Abstract
Vertical stacking of heterogeneous twodimensional (2D) materials has received considerable attention for nanoelectronic applications. In the semiconductor industry, however, the process of integration for any new material is expensive and complex. Thus, first principlesbased models that enable systematic performance evaluation of emerging 2D materials at device and circuit level are in great demand. Here, we propose an ‘atomtocircuit’ modeling framework for all2D MISFET (metal–insulator–semiconductor fieldeffect transistor), which has recently been conceived by vertically stacking semiconducting transition metal dichalcogenide (e.g., MoS_{2}), insulating hexagonal boron nitride and semimetallic graphene. In a multiscale modeling approach, we start with the development of a first principlesbased atomistic model to study fundamental electronic properties and charge transfer at the atomic level. The energy bandstructure obtained is then used to develop a physicsbased compact device model to assess transistor characteristics. Finally, the models are implemented in a circuit simulator to facilitate design and simulation of integrated circuits. Since the proposed modeling framework translates atomic level phenomena (e.g., bandgap opening in graphene or introduction of semiconductor doping) to a circuit performance metric (e.g., frequency of a ring oscillator), it may provide solutions for the application and optimization of new materials.
Introduction
Functionality of an electronic device originates from the interfacial properties of its constituent materials. Advancement of nanofabrication technology has opened up the possibility of realizing interfaces at their ‘ultimatelimit’ by vertical stacking^{1,2,3} or parallel stitching^{4} of 2D materials. Since these new materials inherit diverse electronic and optoelectronic properties, novel device functionalities could be engineered from such atomically thin interfaces.^{5} In such vertically stacked van der Waal’s heterostructures (vdWH),^{6} the individual layers are ‘glued’ together by weak van der Waal’s (vdW) forces of interaction,^{7} whereas the inplane atoms are strongly bound by covalent or ionic bonds. The heterointerfaces thus produced, are atomically sharp and selfpassivated, i.e., free of dangling bonds and trapped charges. These subtle attributes have encouraged the application of vdWHs as a platform for constructing sophisticated nanodevices such as fieldeffect transistors (FETs),^{8,9,10} tunnel devices,^{11,12} photodetector,^{13} lightemitting diode,^{14,15} solar cell,^{16} flexible electronics^{10} etc.
Among all types of FET devices studied theoretically or fabricated for experimental and commercial purposes, the MIS (metal–insulator–semiconductor) structure, which substitutes metal (or highly doped polysilicon) by semimetallic graphene, SiO_{2} (or highK gate dielectric) by insulating hBN, and Si by MoS_{2}, could pave the way for realizing thinnest possible FET. Roy et al.^{9} have demonstrated such experimental device that exhibits ntype behavior with an ON/OFF current ratio >10^{6} and ~33 cm^{2}/Vs electron mobility. The work of Lee et al.^{10} evidences implementation of a flexible and transparent FET claiming fieldeffect mobilities up to 45 cm^{2}/Vs. At the same time, Jeong et al.^{12} have demonstrated the operation of a MIS diode, where carrier tunneling is the principle transport process. In a similar device Vu et al.^{17} have obtained high photocurrent/darkcurrent ratio >10^{5} and ultrahigh photodetectivity of 2.6 × 10^{13} Jones. Using first principlesbased analysis, Zan et al.^{18} have demonstrated the enhancement of interlayer coupling and linear charge transfer between graphene and MoS_{2} layers upon application of homogeneous electric field, and conversely the weakening of interlayer coupling under applied biaxial strain.
It should, however, be noted that introduction of any new material in the process integration phase of technology development in semiconductor industry is an expensive and timeconsuming affair. It is also a difficult task to select appropriate 2D materials from the plethora^{19} without assessing their performance at circuit level. Thus, a modeling framework, that enables systematic performance evaluation of new materials at device and circuit levels before entering into capitalintensive manufacturing phase, is in great demand. Such models must be first principles based so that the assessment could be conducted even before the wafer is available. Despite significant efforts on synthesis and fabrication, community lacks such modeling framework, which can predict integrated circuit performance solely from the crystallographic information of the constituent transistor materials.
