ and $D_{n,th}$ are the thermal diffusion coefficients for holes and electrons, T is the room temperature and $k_B$ is the Boltzman constant. Conduction band $E_c$ and valence band $E_v$ are calculated as follows:

$$E_c=-(V+{\chi }_0)$$

(5a)

$$E_v=-(V+{\chi }_0+E_{g,0})$$

(5b)

with ${\chi }_0$ electron affinity and $E_{g,0}$ energy bandgap of the semiconductor material.

2.1.2. Metal Contacts (Ideal Ohmic and Ideal Schottky)

The potential in ohmic contacts is defined as:

$$V=V_0+V_{eq}$$

(6)

where $V_0$ is the applied potential and $V_{eq}$ is the Fermi level offset in terms of electric potential at a given temperature T.

The inputs for the Schottky contact interface are the metal work function and the effective Richardson constant [19] of the semiconducting material. The effective Richardson constant ($A^{*}$) is given by:

$$A^{*}=\frac{4\pi q{k_B}^2{m}^{*}}{h^3}$$

(7)

where $m^{*}$ is the effective mass for electrons/holes and h is the Planck constant. The Richardson constant is related to thermionic effects. The potential at in the Schottky contact is defined as:

$$V=V_0+{\mathsf{\Phi }}_B-{\chi }_0-V_{eq,adj}$$

(8)

where $V_0$ is the applied potential, ${\mathsf{\Phi }}_B$ is the metal work function and $V_{eq,adj}$ has the same meaning of $V_{eq}$ in Equation (6).

2.1.3. Dielectric Materials and Intrinsic n-Type Behavior

Since insulators are considered dielectric materials, it is sufficient to apply a charge conservation condition according to Gauss’ law for the electric displacement ($\mathbf{D}$) and electric field ($\mathbf{E}$):

$$\mathbf{D}={ϵ}_0{ϵ}_r\mathbf{E}$$

(9)

MoS${}_2$ layers usually behave as n-type doped semiconductors [20,21], thus an analytic doping model has been defined to set doping type and concentration in the channel, with donor concentration $N_d={10}^{18}$ cm${}^{-3}$.

2.1.4. Trap-Assisted Recombination

The trap-assisted recombination interface includes an additional contribute to the carrier current. The trapping model used is the Shockley–Reed–Hall model. This interface implements the following equations:

$$\nabla ·{\mathbf{J}}_n=q{R}_n$$

(10a)

$$\nabla ·{\mathbf{J}}_p=-q{R}_p$$

(10b)

with $R_n$ and $R_p$ electron and holes recombination rates. Recombination rates depend on the carrier lifetimes ${\tau }_n$ and ${\tau }_p$ [22].

2.1.5. Atomistic Simulations Platform

As outlined, we avail of a software platform for the (1) simulations at the atomistic level of the active material (be it MoS${}_2$ or others), (2) the self-consistent derivation of constitutive relations and (3) their insertion in the full-wave solver as permittivity, permeability and/or conductivity. This permits, for example, the inclusions of lattice defects. In the present case (further and more complex cases will be investigated in future works), a single MoS${}_2$ layer was built using the Macromodel MAESTRO suite [23]. Density Functional Theory (DFT) was used with an extended Perdew–Burke–Ernzerhof (PBE) functional combined with a Gaussian type orbital (GTO) basis set 6-311G* to optimize MoS${}_2$ three-dimensional geometry and to extrapolate bandgap values. DFT results were used as a starting point for subsequent computational investigations. Four different models were created, with one, two, three, and four MoS${}_2$ layers, using DFT optimization MoS${}_2$ geometry. A simulation box of 2.24 nm × 2.24 nm × 1.4 nm was prepared for each system. Periodic boundary conditions (PBC) were then set up on simulation boxes along x and y axes, but not on z axes, to avoid the possibility of considering more than four MoS${}_2$ layers (Figure 1). The four systems were minimized using steepest descent and conjugate gradient algorithms, then an initial 200 ps NVT-ensemble of molecular dynamics (MD) simulation was used for the equilibration, following an NPT-ensemble of MD simulation 10 ns long at 298 K and 1 atm pressure. All MD simulations were performed using the GROMACS 5.1.5 suite [24]. PBC and Ewald summation were used to consider the long range electrostatic interatomic interactions.

