Journal of Physics: Condensed Matter

ACCEPTED MANUSCRIPT Quantum of Coupled Two-Dimensional Electron Gases

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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 interact with a metal plate, here we explore the quantum capacitance effect in 19 a unique structure formed by two 2DEG in a 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 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 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

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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 . (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

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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. 1(A). An AlGaN/GaN quantum well with a buried 2DEG 34 separated from the surface by a material with dielectric constant ε1 and thickness d1. The remaining 35 36 epitaxial stack along with the substrate is taken to have a thickness d2 and an effective dielectric constant 37 ε terminated by a metallic electrode. A graphene sheet is transferred on the surface of the quantum well 38 2 39 and contacted using a small metallic contact negligible in comparison with the size of the graphene 40 41 sheet. The graphene sheet is taken to be the forcing terminal, 2DEG and the metal substrate contacts 42 are grounded. The density of states of 2DEG is constant across energy and proportional to the lattice 43 44 electron effective mass (me). The density of states in graphene is approximated to be linear [10] and is 45 plotted in Fig. 1(C) along with 2DEG density for few other commonly used material systems. The 46 47 density of states determines the total energy in the electronic distribution for an imposed charge density 48 and the relative values, as compared in Fig. 1(C), play a crucial role in the charge sharing between the 49 50 different capacitor plates. 51 52 At any potential (V) imposed on the structure shown in Fig. 1(A), the charge density on the graphene 53 54 sheet, quantum well 2DEG and the metal layer is taken to be 휎푔 , σ2D and σm respectively. The neutrality 55 56 condition imposes the relation. 57 58 휎푔 + 휎2퐷 + 휎푚 = 0#2 59 60 A solution of the following form can be assumed.

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1 2 3 2 2 휎2퐷 = ― 휎푔sin 휃 , 휎푚 = ― 휎푔cos 휃#3 4 5 6 where, θ is a variable to be optimized under the total energy minimization criteria and 휎푔 is the imposed 7 carrier density on the graphene sheet for a given potential V. 8 9 The occupation energy in the electron gas systems at a given carrier density can be calculated from the 10 11 density of states (D), which under the effective mass approximation for 2DEG is energy independent 12 13 and is given by [5] 14 15 푔푣푚푒 퐷 = 푛 #4 16 휋ℏ2 17 18 19 where n is the band index. Density of states for different materials systems from Eq. 4 are plotted in 20 Fig. 1(C). If a charge density of σ2D is imposed onto the 2DEG, the total occupation energy in the 2DEG 21 2퐷 22 (퐸푄 ) is then 23 24 ∞ 2퐷 ( 2퐷 ) 25 퐸푄 (휎2퐷) = ∫ 퐸퐷퐹(퐸,휇,푇) = 퐷ℑ1 휇 ,푇 #5 26 0 27 퐸푗푑퐸 28 ∞ 2D Where, F(E,μ,T) is the Fermi-Dirac function, ℑ푗(휇,푇) = ∫ (퐸 ― 휇(푇))/퐾 푇 is the Fermi-integral, and μ 29 0 1 + 푒 퐵 30 is the chemical potential on the 2DEG that depends on the charge density σ, and temperature T, and K 31 B 32 is the Boltzmann constant. 33

34 2푔푣퐸 If a charge density of 휎푔 is imposed on graphene with a linear density of states 퐷퐼(퐸) = 2 under 35 2휋(ℏ푣푓) 36 퐺 37 low carrier densities, the occupation energy in graphene (퐸푄) is 38 39 ∞ 2푔푣 퐸퐺(휎 ) = 퐸퐷 (퐸)퐹(퐸)푑퐸 = ℑ (휇푔,푇)#6 40 푄 푔 ∫ 퐼 2 2 41 0 2휋(ℏ푣푓) 42 푔 43 Where, gv is the valley degeneracy in graphene, 휇 is the chemical potential of graphene sheet and 푣푓 is 44 45 the graphene Fermi velocity. The electric field in the dielectric regions of AlGaN and epitaxial substrate 46 denoted by 휉1 and 휉2 respectively from the Gauss’s law is given by 47 48 휎 49 푖 휉푖 = #7 50 휖푖 51 52 where, 휎1 = 휎푔 and 휎2 = 휎2퐷 and 휖푖 is the dielectric function of the two regions mentioned in Fig. 1(A). 53 54 The total energy in the electric fields of the two regions (E1,2) is 55 56 1 2 57 휖푖휎푖 2 푑푖1 푑푖2 휎푖 푑푖 58 2 퐸푖 = 휖푖휉푖 푑푥 = 2 푑푥 = #8 59 ∫0 2 ∫0 휖푖 2휖푖 60