In this article, we propose hierarchical bottomup modeling methodology for all2D MISFET that bridges between three levels of abstraction viz. material, device, and circuit. It is noteworthy that unlike conventional MOSFETs (Metal Oxide Semiconductor Field Effect Transistors), in this device, the gate electrostatics is dictated by the effects of dipole–dipole interactions,^{20,21,22} prevailing at the interfaces of a typical vdWH. Thus, modeling methodologies reported earlier for 2Dsemiconductor FETs^{23,24,25} are not applicable here. Apart from this, the bandgap opening in graphene due to vdW interactions poses additional complexity to the device physics. To capture all these intricate atomic level phenomena at circuit level, we adopt several sophisticated techniques, e.g., density functional theory (DFT)based atomistic model, tightbinding Hamiltonian for graphene encompassing the effect of sublattice symmetry breaking, FermiDirac (FD) distribution of mobile charge carriers, driftdiffusion (DD) formalism^{26} with biasdependent diffusivity, and piecewise charge linearization (PWCL) technique.^{27} Finally, we arrive at the closedform expressions for drain current and terminal charges for the all2D MISFET, which are implemented in professional circuit simulator using VerilogAMS interface in order to conduct static and transient simulation of integrated circuits. We induce bandgap variation in graphene by tuning the interlayer distance between hBN and graphene^{28}, and observe its effect at transistor and circuit level. Since doped MoS_{2} is now commercially available,^{29} we further demonstrate a materialdevicecircuit coassessment scheme based on the semiconductor doping. The VerilogAMS module developed in this work connects the material modeling tools^{30,31,32} with industrial electronic design automation tools^{33,34} and thus promises to provide a solution to the DTCO (designtechnology cooptimization) challenges for new materials.
Results
The proposed ‘atomtocircuit’ modeling framework has been synopsized in Fig. 1. We start with the DFTbased atomistic model development of vdWH in order to probe the energy band structure and interlayer charge transfer. Dielectric constant values and carrier mobilities of the materials have been acquired from reported first principlebased calculations.^{35,36} The material level attributes thus obtained are then used to develop physicsbased compact device model to understand the properties of MIS capacitor and FET. Limited by the computational budget, we consider only two layers of hBN for developing the atomistic model, although experiments^{12,17} were conducted using several layers of it. However, our device model parameters could easily be calibrated to any number of hBNlayer based vdWH. Finally, those closedform expressions of current and charges are implemented in professional circuit simulator to enable design and simulation of a digital logic inverter and a 15stage ring oscillator.
Energy band structure and interlayer charge transfer
The geometrically optimized atomic structure of graphenehBNhBNMoS_{2} is shown in Fig. 2(a) (see methods for details). The equilibrium spacings between graphenehBN (d_{1}), MoS_{2}hBN (d_{2}) and hBNhBN (d_{3}) were found to be 3.2, 4.9 and 3.1 Å respectively. It is wellknown^{28,37,38,39,40,41,42} that the energy gap at Dirac point of graphene is strongly influenced by the substrates and the stacking pattern of the layers. This is due to the fact that there is always an interplay between the strain energy of graphene, when placed on a substrate^{38} and the reduction of symmetry from C_{6v} to C_{3v} point group due to breaking of the chemical equivalency of carbon lattice sites.^{21} For graphene on top of hBN (CB configuration), there is an effective charge transfer from C_{0} (C atom on top of BN hexagon center) atom to C_{B} (C atom on top of B atom) atom owing to the difference in electronegativities of B and N atoms of hBN.^{41} It results into an onsite energy difference of C atoms at different sublattices. Thus, sublattice symmetry is broken and the energy levels of p_{z} orbitals of carbon atoms shift to open a bandgap, exactly equal to the onsite energy difference at the Dirac point of otherwise gapless semimetallic graphene. The bandgap opening in graphene (analogous to depletion in polysilicon gate of Si MOSFETs) is expected to play a very crucial role in dictating the MIS capacitor and transistor characteristics. Since this bandgap might be perturbed due to material defects, externally applied strain or any uncertainties in process variations, we emulate such bandgap variation by changing the graphene–hBN interlayer distance d_{1} (3.2 ± 0.2 Å) of the graphenehBNMoS_{2} system.
Figure 2(b) illustrates the band structures of the vdWH with d_{1} varying from 3.0 to 3.4 Å with an interval of 0.2 Å. The Fermi level (E_{F}) for the band structure is referenced to 0 eV. As depicted in Fig. 2(b), graphene has a bandgap of 0.06 eV at equilibrium separation and the bandgap almost doubles at 0.2 Å decrease of d_{1}. Besides, the energy dispersion near Dirac point of graphene resembles more of a parabolic nature^{28} rather than linear one and the parabolicity increases with the reduction of d_{1}. It also reveals that the otherwise pristine MoS_{2} becomes ptype in nature due to vdW stacking at equilibrium spacing (d_{1} = 3.2 Å), which is in agreement with the reported hBNMoS_{2} heterostructure.^{43} We further observe that the ptype nature becomes more evident with decreasing values of d_{1}, whereas it tends to behave like ntype semiconductor while d_{1} increases from its equilibrium value. However, the bandgap of MoS_{2} remains unaltered with a value of 1.71 eV. The absolute numeric values of Δ_{GC}, Δ_{GV}, Δ_{MC}, Δ_{MV}, which represent the shifts in energy of grapheneCBM (conduction band minima), grapheneVBM (valence band maxima), MoS_{2}CBM and MoS_{2}VBM respectively, as measured from E_{F} and the values of effective masses (viz. \(m_{{\mathrm{e}}{\mathrm{G}}}^ \ast\), \(m_{{\mathrm{h}}{\mathrm{G}}}^ \ast\), \(m_{{\mathrm{e}}{\mathrm{M}}}^ \ast\), \(m_{{\mathrm{h}}{\mathrm{M}}}^ \ast\); where the subscripts denote e: electron, h: hole, G: graphene, M: MoS_{2}) as calculated from band structures, are presented in Table 1 (m_{0} being rest mass of electron).