The CLAYFF force field interatomic potentials [25] was used to describe the MoS${}_2$ layers along MD simulation after a previous enrichment with new MoS${}_2$ parameters determined at the DFT level. Visual molecular dynamics (VMD) [26] and Chimera [27] software were used for trajectory visualization and analyses, while Xmgrace (Grace 5.1.21 GNU public license, Cambridge, MA, USA) was used for generating plots.

2.2. Model Validation

2.2.1. MoS${}_2$ FET with n+ Si Back Gate

The first model used for the validation is the one reported by Howell et al. [28]. Here, the simulation settings both for a monolayer and a 4-layer MoS${}_2$ FET are shown. A schematic view of the device is shown in Figure 2. All the parameters used for the simulations are listed in

Table 1. Simulation parameters. Material properties are from the supporting information provided in attachment to [28].

Parameter	Value	Parameter	Value
Thickness of MoS${}_2$	$0.7$$\mathrm{n}$$\mathrm{m}$/layer	Electron effective mass	0.5 m${}_0$
Bandgap 1L MoS${}_2$	$2.76$$\mathrm{e}\mathrm{V}$	Hole effective mass	0.5 m${}_0$
Bandgap 4L MoS${}_2$	$1.6$$\mathrm{e}\mathrm{V}$	Thickness gold contact	75 $\mathrm{n}$$\mathrm{m}$
Electron affinity 1L Mo${}_2$	$4.7$$\mathrm{e}\mathrm{V}$	Length MoS${}_2$	$3.5$$\mathsf{\mu }$$\mathrm{m}$
Electron affinity 4L MoS${}_2$	4 $\mathrm{e}\mathrm{V}$	Silicon thickness	2 $\mathsf{\mu }$$\mathrm{m}$
Relative permittivity 1L	4.2	SiO${}_2$ thickness	300 $\mathrm{n}$$\mathrm{m}$
Relative permittivity 4L	11	Width	$6.8$$\mathsf{\mu }$$\mathrm{m}$
Mobility 1L	6 cm2 V−1 s−1	Work function of gate	$4.05$$\mathrm{V}$
Mobility 4L	25 cm2 V−1 s−1	SiO${}_2$ Relative Permittivity	3.9
Drain and Source contact type	Ideal ohmic	Donor concentration (N_D)	$1\times {10}^{18}$ cm${}^{-3}$

.

Drain and source metal contacts have been placed at the boundary between gold (orange) and MoS${}_2$ (magenta). The semiconductor material interface is defined in the active region (magenta), while the charge conservation is applied in the insulator region (green). The gate contact is modeled with a terminal physics interface placed between the gate oxide and the gate region (plum).

The last interface mentioned is used for connections to outer circuits and requires a metal work function to be properly modeled. The doping concentration is specified through an analytic doping model defined in the active region.

2.2.2. MoS${}_2$ Transistor with HfO${}_2$

The second structure analyzed is presented by Radisavljevic et al. [10]. The main differences with respect to the previous model are the presence of a gold top gate with a 30 nm thick HfO${}_2$ insulator, the type of metal contact chosen for the drain, source and back gate contacts (Schottky) and finally the method used to model the back gate. In this case the silicon back gate is modeled as a degenerate semiconductor by defining a high doping level in the gate region, which is contacted with a Schottky metal contact. All the remaining regions are modeled in the same way as the first model presented. In Figure 3, we can see the metal contacts (orange), the HfO${}_2$ top gate insulator (light green), the monolayer MoS${}_2$ active region (magenta), the SiO${}_2$ back gate insulator (green), finally a n+ Si back gate contact. All the simulation parameters used for the structure modeling are shown in

Table 2. Simulation parameters. All the data are taken from [10,28].

