Study of an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {MoS}}_{2}$$\end{document}MoS2 phototransistor using a compact numerical method enabling detailed analysis of 2D material phototransistors

Research on two-dimensional material-based phototransistors has recently become a topic of great interest. However, the high number of design features, which impact the performance of these devices, and the multi-physical nature of the device operation make the accurate analysis of these devices a challenge. Here, we present a simple yet effective numerical framework to overcome this challenge. The one-dimensional framework is constructed on the drift-diffusion equations, Poisson’s equation, and wave propagation in multi-layered medium formalism. We apply this framework to study phototransistors made from monolayer molybdenum disulfide (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {MoS}}_{2}$$\end{document}MoS2) placed on top of a back-gated silicon-oxide-coated silicon substrate. Numerical results, which show good agreement with the experimental results found in the literature, emphasize the necessity of including the inhomogeneous background for accurately calculating device metrics such as quantum efficiency and bandwidth. For the first time in literature, we calculate the phase noise of these phototransistors, which is a crucial performance metric for many applications where precise timing and synchronization are critical. We determine that applying a low drain-to-source voltage is the key requirement for low phase noise.

The Fermi-Dirac integral, is a mathematical function that describes the distribution of electrons in a system at thermodynamic equilibrium, based on Fermi-Dirac statistics.This well-known integral plays a crucial role in semiconductor physics.
Calculating F 1/2 (η) for a given η, where η is the ratio of the Fermi energy to the product of the Boltzmann constant and temperature, via numerical integration is not a trivial task but the inverse operation (finding η for a given F 1/2 value) is.Hence we have created two simple yet accurate formulas, Eqs. ( 2) and (3), to compute these forward and inverse operations efficiently. ) ,where g(η) = −10.39η 4 + 1630η , where x = log 10 (F 1/2 (η)). (3) As shown in Supp.

PLANE WAVE PROPAGATION IN MULTI-LAYERED MEDIA
To determine the electric field intensity inside the monolayer MoS 2 , we use the plane wave propagation in multilayered media formalism.In this approach, the field intensities in each layer are calculated recursively, starting from the first layer, where the intensity of the forward traveling wave is known from the incident power, P .Figure 6 (b) shows the electric field intensity in the middle of the MoS 2 layer as a function of the excitation wavelength.We observe that the intensity reaches its maximum value around 561 nm and makes two dips around 420 nm and 830 nm.

General Formulation
Assume a multi-layered medium with N + 1 planar layers aligned parallel to the xy-plane.Each layer is defined with its electrical permittivity (ϵ ℓ = ϵ 0 ϵ r,ℓ ), magnetic permeability (µ ℓ = µ 0 µ r,ℓ ), thickness (h ℓ ) for ℓ = 0, 1, • • • , N , and h 0 = h N +1 = ∞, and ϵ 0 and µ 0 are the electrical permittivity and magnetic permeability of vacuum.When this multi-layered medium is excited from the first layer with a transverse-electric (TE) or transverse-magnetic (TM) wave, the fields in each layer can be written as a sum of two components: forward traveling and backward traveling.We can define a global reflection coefficient as the ratio of these two parts at the interfaces.For example, the field in layer i can be written as where R i,i+1 is called the global reflection coefficient between layer i and layer i + 1, and where z N = z N −1 is assumed but this assumption does not affect the results (in reality, z N → ∞).The two terms in (4) will be denoted as the F W 1 and BW 1 waves, corresponding to forward-traveling and backward-traveling waves, respectively, in the i th layer.
Similarly, the field in layer i + 1 can be written as in ( 4) The two terms in (6) will be denoted as the F W 2 and BW 2 waves in the (i + 1) st layer.Now consider the field in layer i + 1 at the interface z = z i .The forward-traveling wave F W 2 has two contributions: (i) the local transmission (T i,i+1 ) of F W 1 , and (ii) the local reflection (R i+1,i ) of BW 2 .Therefore from ( 4) and (6) we have where is the "propagator" in layer i + 1.
Similarly, consider the backward-traveling wave in layer i at the interface z = z i .The backward-traveling wave BW 1 also has two contributions: (i) the local transmission (T i+1,i ) of BW 2 , and (ii) the local reflection (R i,i+1 ) of F W 1 .Therefore, Solving ( 7) and ( 8) yields where we have assumed that z N = z N −1 (thus P N = exp[−jk N z (z N − z N −1 )] = 1), and The global transmission coefficient is In the above, P i is called the propagator for layer i since it denotes the propagation from z i−1 to z i .Equations ( 9) and ( 10) give the recursive relations for R i,i+1 and A i+1 .Obviously, the "initial" conditions are A 1 = E 10 e −jkxx e −jk1zz1 , for TE z H 10 e −jkxx e −jk1zz1 , for TM z From ( 13) and ( 9), we can find R N −2,N −1 , • • • , R 1,2 .Similarly, from ( 14) and (10), we can find A 2 , • • • , A N .
To find the transmission wave amplitude, we use ( 14) and (10) repeatedly and obtain where P N = 1, and is called the global transmission coefficient between layer 1 and layer N .Given the incident field, then the field in all layers can be obtained by ( 16).Note that if the last layer is air, then R N,N +1 = 0 and if it is a PEC, then R N,N +1 = −1.
Figs. 1 (a) and (b), these formulas provide us with very accurate approximations of both sides of Eq. (1) enabling us to calculate the energy band diagram of the device.ADDITIONAL NUMERICAL ANALYSIS Supp.Fig. 2. Quantum efficiency (Q eff ) as functions of gate voltage (Vg), drain-to-sourve voltage (V d ), and normalized incident power P/Pmax.λ = 561 nm.Observation: Q eff increases with increasing voltages and decreasing incident power.Supp.Fig. 3. Output current as functions of gate voltage (Vg), drain-to-sourve voltage (V d ), and normalized incident power P/Pmax.λ = 561 nm.Observation: Output current increases with increasing voltages and incident power.
Supp.Fig. 6.(a) The phototransistor consists of 4 layers: 0.65 nm thick MoS2, 270 nm thick SiO2, 1 µ thick Si, and perfectly electrical conductor (PEC).(b) The electric field intensity in the middle of the MoS2 layer as a function of the excitation wavelength.