The charge transfer from C_{0} to C_{B} atom can be quantified by the Mulliken population analysis as listed in Table 2. It reveals the fact that the amount of charge transfer (Δ_{q}) increases with decreasing values of d_{1}, indicating wider bandgap opening in graphene. To further investigate the charge redistribution at the heterointerfaces, we calculate the electron density difference (EDD), Δρ, for three different values of d_{1}, averaged along the z direction as shown in Fig. 2(c). The EDD is computed as \(\Delta \rho = \rho _{G + hBN + MoS_2}  \rho _G  \rho _{hBN}  \rho _{MoS_2}\), where ρ represents electron density. In Fig. 2(c), both charge accumulation (positive peaks) and depletion (negative peaks) regions are found at the interfaces. Clearly, the charge redistribution is more pronounced at the graphene–hBN interface rather than hBN–hBN or hBN–MoS_{2} interfaces and it increases at the graphene–hBN interface with decreasing d_{1}, while remains almost unaltered at the other two interfaces. When d_{1} < 3.2 Å, the surface charge repulsion (Pauli repulsion) dictates the charge redistribution in graphene–hBN interface, whereas for d_{1} > 3.2 Å, an accumulation region at the middle of the interlayer spacing indicates strong orbital hybridization^{22} between graphene and hBN. Quite obviously, there is a concoction of Pauli repulsion and orbital hybridization prevalent at d_{1} = 3.2 Å. This orbital hybridization originates from the orbital contributions of distinct atoms to the band structure (detailed in Supplymentary Information). The VBM and CBM of graphene at Kpoint in Brillouin zone are contributed by the p_{z} orbitals (πbonding and π^{*}antibonding states respectively) of C atoms, which are prone to be perturbed by the interactions with πelectron clouds of hBN, localized around N centers. But this should not be the case for MoS_{2} since its CBM and VBM are mostly composed of \(d_{z^2}\) and \(d_{x^2  y^2}\) orbitals of Mo atoms; whose interactions with s and p orbitals of B and N atoms are very limited. However, a comparatively small accumulation region at MoS_{2}–hBN interface indicates orbital hybridization between s and p orbitals of hBN and nearer S atoms. Interestingly, there is a dipole formation at MoS_{2}–hBN interface which in turn causes the workfunction modification,^{22} signifying band alignment of MoS_{2} and hBN. This effectuates a charge transfer between MoS_{2} and hBN, thereby dictating the ptype character of MoS_{2}. To further quantify the charge redistribution, we have calculated the area under EDD curve at MoS_{2}–hBN (between hBN and nearest S atoms) and graphene–hBN interfaces as detailed in Table 3. The higher the value, more pronounced is the charge redistribution with more chemical interactions occurring at the interface. As depicted, the comparatively smaller area under EDD at MoS_{2}–hBN interface remains almost unaltered with variation of d_{1}, while for graphene–hBN interface it increases with decreasing d_{1}. The threshold voltage of MIS capacitor is expected to increase with the bandgap broadening in graphene and the upward shifting of MoS_{2} band structure. Furthermore, since the charge distribution at the interlayer spacing can be perceived as a way of energy storing, the increment of EDD area at the graphene–hBN interface with decreasing d_{1} signifies possible enhancement of the quantum capacitance of graphene as will be elucidated in the next section.
The electronic band structure of the aforesaid vdWH comprising of ntype and ptype MoS_{2} monolayer is depicted in Fig. 2(d). The downward or upward shifting of the bands with respect to the Fermi level is clearly visible for ntype and ptype doping respectively; although the band gaps and effective masses of graphene and MoS_{2} remain unaltered. The values of effective masses and absolute positions of conduction and valence band edges of graphene and MoS_{2} (n or ptype) in the energy scale are provided in Table 1. Figure 2(d) reveals that even though the graphene layer is kept pristine and only MoS_{2} layer undergoes electrostatic doping, due to interlayer charge transfer and thereby charge redistribution, graphene also becomes effectively ntype or ptype according to the doping in MoS_{2}. Since this is mediated by interlayer charge transfer, graphene should eventually tend to remain pristine if we increase the number of hBN layers in between graphene and MoS_{2}.