Parameter	Value	Parameter	Value
Thickness of MoS${}_2$	$0.65$$\mathrm{n}$$\mathrm{m}$	SiO${}_2$ Relative Permittivity	3.9
Bandgap MoS${}_2$	$1.8$$\mathrm{e}\mathrm{V}$	Electron effective mass	0.5 m${}_0$
Electron affinity MoS${}_2$	5 $\mathrm{e}\mathrm{V}$	Hole effective mass	0.5 m${}_0$
Relative permittivity MoS${}_2$	$4.2$$\mathrm{e}\mathrm{V}$	Gold contact length	500 $\mathrm{n}$$\mathrm{m}$
Relative permittivity HfO${}_2$	25	Source-gate spacing	500 $\mathrm{n}$$\mathrm{m}$
Mobility	217 cm2 V−1 s−1	Gate-drain spacing	500 $\mathrm{n}$$\mathrm{m}$
SRH lifetimes	$1.5$$\mathrm{n}$$\mathrm{s}$	Thickness gold contact	50 $\mathrm{n}$$\mathrm{m}$
Metal work function of top gate	$4.5$$\mathrm{V}$	SiO${}_2$ thickness	270 $\mathrm{n}$$\mathrm{m}$
Work function of bottom gate	$4.05$$\mathrm{V}$	HfO${}_2$ thickness	30 $\mathrm{n}$$\mathrm{m}$
Metal work function source	$5.1$$\mathrm{V}$	Width	4 $\mathsf{\mu }$$\mathrm{m}$
Metal work function drain	$5.1$$\mathrm{V}$	Donor concentration (N_d)	$1\times {10}^{18}$ cm${}^{-3}$

. Since the MoS${}_2$ electron affinity is not provided in [10], it has been tuned in order to fit the results from the just mentioned paper.

2.3. MoS${}_2$ Transistor with Hf${}_{0.3}$Zr${}_{0.7}$O${}_2$

In this section, on the basis of the previous MoS${}_2$ models, and taking into account the remarkable insulating properties of the HfO${}_2$, we present a concept model and simulations of an FeM-FET device (Figure 4). This model should pave the way for the fabrication of novel kinds of high performance MoS${}_2$ based devices. Simulations have been performed starting from the model described in Section 2.2.2 and adding a 6 $\mathrm{n}$$\mathrm{m}$ thick layer of Hf${}_{0.3}$Zr${}_{0.7}$O${}_2$.

The values of the permittivity in function of the applied potential are taken from [17] and shown in Figure 5. The $ϵ$-V curve is interpolated with a linear method, extrapolation is performed using the nearest function method.

Table 3. Simulation parameters. All the data are taken from [10,28].

Parameter	Value	Parameter	Value
Thickness of MoS${}_2$	$0.65$$\mathrm{n}$$\mathrm{m}$	Hf${}_{0.3}$Zr${}_{0.7}$O${}_2$ thickness	6 nm
Bandgap MoS${}_2$	$1.8$$\mathrm{e}\mathrm{V}$	Electron effective mass	0.5 m${}_0$
Electron affinity MoS${}_2$	5 $\mathrm{e}\mathrm{V}$	Hole effective mass	0.5 m${}_0$
Relative permittivity MoS${}_2$	$4.2$$\mathrm{e}\mathrm{V}$	Gold contact length	500 $\mathrm{n}$$\mathrm{m}$
Relative permittivity HfO${}_2$	20	Source-gate spacing	500 $\mathrm{n}$$\mathrm{m}$
Mobility	217 cm2 V−1 s−1	Gate-drain spacing	500 $\mathrm{n}$$\mathrm{m}$
SRH lifetimes	$1.5$$\mathrm{n}$$\mathrm{s}$	Thickness gold contact	50 $\mathrm{n}$$\mathrm{m}$
Metal work function of top gate	$4.5$$\mathrm{V}$	SiO${}_2$ thickness	270 $\mathrm{n}$$\mathrm{m}$
Work function of bottom gate	$4.05$$\mathrm{V}$	HfO${}_2$ thickness	30 $\mathrm{n}$$\mathrm{m}$
Metal work function source	$5.1$$\mathrm{V}$	Width	4 $\mathsf{\mu }$$\mathrm{m}$
Metal work function drain	$5.1$$\mathrm{V}$	Donor concentration (N_d)	$1\times {10}^{18}$ cm${}^{-3}$

lists the parameter values used for this simulation run.

3. Results and Discussion

3.1. Atomistic Simulations Results

In the following, we will consider the MoS${}_2$ without any substrate or superstrate material. From DFT results, the intrinsic electronic bandgap of 1L MoS${}_2$ was determined to be 2.4 eV, decreasing to 2.1 eV for 2L MoS${}_2$. 3L MoS${}_2$ showed a bandgap value of 1.75 eV, while 4L MoS${}_2$ presented a lower value as 1.43 eV. Data revealed that MoS${}_2$ bandgaps decreased with increasing layers’ number (Figure 6a). This is caused by the quantum confinement effect, which is due to changes in the atomic structure as a result of direct influence of ultra-small length scale on the energy band structure [29].

