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Quantum Capacitance of Coupled Two-Dimensional Electron Gases Journal of Physics: Condensed Matter ACCEPTED MANUSCRIPT Quantum Capacitance of Coupled Two-Dimensional Electron Gases To cite this article before publication: krishna bharadwaj Balasubramanian 2021 J. Phys.: Condens. Matter in press https://doi.org/10.1088/1361- 648X/abe64f Manuscript version: Accepted Manuscript Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors” This Accepted Manuscript is © 2021 IOP Publishing Ltd. 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Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. View the article online for updates and enhancements. This content was downloaded from IP address 14.139.38.199 on 02/03/2021 at 09:06 Page 1 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPCM-117480.R2 1 2 3 4 5 Quantum Capacitance of Coupled Two-Dimensional 6 7 Electron Gases 8 9 Krishna Balasubramanian 10 11 Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 12 13 Abstract: 14 15 Quantum capacitance effect is observed in nanostructured material stacks with reduced density 16 of states. In contrast to conventional structures where two-dimensional electron gases (2DEG) with 17 18 limited density of states interact with a metal plate, here we explore the quantum capacitance effect in 19 a unique structure formed by two 2DEG in a graphene sheet and AlGaN/GaN quantum well. The total 20 21 capacitance of the structure depends non-linearly on the applied potential and the linear density of states 22 23 in graphene leads to enhanced electric field leakage into the substrate causing a dramatic 50% drop in 24 the overall capacitance at low bias potentials. We show theoretical projections of the quantum 25 26 capacitance effect in the proposed stack, fabricate the structure and provide experimental verification 27 of the calculated values at various temperatures and applied potentials. The wide swing in the total 28 29 capacitance is sensitive to the chemical potential of the graphene sheet and has multiple applications in 30 molecular sensing, electro-optics, and fundamental investigations. 31 32 33 Introduction: 34 35 Capacitance measurement is an invaluable tool to inspect the quantum nature of the electronic 36 subsystem at nanoscales [1]. Deviations from the conventional parallel plate capacitance 37 38 approximation, called the quantum capacitance effect, are observed due to limited density of states in 39 low dimensional electron gases [2], strong electron-electron correlation [3], and topological character 40 41 [4]. It was first discussed in the context of quantum well systems where the two-dimensional electron 42 43 gas (2DEG) incompletely screens the electric field from the top metal electrode, leading to field spillage 44 across the substrate and induced charges at the substrate bottom [2]. The effective total capacitance 45 46 value (Ctot) of a metal-2DEG capacitor is lower than its conventional geometric capacitance (metal 47 parallel plate capacitance) and is expressed as a series combination of the geometric metal-2DEG 48 49 capacitor (C1) and the bulk-substrate capacitor (C2) modulated by the extent of induced charges at the 50 substrate bottom (called quantum capacitance (C )) as given in equation 1. 51 Q 52 1 1 1 53 퐶 = 퐶 + 퐶 + 퐶 #1 54 푡표푡 1 2 푄 55 56 57 58 59 60 Accepted 1Manuscript AUTHOR SUBMITTED MANUSCRIPT - JPCM-117480.R2 Page 2 of 13 1 2 3 The density of states (D) of the 2DEG in quantum well structures is independent of the particle energy 4 5 [5] and the effect of quantum capacitance on the overall capacitance for the degenerate case can be 6 2 푔푣푚푒푒 analytically evaluated to be 퐶 = 퐷 = where is the valley degeneracy, me is the electron 7 푄 휋ℏ2 8 9 effective mass, e is the electronic charge and 푔푣 is the reduced Plank’s constant. ℏ 10 11 Quantum capacitance has been experimentally observed in several material systems such as Si 12 13 nanoscale transistors [6], quantum wells of Arsenides and Nitrides [7], and quantum dots [8,9]. Layered 14 2D materials also display quantum capacitance effects and experimental measurements with reduced 15 16 total capacitance have been observed in graphene, carbon nanotubes and transition metal 17 18 dichalcogenides [10,11]. The quantum capacitance of graphene devices with approximated linear 19 density of states [12] can be numerically evaluated and is reported to match well with experiments on 20 21 large area sheets and ribbons [13]. However, up until now, the quantum electronic systems with limited 22 density of states interface with metallic electrodes, with no observable limitations in the charge density. 23 24 Hence, the quantum capacitance effect can wholly be attributed to the 2DEG. Following the recent 25 advancements in 2D material synthesis techniques, one can envision a unique quantum capacitor with 26 27 both the plates having constrained density of states [14,15]. The charge distribution scheme and electric 28 29 field penetration into the substate bulk in such structures are more interesting than the conventional 30 metal-2DEG quantum capacitor, leading to significant and readily observable quantum capacitance 31 32 effect. The structure is easily accessible to traditional nanofabrication, and an excellent workbench to 33 study the quantum nature, interaction, and correlation between the two fundamentally different low 34 35 36 37 38 39 40 41 42 43 44 45 46 47 Figure 1: (A) Schematic representation of device under consideration. A CVD grown large area graphene film (black rectangle) is 48 taken as the forcing terminal, 2DEG (red block) at the interface of AlGaN/GaN quantum well and a bottom metal contact for the 49 stack are grounded. (B) Equivalent electrical circuit of the structure showing different capacitances. (C) Comparison of 2D D(E) in 50 graphene and other common 2DEG material systems plotted from equations 4 and 6 with appropriate effective masses. Linear energy 51 dependent density of states in graphene strongly affects the charge sharing between the plates. Steps in the DOS of GaAs and InAs 52 systems are intentionally not shown for the sake of clarity. In GaN, the first step typically appears higher than the energy ranges 53 considered here. 54 dimensional electronic systems. 55 56 57 Developments in large-scale synthesis of two-dimensional (2D) materials and advancements in 58 transferring techniques to arbitrary substrates permit facile vertical integration of several 2D layers with 59 60 conventional materials [16]. Hence, it is practically simple to realize a capacitor with a large area Accepted 2Manuscript Page 3 of 13 AUTHOR SUBMITTED MANUSCRIPT - JPCM-117480.R2 1 2 3 graphene sheet as one electrode and 2DEG arising in a quantum well such as AlGaN/GaN as another 4 5 electrode. A schematic of the proposed structure is presented in Fig. 1(A) with the substrate bottom 6 grounded with a metal plate. An equivalent electrical circuit showing the capacitances of the structure 7 8 is presented in Fig. 1(B). The unexplored question is the charge sharing scheme and effective 9 10 capacitance of this three-plate structure having a quantum state of the electron gas on two plates. Though 11 a graphene gated AlGaN/GaN transistor was previously demonstrated, no attempt to investigate the 12 13 actual gate capacitance was reported [17]. It must also be noted that this structure is fundamentally 14 different from multi-quantum well structures (super-lattices) [18], where seemingly similar case can 15 16 occur, as the two electronic systems forming the capacitor plates in the structure under consideration 17 have different density of states characteristics. Here we discuss, for the first time, the charge density 18 19 distribution between graphene and AlGaN/GaN based 2DEG at various temperatures and imposed 20 21 charge densities, and calculate the overall device capacitance. We show that effective capacitance of a 22 graphene-2DEG capacitor is lower than the previously known metal-2DEG capacitance under all 23 24 conditions indicating a much stronger quantum capacitance effect. Ctot drops dramatically to half the 25 geometric capacitance under low bias and non-degenerate conditions. Finally, we fabricate the proposed 26 27 device and show that the total capacitance at various temperatures and applied potentials match closely 28 with the theoretical predictions. 29 30 31 Results and Discussion 32 33 Let us consider a structure as shown in Fig.
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