MIS capacitor and transistor characteristics
It is worth noting that in previous studies^{18} the electric field within the material system was considered to be homogeneous. The charge distribution inside a material system under spatially varying electric field is beyond the scope of pure DFT calculations.^{44} The proposed model is thus useful to understand the CV characteristics of the vdWH system. The transistor schematic of the concerned vdWHbased MISFET is illustrated in Fig. 3(a), where ‘S’ and ‘D’ denotes source and drain (deemed to be ‘ideal’) respectively, V_{GS} and V_{DS} refer to gatetosource and draintosource voltage respectively. The nonideal effects arising from the charges, unintentionally present in semiconductor substrate have been neglected in this work. The coined device symbol is portrayed in inset of Fig. 3(a). The electrical equivalent circuit of this MIS capacitor is shown in Fig. 3(b) that consists of three capacitors in series connection viz. quantum capacitance of graphene (C_{qG}), insulator capacitance of bilayer hBN (C_{I}) and quantum capacitance of MoS_{2} (C_{qM}). Considering the semiconductor body to be grounded, if we apply a gate voltage V_{G} in graphene, equal amount of charge will be stored in all three capacitors, thereby causing a potential drop of Ψ_{G} across C_{qG}, Ψ_{I} across C_{I} and Ψ_{S} across C_{qM}. For convenience, the relevant band structure parameters used in our model are depicted in Fig. 3(c). The calibrated tightbinding Hamiltonian (see methods section) and corresponding energy dispersion relations, used for device model development, are found to be in good agreement with DFT calculation as shown in Fig. 3(d), where \(m_{eG}^ \ast = \Delta _{GC}/v_F^2\) (v_{F} being the reduced Fermi velocity of graphene over hBN) and \(m_{hG}^ \ast = \Delta _{GV}/v_F^2\). For a given bias condition, the variation of the inversion charge density in MoS_{2} layer (\(Q_I^\prime\)) along the MISFET channel as a function of surface potential Ψ_{S} is demonstrated in Fig. 3(e). Similar to conventional SiMOSFETs, \(Q_I^\prime\) holds a linear relationship with Ψ_{S} as long as the band gap opening in graphene is small. However, significant nonlinearity creeps in as the bandgap broadens, which necessitates PWCL technique based drain current and terminal charge modeling as discussed in methods section.
Figure 4(a) and (b) respectively depicts the plots of C_{qG} and C_{qM} as a function of V_{GS} (swept from 0 V to 1.5 V) with V_{DS} set to 0 V. It is found that both C_{qG} and C_{qM} operate within the quantum capacitance limit (i.e., q^{2}g_{2D}, where q is fundamental electronic charge and g_{2D} is 2D density of states). The value of C_{qG} increases as we decrease the interlayer separation due to the increased charge redistribution at graphene–hBN interface as mentioned before. However, unlike C_{qG}, C_{qM} doesn’t even tend to saturate at large gate bias of 1.5 V. The total gate capacitance of the device as seen from the gate terminal, i.e. C_{gg}, can be calculated as \(C_{gg} = \left( {1/C_{qG} + 1/C_I + 1/C_{qM}} \right)^{  1}\) using the value of C_{I}, computed to be ~26.5 fF considering the relative permittivity of bilayer hBN, i.e. ε to be 1.9.^{35} The variation of C_{gg} as a function of V_{GS} has been plotted in Fig. 4(c) keeping V_{DS }= 0 V. It reveals that saturated C_{gg} increases with increasing bandgap of graphene and the crossing between individual graphs is a direct consequence of threshold voltage increment of the device. In the overall capacitancevoltage characteristics, C_{gg} is mostly dominated by C_{qG}.