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1 2 3 where, d1,2 is the dielectric thickness of two regions. The total energy (ET) is then given by 4 5 퐸 = 퐸 + 퐸 + 퐸2퐷 + 퐸퐺 #9 6 푇 1 2 푄 푄 7 8 The charge density of individual plates (휎푔 , σ2D and σm) are obtained after solving for the chemical 9 potential of the individual systems by minimizing the total system energy for all applied electric 10 11 potentials and temperature as given below 12 13 2 2 휎푔푑1 휎푚푑2 2푔푣 14 ( 2퐷 ) ( 푔 ) 훿 + + 퐷ℑ1 휇 ,푇 + 2ℑ2 휇 ,푇 = 0#10 15 ( 2휖1 2휖2 2휋(ℏ푣 ) ) 푓 16 17 18 For any given charge density 휎푔 on graphene, the Eq. 10 can be numerically evaluated with Eq. 3 to 19 obtain the variable θ, and subsequently, the distribution of charges in the other two plates. 20 21 22 The carrier distribution on the 2DEG and the metal were calculated for typical graphene carrier 10 -2 12 -2 23 densities (10 cm - 10 cm ) at 0K using the equations above and the result is presented in Fig. 2(A). 24 25 The charge density on the bottom metal plate is about three orders lower than that on the 2DEG and

26 휎2퐷 linearly increases with increasing 휎 . In the degenerate limit, the ratio is a constant at all charge 27 푔 휎푔 28 29 densities following the results obtained previously by Luryi [2]. The negligible density on the metal, 30 in-comparison with the 2DEG density, also indicates vanishingly small quantum effects on the overall 31 32 capacitance at low temperatures. The above set of equations was also evaluated at temperatures from 33 0K to 300K and pertinent metal charge densities (σ ) at various temperatures are presented in Fig. 2(B). 34 m 35 The system was treated quantum mechanically under all temperature ranges and solved numerically as 36 the threshold for the classical Maxwell-Boltzmann approximation to hold true will depend on the 37 38 applied potential. Consistently higher charge densities are observed in the bottom metal plate at higher 39 40 temperatures causing additional electric field leakage into the substrate. Particularly, the charge density 41 on the metal is highest and of the same order as the graphene density close to the room temperature. It 42 43 is also important to note that at high temperatures (~room temperature), the imposed charge density on 44 graphene terminal induce relatively small changes in the metal charge density (flatter lines are observed 45 46 at higher temperatures in Fig. 2(B)). 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Figure 2 (A) Calculated carrier density on the metal and 2DEG for changes in imposed carrier density on graphene under degenerate conditions displayed in log-log plot. The relation is linear due to the constant density of states of 2DEG. (B) 20 Temperature dependent variations in the metal charge density for variations in graphene charge density. Significant effect of 21 temperature at low densities is apparent. 22 23 The potential on the capacitor plates due to imposed charge densities can be obtained by summation of 24 25 the Galvanic potential from the field in the dielectric and a quantum potential arising from the non- 26 negligible rise in the chemical potential of graphene and 2DEG. Thus, the potential difference due to 27 28 the imposed charge density (휎푔 ) on graphene is 29 30 휎푔푑1 휎푚푑2 푉 = + + Δ휇푔 + Δ휇2퐷 #11 31 휖 휖 32 1 2 33 34 The metal with extremely high density of states near the is assumed to have no increase in 35 the Fermi energy. The increase in the 2DEG and graphene Fermi energy (Δ휇2퐷 and Δ휇푔 respectively) at 36 37 any temperature can be obtained by solving the following Fermi-integrals numerically [19]. 38 39 2푔푣 ( 푔 ) 40 휎푔 = 2ℑ 휇 ,푇 41 2휋(ℏ푣푓) 2 2퐷 42 휎2퐷 = 퐷ℑ1(휇 ,푇)#12 43 44 The overall capacitance (C )for the structure can then be evaluated using the differential 45 tot 46 푑휎 47 푔 퐶푡표푡 = #13 48 푑푉 49

50 The calculated Ctot (red dots) normalized against the geometric capacitance (green stars) for the 51 52 degenerate case is plotted in Fig. 3(A) against the potential applied. The conventional metal-2DEG 53 capacitor values (obtained by replacing graphene by a metal plate) is given by blue squares in the same 54 55 figure (Fig 3(A)). The total capacitance value of the graphene-2DEG capacitor is non-linear with the 56 applied potential and the value at low bias potentials drops by 20% of the geometric capacitance. The 57 58 total capacitance at any potential is lower than the 2DEG-metal quantum capacitor with values 59 60 saturating at 95% of the geometric capacitance even under high bias conditions. The calculated total

