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An Intuitive Equivalent Circuit Model for Multilayer Van Der Waals Heterostructures Abhinandan Borah, Punnu Jose Sebastian, Ankur Nipane, and James T

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An Intuitive Equivalent Circuit Model for Multilayer Van Der Waals Heterostructures Abhinandan Borah, Punnu Jose Sebastian, Ankur Nipane, and James T 1 An Intuitive Equivalent Circuit Model for Multilayer van der Waals Heterostructures Abhinandan Borah, Punnu Jose Sebastian, Ankur Nipane, and James T. Teherani Abstract—Stacks of 2D materials, known as van der Waals (vdW) heterostructures, have gained vast attention due to their Layer 1 interesting electrical and optoelectronic properties. This paper Dielectric 1 1 presents an intuitive circuit model for the out-of-plane electrostat- Layer 2 � ics of a vdW heterostructure composed of an arbitrary number of Dielectric 2 2D material / layers of 2D semiconductors, graphene, or metal. We explain the mapping between the out-of-plane energy-band diagram and the � metal elements of the equivalent circuit. Though the direct solution of n dielectric the circuit model for an -layer structure is not possible due to the Dielectric variable, nonlinear quantum capacitance of each layer, this work uses the intuition gained from the energy-band picture to provide Layer Dielectric � − an efficient solution in terms of a single variable. Our method �−1 employs Fermi-Dirac statistics while properly modeling the finite Layer � − � density of states (DOS) of 2D semiconductors and graphene. � − The approach circumvents difficulties that arise in commercial � �� TCAD tools, such as the proper handling of the 2D DOS and the simulation boundary conditions when the structure terminates Fig. 1. An n-layer vdW heterostructure with voltages applied to each layer. with a nonmetallic material. Based on this methodology, we have developed 2dmatstacks, an open-source tool freely available on nanohub.org. Overall, this work equips researchers to analyze, understand, and predict experimental results in complicated vdW [11], [13], [21]. For multilayer vdW heterostructures, however, stacks. Poisson’s equation results in several difficulties: Index Terms—2D materials, TMDC, graphene, van der Waals (i) there is no easy way to incorporate a 2D DOS; heterostructures, electrostatics, quantum capacitance, density of (ii) it is difficult to place grid points in the out-of-plane states, DOS direction due to the discrete nature of 2D materials; and (iii) boundary conditions are poorly defined when the struc- I. INTRODUCTION ture terminates with a nonmetallic material [22]. TRUCTURES fabricated from 2D materials have evolved Equivalent circuit models composed of geometric and quan- S from single-layer transistors to complex van der Waals tum capacitances have been proposed for devices composed of (vdW) heterostructures that have yielded exciting physics and a single layer or two different layers of 2D semiconductors [9], applications [1]–[6]. These vdW stacks consist of multiple 2D [12], [13]; however, these circuit models cannot be easily ex- layers on top of each other separated by either an insulator panded to n layers (as shown in Fig. 1) since they incorporate or a vdW gap (i.e., the innate atomic spacing between 2D the work function difference between the layers as a flatband layers) [7], [8]. With an increasing number of experiments voltage shift of the gate potential [9]. A single flatband voltage on novel vdW structures, tools are needed to explain their shift cannot accommodate the work function differences of an device characteristics. The circuit models developed so far arbitrary number of layers. Similarly, their model’s definition have focused on transistor characteristics of devices made of the voltage across the quantum capacitor (EF = qVC ) does from a single 2D material [9]–[13]. Research on 2D materials, not extend to multiple layers. however, extends beyond lateral transport in 2D FETs. Band- Here, we derive an equivalent circuit model for the vertical to-band tunneling [14]–[16], the hot-electron transistor [17]– electrostatics of a vdW heterostructure from the out-of-plane [19], and metal-to-insulator transitions [20] are examples of energy-band diagram, which provides an intuitive explanation phenomena where the vertical electrostatics of complex vdW of the circuit elements. We show that, although the direct heterostructures must be understood to explain the device solution of the equivalent circuit for an n-layer structure is not physics. possible due to the variable, nonlinear quantum capacitance Existing techniques to calculate the electrostatics — the of each layer, an efficient method for solving the circuit is charge density, Fermi level, and electrostatic potential — possible using a physics-based approach. Overall, the method of structures made from 2D materials solve either Poisson’s presented in this paper provides equation [10] or equivalent circuit models for the devices [9], (i) the ability to incorporate an arbitrary number of layers The authors are with the Department of Electrical Engineering, Columbia of 2D semiconductors, graphene, or metals with different University, New York, NY 10027, USA, (e-mail: [email protected]) work function into a vdW structure; Digital Object Identifier: 10.1109/TED.2018.2851920 1557-9646 c 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. 