Figure 4(d) and (e) respectively provides the transfer and drain characteristics of the MISFET, where I_{DC} denotes the dc drain current flowing in the channel. The transfer characteristics of Fig. 4(d) clearly points out the threshold voltage increment of the transistor as bandgap opening in graphene becomes more pronounced with decreasing d_{1}. However, the subthreshold slope remains almost unaltered in all three cases. The subthreshold swings were calculated to be ~60.48 mV/decade for all cases. This is because in subthreshold regime C_{qM} is found to be much smaller than C_{qG} and C_{I}, which makes the subthreshold slope factor nearly unity. In the drain characteristics, delineated in Fig. 4(e), I_{DC} attains a maximum value of ~60 μA/μm for d_{1} = 3.2 Å. Since the threshold voltage is maximum for d_{1} = 3.0 Å, it bears the lowest pinchoff voltage (V_{P}) among all three cases and that’s why the corresponding drain current saturates at earliest having smallest saturation drain current value (I_{D,sat}). From similar analogy, the transistor with d_{1} = 3.4 Å has highest V_{p} and therefore it should have maximum I_{D,Sat}. However, this is not the case due to the tradeoff between V_{p} reduction and C_{gg} increment. The higher capacitance value for d_{1} = 3.2 Å drives the drain current of the transistor with d_{1} = 3.2 Å to be greater than that with d_{1} = 3.4 Å. On the other hand, upon introduction of ntype doping in MoS_{2}, the threshold voltage of the device decreases and accordingly I_{DC} attains a much higher value (~105 μA/μm) at saturation (see Supplementary Information). Conversely, the threshold voltage of the transistor with ptype MoS_{2} as channel material increases and I_{DC} saturates with a value as small as ~17 μA/μm (see Supplementary Information). We also plot the nature of two important transcapacitances: C_{gd} (calculated at V_{GS }= 1.5 V) and C_{dg} (calculated at V_{DS} = 1.5 V) in Fig. 4(f), which follow the similar trends as that of C_{gg}. For all these characteristics, the mobility value is taken to be 400 cm^{2}/Vs as estimated by the DFT + Bolzmann formalism.^{36}
Digital logic performance
The equations for proposed compact device model are shown in Fig. 5 (where V_{CB} is channel potential/imref; W and L—both taken to be 1 μm, specifies the channel width and length respectively; μ and D, respectively denotes lowfield mobility and fielddependent diffusivity; Q_{G}, Q_{D}, Q_{S} and i_{G}, i_{D}, i_{S} refers to terminal charges and currents at gate, drain and source respectively; and finally \(Q_{I_n}^\prime\) and \(\psi _{S_n}\) symbolizes the values of \(Q_I^\prime\) and Ψ_{S} at n^{th} breakpoint). The surface potential equation ζ = 0 (Eqn. (F1) in Fig. 5) is a function of gate bias, imref and material parameters, which are obtained from the DFT calculations. We solve this equation numerically for different bias conditions to calculate the surface potentials and inversion charge densities, and thereby the dc drain current (I_{DC}) and terminal charges (Q_{G}, Q_{D}, and Q_{S}) become explicit polynomials of them. Efficient algorithms^{45,46} could be developed in this regard to solve such equations inside the circuit simulator. We implement this model in professional circuit simulator^{34} or SPICE (simulation program for integrated circuit emphasis) using its Verilog–AMS interface in order to conduct static and transient simulation of integrated circuits. As shown in Fig. 5, SPICE engine assigns terminal voltages (viz. V_{G}, V_{D} and V_{S}) to the VerilogAMS module, which computes terminal currents (viz. i_{G}, i_{D} and i_{S}) according to Eqn. (F1)–(F5), and then returns them to the individual terminals to be further processed by the SPICE engine. A direct implementation of these equations in SPICE without any numerical pitfall is very tedious and timeconsuming process. For simulation of basic logic circuits, however, a lookup table approach will suffice and be followed in this work. First, we have designed a resistiveload inverter (see Fig. 6(a)) and simulated its voltage transfer characteristics (VTC) with the input voltage varying from 0 V to 1.5 V, as plotted in Fig. 6(b). It reflects the effect of bandgap broadening of graphene by indicating the positive shift of threshold voltage of the inverter, designed with the MISFET having smaller d_{1}. Also plotted in Fig. 6(b) are the VTC of the logic inverters designed with the MISFETs having n and ptype MoS_{2} monolayers as semiconducting channel materials. As depicted in Fig. 6(b), the inverter corresponding to ntype MoS_{2} channel experiences a smaller threshold voltage, whereas the inverter with ptype MoS_{2} has a greater threshold voltage. Clearly, this effect of threshold voltage modification is quite similar to the effect of band gap modulation of graphene. Subsequently, we have designed 15stage ring oscillator circuits with these resistiveload inverters to study the transient response. The output voltage waveforms of three ring oscillators (each designed with the MISFETs with different values of d_{1}) are shown in Fig. 6(c). It exhibits moderate changes in oscillation frequencies as mentioned in the figure itself. To be precise, variation of d_{1} by ± 0.2 Å results in increment of the oscillating frequency by a factor of 1.085 and conversely, a decrement of the same by a factor of 0.85. The ring oscillator featuring d_{1} = 3.0 Å possesses the lowest frequency of oscillation among all. The waveforms of the oscillators featuring MISFETs with n and ptype MoS_{2} channel have also been plotted in Fig. 6(c) and similar trends of increment and decrement of oscillating frequencies are observed. For ntype doping, the frequency increases to 1.125f_{0} and it decreases to 0.85f_{0} for ptype doping, with f_{0} = 5.75 MHz being the frequency of the ring oscillator featuring undoped MoS_{2} channel and d_{1} = 3.2 Å. Good convergence of the simulations in all cases advocates the practicality and applicability of the proposed model for largescale circuit simulation in order to assess the device performance at circuit level at the early stage of technology development.