Numerical values of dielectric constant were extrapolated from MD simulation of MoS${}_2$ systems through a combined use of gmx_dipoles and gmx_dielectric GROMACS tools. The 4.3 value of 1L MoS${}_2$ was increased to 6.5 for 2L, while 8.9 and 11.3 were the dielectric constant values obtained for 3L and 4L MoS${}_2$, respectively (Figure 6b). A direct correlation between the number of layers and the dielectric constant value was observed.

3.2. MoS${}_2$ FET with n+ Si Back Gate Results

Figure 7 and Figure 8 show a comparison between the results reported in literature [28] and the COMSOL simulations. We can observe a general good agreement both in terms of behavior and order of magnitude; the mismatch is almost due to the doping variations in the synthesis of the different MoS${}_2$ samples, that is an intrinsic, not predictable, fabrication characteristic. The ohmic nature of gold contacts is visible in Figure 7b since the drain current has a linear behavior for small voltages.

The original structure showed by Howell et al. [28] presents side contacts. However, in order to find a better matching between COMSOL simulation and experimental results and to take into account possible imperfections during the fabrication process, we tried a top contact configuration (Figure 2). The latter led to not substantially different results. From this consideration we can assume that in this particular case, the contacting method has no influence on the structure.

3.3. MoS${}_2$ Transistor with HfO${}_2$ Top Gate Insulator

The gating characteristics of the transistor is shown in Figure 9a and this is typical of FET devices with an n-type channel. The source current versus source bias characteristics (Figure 9b) is linear in the $\pm 50 \mathrm{m}\mathrm{V}$ range of voltages.

In Figure 9b, it can see that the drain current behaves as also shown in Figure 7b, this means that contacts are ohmic, even though we used Schottky contacts to better fit the results from our simulation.

From overall evaluations, we can state that our model provided good results also for this different kind of structure.

3.4. MoS${}_2$ Transistor with Hf${}_{0.3}$Zr${}_{0.7}$O${}_2$—Simulation Results

Figure 10a shows the $I_d$ − $V_{tg}$ curve with $V_{ds}=10 \mathrm{m}\mathrm{V}$, the silicon substrate, which is also considered as bulk, is grounded. The Figure 10b shows the $I_d-V_{ds}$ curve with $V_{bg}=0$ V for $V_{tg}=-2$ V, 0 V and 5 V. In this case the maximum drain current is about 25 $\mathsf{\mu }$A obtained for $V_{tg}=5$ V. In the resistive region the slope is higher than the previous study from Section 2.2.2 but the maximum current is lower.

Figure 10c indicates that for $V_{tg}=-2.5\mathrm{V}$ the device in still on, while in the same conditions the device is completely turned off in Figure 9d, also we can predict an ohmic behavior of the drain and source contacts.

Figure 11 shows a worse $I_{on}/I_{off}$ ratio than Figure 9c. With a ferroelectric material we have an $I_{on}/I_{off}$ ratio of ${10}^5$ for $V_{ds}=500 \mathrm{m}\mathrm{V}$ and about ${10}^3$ for $V_{ds}=10 \mathrm{m}\mathrm{V}$ while in [10] for $V_{ds}=500 \mathrm{m}\mathrm{V}$, the $I_{on}/I_{off}$ ratio is ${10}^8$ and for $V_{ds}=10 \mathrm{m}\mathrm{V}$, the $I_{on}/I_{off}$ ratio is ${10}^6$.

4. Conclusions

In this work, we introduce a full-wave a model of a MoS${}_2$-based FET, by using COMSOL Multiphysics. A remarkable issue, that is also a research route for further works, relies on the fact that we first analyze the 2D active material (in the actual case MoS${}_2$) at the atomistic level. The ab-initio (atomistic) simulations are based on a combination of the DFT and molecular dynamics techniques. From the atomistic simulations we derive the complete electronic band structure, as well as effective mass, permittivity, permeability and/or conductivity to be used as material constitutive relations in the subsequent full-wave simulations. The combination of atomistic vs. full-wave techniques gives high efficiency and versatility for the analysis of very different structures, devices and systems, ranging from the ballistic to the diffusive regime [30]. Then, we present examples of FET simulations and compare, for the devices described in Section 2.2.1 and Section 2.2.2, the theoretical results to the experimental ones from the literature [10,28], showing very good agreement.