2 (ii) the ability to apply an independent potential bias to each (a) layer; Layer 1 (b) Layer 1 Dielectric (iii) the proper DOS for 2D semiconductors and graphene Layer 2 Layer 2 �1 that captures the quantum effects arising from the atomic thickness; � (iv) the handling of non-degenerate and degenerate carrier Layer Layer���� 2 Layer�� Dielectric Layer���� concentrations through the use of Fermi-Dirac statistics; �� and (v) a computationally efficient methodology that yields the Eq. 1 1 energy-band diagram in the out-of-plane direction — 1� � � � providing a clear, intuitive picture of the charge, potential, � � 1� � � � and Fermi level across the structure. � −� � −� ��−�� � � 1 (c) � −� ��1 − � II. EQUIVALENT CIRCUIT MODEL - + �,1 1�/� + - A. Two-Layer Structure � 2D material/ + 1 metal To derive the equivalent circuit for an n-layer vdW het- - Eq. 6 � erostructure, we begin by considering only two layers, as ��,1 dielectric ��� ��� + - shown in Fig. 2. The layers are either a 2D semiconductor, ��, graphene, or metal, while the dielectric between the layers is - + /q � either an insulator or a vdW gap [23]. Here, a vdW gap is treated as a dielectric with vacuum permittivity and atomic ��, � thickness (∼5 A).˚ In isolation, layers 1 and 2 have different Fig. 2. Energy-band diagrams of (a) two layers in isolation and (b) two work functions, indicated by W1o and W2o in Fig. 2(a). layers in contact with one another. Evac is the local vacuum level, Ec is the Figure 2(b) shows the out-of-plane energy-band diagram after conduction-band edge, Ef is the Fermi level, W is the work function, χ is the electron affinity, and Vox is the potential drop across the dielectric. The the layers are brought together in a stack, and voltages V1 and subscript o indicates parameters for a layer in isolation. (c) Equivalent circuit V2 are applied. From the energy-band diagram, the potential of the structure in (b). CQ is the quantum capacitance of a layer. drop across the dielectric (Vox) can be expressed as − − − − − qVox = χ2 χ1 +(Ec Ef )2 (Ec Ef )1 +q(V1 V2), (1) With the definition of VQ in Eq. (3), we rewrite Eq. (2) as where q is the elementary charge, χ is the electron affinity, W1o W2o − 0= V1 − − VQ,1 − Vox + VQ,2 + − V2, (6) Ec Ef is the difference between the conduction-band edge q q and the Fermi level, and V is the applied voltage. Using the which represents Kirchhoff’s voltage law (KVL) for the loop definition that Wo = χ +(Ec − Ef )o, Eq. (1) can be rewritten as shown in Fig. 2(c). The terms VQ and Vox are represented by capacitances in the circuit model since these voltages are qVox = W2o − W1o + [(Ec − Ef )2 − (Ec − Ef )2o] (2) dependent on the net charge in the respective layers. − − − − − [(Ec Ef )1 (Ec Ef )1o]+ q(V1 V2), CQ is indicated as a variable capacitance since its capaci- − where W is the work function, and the subscript o indicates tance changes as a function of VQ (or, equivalently, Ec Ef ). parameters for a layer in isolation. For each layer in a vdW A formal definition is provided in Appendix B. stack, we represent the change in Ec − Ef from its value in The constant voltage sources equal to the work function of isolation by defining a new term: the layers obviate the need for fitting parameters to account for the initial band bending across the device. The presence − − − qVQ =(Ec Ef ) (Ec Ef )o. (3) of these constant voltage sources is due to the definition of A change in VQ induces a change in the net charge (Q) on VQ in Eq. (3). [One could choose to redefine VQ such that the a layer due to the change in Ec − Ef . This represents the constant voltage offset is included as a part of the potential quantum capacitance of the layer, which is defined as across the quantum capacitance; however, by doing so, the ∂Q values of VQ for different layers would be shifted by the CQ = , (4) electron affinities, obscuring the physical significance of V ∂V Q Q since its sign would no longer be related to the net charge where VQ is the voltage across the quantum capacitance. We on the layer.] Moreover, our definition of VQ highlights the choose our definition of VQ such that the net charge on a layer flatband condition for a multilayer structure — when the − is zero when VQ = 0 or, equivalently, when (Ec Ef ) = voltage applied to every layer is equal to its work function, − (Ec Ef )o. The sign of VQ indicates the sign of the net the net charge on every layer is zero since all VQ and Vox are 1 charge on the layer. A positive Q yields a positive VQ.
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