Discussion
In this article, we propose a first principlebased ‘atomtocircuit’ modeling methodology, which can assess the impact and predict the performance of a material or materialsystem at device and circuit level even in the absence of any experimental data. We demonstrate two applications of the proposed model: (i) effect of bandgap fluctuation of graphene on the circuit performance and (ii) materialdevicecircuit coassessment scheme considering semiconductor doping as a design parameter. A schematic view as depicted in Fig. 6(d) captures the basic philosophy of the proposed modeling scheme, which bridges between first principlebased material modeling tools and industry standard circuit simulators. The model equations could further be simplified to be applicable for any 2Dmaterialchannel MOSFET with conventional gate stack. Furthermore, since we use industrystandard DD formalism for SPICE model development, the proposed model is ‘core’ in nature and standard techniques (i.e., precorrection) for including several small geometry effects (such as drain induced barrier lowering, velocity saturation etc.) could easily be conjoined with it. Henceforth, this stupendous flexibility and extensive scope of applicability should encourage our modeling framework to stand high.
Methods
Atomistic model development
In order to carry out the first principlesbased calculations, the DFT code as implemented in atomistix tool kit (ATK)^{30} is employed in conjunction with generalized gradient approximation (GGA) exchange correlation and the PerdewBurkeErnzerhof (PBE) functional.^{47} Apart from that, we have used the OpenMX (Open Source package for Material eXplorer) normconserving pseudopotentials^{48,49} as implemented in ATK database with pseudoatomic orbitals (PAO) and ‘medium’ basis sets for the constituent atoms. To be precise, the LCAO (linear combination of atomic orbitals) basis sets for ‘C’, ‘B’, ‘N’ and ‘S’ atoms are adopted to be s^{2}p^{2}d^{1}, and s^{3}p^{2}d^{1} for ‘Mo’ atom. The kpoints in the MonkhorstPack grid^{50} were set to 9 × 9 × 1 along with the density mesh cutoff of 90 Hartree. The maximum iteration steps were set to 200 using Pulay mixer algorithm and the Poisson solver we followed was the fast Fourier transform (FFT). To account for the vdW interactions in the heterostructures, we have incorporated Grimme DFTD2 semiempirical correction^{51} as defined in ATK database in combination with counterpoise correction^{52} that deals with the basis set superposition error (BSSE) of LCAO basis sets. Furthermore, for all the structures, we have provided sufficient vacuum in the perpendicular, i.e., normaltotheplane direction to avoid spurious interactions between periodic images. The geometry optimization of the unit cells of graphene, hBN, and semiconducting 2HMoS_{2} were performed using the LBFGS (Limitedmemory BroydenFletcherGoldfarbShanno) algorithm^{53} with maximum stress tolerance value of 0.001 eV/Å^{3} and force tolerance of 0.01 eV/Å. Keeping in mind the commensurability condition, the interfaces between these constituent materials were formed by 4 × 4 MoS_{2} supercell (lattice parameter = 12.77 Å), 5 × 5 hBN supercell (lattice parameter = 12.63 Å) and 5 × 5 graphene supercell (lattice parameter = 12.36 Å), leaving 0.75% and 1.88% mean absolute strain on hBN and graphene respectively, which are reasonably small. In order to model n and ptype doped MoS_{2} monolayers, we have exercised the method of electrostatic doping (using n and ptype atomic compensation charges^{54,55}) available in ATK^{30} instead of explicitly substituting the Mo atoms by elemental dopant atoms (e.g., Nb for ptype^{56} and Au, Re for ntype^{57}). This is because in order to achieve a practical n or ptype doping concentration in MoS_{2} (typically 1 × 10^{17} − 1 × 10^{19} cm^{−3})^{58} by substitutional doping, a very large supercell is needed, which demands very large computation cluster. Such electrostatic doping scheme is very effective and advantageous since it does not depend on exact detail of dopant atoms.^{54} Here, the doping concentration is set to 1 × 10^{19} cm^{−3} for both n and ptype doping.
Development of energy dispersion relations
We start with the 2band Hamiltonian of graphene that incorporates the effect of sublattice symmetry breaking.^{42,59,60} In nearestneighbor π– electron tightbinding parlance, where sublattice symmetry breaking can be parameterized by a mass term, the Hamiltonian describing electronic properties of graphene near the Fermi level can be approximated as^{59}:
where, k is wave vector relative to Dirac (K or K′) point, ħ is modified Planck’s constant and Δ is the onsite energy difference between two sublattices A and B, which in turn equals to \(m^ \ast v_{\mathrm{F}}^2\) (\(m^ \ast\) = \(m_{{\mathrm{eG}}}^ \ast\) or \(m_{{\mathrm{hG}}}^ \ast\)). This Hamiltonian operates on spinors \(\Psi = \left( {\begin{array}{*{20}{c}} {\Phi _{\mathrm{A}}} \\ {\Phi _{\mathrm{B}}} \end{array}} \right)\) (where Φ_{A} and Φ_{B} are the amplitudes of the wavefunctions of two sublattices), to produce the energy eigenvalues of the form: \(E = \pm \sqrt {\Delta ^2 + \left( {\hbar v_{\mathrm{F}}k} \right)^2}\). For graphene on top of hBN, Δ ≠ 0 and the energy dispersions near the Dirac points can be approximated to be parabolic having the form^{28}:
Here, ‘±’ sign corresponds to the conduction band (CB) with Δ = Δ_{GC} and valence band (VB) with Δ = Δ_{GV} respectively, defining the nonzero bandgap introduced at Dirac point as: E_{gG} = Δ_{GC} + Δ_{GV}. Although for ±0.2 Å variation of d_{1}, the bandgap opening is symmetric (i.e., Δ_{GC} = Δ_{GV}) about E_{F}, in general Δ_{GC} may not be equal to Δ_{GV} for further reduction of d_{1}. On the other hand, the band structure of MoS_{2} can also be approximated to be simple parabolic in nature, written as^{23,25}:
for CB (VB) (where ‘+’: CB and ‘–’: VB), featuring a bandgap of E_{gM} = Δ_{MC} + Δ_{MV}.
Thereafter, we obtain the expressions for density of states (DOS) of conduction and valence bands of graphene and MoS_{2} as: \(g_{{\mathrm{Gn}}} = \frac{{g_{{\mathrm{sG}}}g_{{\mathrm{vG}}}\Delta _{{\mathrm{GC}}}}}{{2\pi \left( {\hbar v_{\mathrm{F}}} \right)^2}}\) (for graphene CB), \(g_{{\mathrm{Gp}}} = \frac{{g_{{\mathrm{sG}}}g_{{\mathrm{vG}}}\Delta _{{\mathrm{GV}}}}}{{2\pi \left( {\hbar v_{\mathrm{F}}} \right)^2}}\) (for graphene VB), \(g_{{\mathrm{Mn}}} = \frac{{g_{{\mathrm{sM}}}g_{{\mathrm{vM}}}m_{{\mathrm{eM}}}^ \ast }}{{2\pi \hbar ^2}}\) (for MoS_{2} CB) and \(g_{{\mathrm{Mp}}} = \frac{{g_{{\mathrm{sM}}}g_{{\mathrm{vM}}}m_{{\mathrm{hM}}}^ \ast }}{{2\pi \hbar ^2}}\) (for MoS_{2} VB). Here, g_{sG}, g_{sM} are spin degeneracies (both taken as 2) and g_{vG}, g_{vM} are valley degeneracies (both taken as 2) of graphene and MoS_{2} respectively.
MIS capacitor modeling
Employing FermiDirac statistics, we first calculate the intrinsic carrier concentrations in graphene (MoS_{2}) as:
where, n_{G} (n_{M}) and p_{G} (p_{M}) stands for electron and hole concentrations in graphene and MoS_{2} respectively, K is the Boltzman constant and T is temperature (taken as 300 K). Now, if we symbolize net electron and hole concentrations in graphene and MoS_{2} (upon application of V_{G}) by n_{netG}, p_{netG}, n_{netM} and p_{netM} respectively, then the total charge densities in these two layers can be written as \(Q_{\mathrm{G}}^\prime = q\left( {p_{{\mathrm{netG}}}  n_{{\mathrm{netG}}}} \right)\) for graphene and \(Q_I^\prime = q\left( {p_{{\mathrm{netM}}}  n_{{\mathrm{netM}}}} \right)\) for MoS_{2}, where
Here, we consider the charge distribution inside MoS_{2} to be an ideal 2D sheet having no spatial variation, which removes the burden of solving Poisson’s equation beforehand. On the other side, Ψ_{G}, Ψ_{I} and Ψ_{S} sum up to satisfy the potential balance equation, which is:
where Ψ_{I} can be calculated as:
Here, we neglect any leakage current through the hBN layers. Now using Eqns (5)–(7), the charge balance equation can be solved numerically for Ψ_{S}, which can be written as:
that ultimately takes the form of Eqn. (F1) of Fig. 5 with V_{CB} set to zero.
Once we get the numeric values of Ψ_{S}, the other two potentials Ψ_{G} and Ψ_{I} can be easily computed, thereby facilitating the computation of quantum capacitances viz. C_{qG} and C_{qM} using the following expressions:
For a similar device with more than two hBN layers, the model equations will remain all the same, however, the parameters viz. Δ_{GC}, Δ_{GV}, Δ_{MC}, Δ_{MV}, v_{F}, \(m_{{\mathrm{eM}}}^ \ast\), \(m_{{\mathrm{hM}}}^ \ast\) and ε need to be recalculated using DFT. Similarly, for the heterostructures with n or ptype MoS_{2} monolayers, the whole set of equations will essentially remain same, provided the aforesaid parameters are obtained from DFT calculations of that particular heterostructure. This is because the effect of doping has been effectively captured here through the positions of the band extrema in the energy scale and corresponding effective masses and that basically rules out the necessity of introducing a doping term explicitly in the charge neutrality equation. It is noteworthy that the band structure obtained from DFT calculation is qualitatively similar to the previously reported work,^{18} which uses different simulation toolkit and deals with single hBN layer. In this aspect, the proposed model equations are independent of the exchangecorrelation functionals or pseudopotentials; only the relevant model parameters need to be calculated accordingly.
MISFET modeling
To formulate the drain current and terminal charges, we adopt the semiclassical DD formalism,^{26,61} which has been practiced in industrial top–down compact modeling methodology over the years. We can take the benefits arising from wellestablished techniques available to add different small geometry effects to the core model as future work. It also allows us to develop the model without using any unphysical model parameters or interpolating function. Since the proposed model involves only material parameters (CBMs, VBMs, effective masses, dielectric constant, and mobility), which could be calculated by first principlesbased methods, it can predict the device and circuit characteristics just from the crystallographic information of the constituent materials.
In DD formalism, the dc drain current equation reads^{61}:
where, Ψ_{S0}, Ψ_{SL} and \(Q_{I0}^\prime\), \(Q_{IL}^\prime\) are the values of Ψ_{S} and \(Q_I^\prime\) at source (x = 0) and drain (x = L) end of the transistor respectively, with x being considered as the direction of transport. Since we explicitly use FD statistics to describe carrier occupation probability, instead of using typical Einstein relation, we treat the diffusivity coefficient D to be bias dependent.^{62} We express D as^{63} \(\left. {D = \mu Q_I^\prime \left( {d\it\Psi _{\mathrm{S}}/dQ_I^\prime } \right)} \right_{V_{{\mathrm{DS}}} = 0}\) to ensure zero drain current at V_{DS} = 0. Due to complex nature of ζ, an analytical solution of Eq. (9) is not possible. Hence, we apply charge linearization technique^{27} to obtain a closed form expression of the drain current. In case of conventional SiMOSFET, for a given bias condition, \(Q_I^\prime\) changes quasilinearly with Ψ_{S} along the channel and hence \(Q_I^\prime\) is approximated as a linear function of Ψ_{S} in Eqn. (10), which results in a quadratic relationship between I_{DC} and Ψ_{S}. Such relationship is very useful to obtain closed form expression of terminal charges under WardDutton (WD)^{64} charge partitioning scheme. However, as shown in Fig. 3(e), due to the bandgap opening in graphene, \(Q_I^\prime\) does not always maintain a linear relationship with Ψ_{S} in case of vdWH MISFET. Thus, in this work we adopt PWCL technique^{27} (see Supplementary Information). Using the expression of drain current, the terminal charges could be computed under WD charge partitioning^{64} scheme as: \(Q_{\mathrm{G}} =  W\mathop {\int}\limits_0^L {Q_{Ix}^\prime \left( x \right)dx}\), \(Q_{\mathrm{D}} = W\mathop {\int}\limits_0^L {\frac{x}{L}Q_{Ix}^\prime \left( x \right)dx}\) and Q_{S} = −(Q_{G }+ Q_{D}), where \(Q_{Ix}^\prime\) is the value of \(Q_I^\prime\) at any position x in the channel. Similar to stateoftheart surface potentialbased SiMOSFET models,^{65} Eqns. (F2)–(F5) are ‘singlepiececontinuous’ equations, valid for all regimes of transistor operation.
Data availability
The authors declare that the main data supporting the findings of this study are available within the paper and its Supplementary Information file. MATLAB and VerilogAMS codes are freely available at osf.io/me236. Other relevant data are available from the corresponding author upon request.
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Acknowledgements
This work was supported by Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India, under Grant SB/S3/EECE/0209/2015. We would like to thank G.N. Kadloor, DESEIISc, for his help in preparing Fig. 6(d).
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B.D. performed the DFT calculations, developed the compact device model, conducted SPICE simulations, and analyzed final results. S.M. conceived the problem statement and overall supervised the work. All authors contributed in the writing.
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Das, B., Mahapatra, S. An atomtocircuit modeling approach to all2D metal–insulator–semiconductor fieldeffect transistors. npj 2D Mater Appl 2, 28 (2018). https://doi.org/10.1038/s4169901800733
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DOI: https://doi.org/10.1038/s4169901800733
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