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1 2 3 capacitance as a function of both temperature and applied potential is shown in Fig. 3(B). The Ctot 4 5 dramatically drops to about half the geometric capacitance value at 300K and displays maximum 6 sensitivity to potential changes in the region marked by green dashed line in the Fig. 3(B). Sharp 7 8 changes in Ctot with chemical potential of the electronic subsystem, which is also sensitive to external 9 10 perturbations such as dopants, temperature and the charge configuration of the environment, make the 11 proposed structure ideal for many applications in chemical and physical sensing, quantum memories, 12 13 photon detectors and such. The double 2DEG quantum capacitor displays much higher sensitivity to 14 chemical potential changes than graphene quantum and conventional metal-2DEG capacitor 15 16 [2,19] as seen in the comparisons presented in Fig. 3(A). The calculated quantum capacitance values 17 (C ) using Eq. 1 is plotted as a function of the applied potential for temperatures (0 – 350K) in the inset 18 Q 19 of Fig. 3(B). CQ of an ideal metal capacitor is infinite, and Ctot approaches the geometric value for high 20 21 CQ values [2]. Reduced density of states in a quantum system leads to reduction in CQ and subsequently 22 affects the total capacitance Ctot as can be noted in the similarity between the CQ plot provided in Fig. 23 24 3(B) inset and Ctot presented in Fig. 3(A). CQ, plotted in the inset also displays maximum change at 25 higher temperatures and applied potential close to the graphene charge neutrality point. 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Figure 3(A) Total effective capacitance normalized with the geometric capacitance under degenerate conditions is plotted 43 with red circles. (B) Normalized total capacitance at temperatures from 0K to room temperature is shown illustrating that the 44 total capacitance drops to half the geometric value at room temperature and low bias conditions. Inset: CQ plotted for different 45 applied potentials for different temperatures seen to closely follow Ctot plot in (A). 46 Electron-electron correlations in 2D systems can have a strong influence on the device response. The 47 48 electron correlation parameter (푟푠 ) is the ratio of average potential energy to the kinetic energy of the 49 50 electron [20]. In graphene, the average potential energy from the Coulomb repulsion between the 51 푒2 electrons is expressed by 〈푃퐸〉 = , where 푟0 = 1/ 휋푛 is the average spacing between electrons 52 〈4휋휖푟0〉 53 54 in the 2D system, n is the electron density and  is the average dielectric constant of the environment 55 [21]. The average electron kinetic energy at the Fermi level for graphene with approximated linear 56 57 dispersion is given by 〈퐾퐸〉 = ℏ푣퐹 휋푛. This gives a carrier density independent correlation parameter 58 2 59 (푟푆 = 푒 /휖ℏ푣푓) in graphene, in contrast to the inverse square root relation in conventional 2D systems 60

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1 2 3 [22]. Large Fermi velocity (푣푓 ) of the electron weakens the electron-electron correlation in graphene 4 5 significantly and the high dielectric constant employed here (휖푟≅ 9 for GaN) degrades the correlation 6 7 even further. This gives a 푟푠~0.1 ― 0.2 for graphene on GaN/AlGaN quantum well system indicating 8 negligible electron correlation effects and explaining the excellent match of the first order calculations 9 10 with experimental measurements. The density of states of the two materials is assumed to be 11 independent of each other and is represented only by the material properties forming the 2DEGs [19,2]. 12 13 While the density of states of ideal graphene at the Dirac point tends to zero, the potential fluctuations 14 15 due to impurities in any practical 2D sheet modifies the density of states [23]. The fluctuations cause 16 additional density of states leading to charge puddles and can be taken to have a Gaussian form in 17 18 graphene as described below [21] 19 20 1 푉2 푃(푉) = exp ― #14 21 2휋훾2 ( 2훾2) 22 23 where, V is the fluctuation potential, and 훾 is a Gaussian disorder parameter. The ensuing density of 24 25 states deviates from the ideal linear density (퐷퐼 ) and can be obtained from the overlap integral [24] 26 27 퐸 퐸 퐸 훾 퐸2 28 퐷(퐸) = 퐷퐼(퐸 ― 푉)푃(푉)푑푉 = 퐷퐼 푒푟푓푐 ― + exp ― 2 #15 29 ∫ ―∞ [ 2 ( 훾 2) 2휋 ( 2훾 )] 30 31

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Figure 4 (A) Calculated graphene density of states in the presence of disorder potential (V) for different Gaussian parameter 50 훾. Additional density of states is observed close to the Dirac point energy (ED). (B) Normalized total capacitance 51 calculated with disorder for two extreme temperatures of 0.1K and 300 K. While influence of disorder is easily 52 observable at low temperatures, at room temperature no significant impact is noticed. 53 54 The calculated density of states of practical graphene films for various disorder parameter value (훾 = 0, 55 10, 15 and 30 meV) are plotted in Fig. 4(A). As intended, an increase in additional density of states is 56 57 observed close to the Dirac point energy for corresponding increase in disorder value 훾. The total 58 59 capacitance of the proposed double quantum well capacitor is calculated for the modified density of 60 states at two extreme temperatures (0.1K and 300K) and are plotted in Fig. 4(B). If we compare the

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1 2 3 curves for different 훾 values at 0.1 K (the red, blue, and green lines of Fig. 4(B)), disorder is observed 4 5 to have a marked impact on the total capacitance at low bias potentials. Increase in the density of states 6 close to Dirac point leads to reduced quantum capacitance effect and the minimum total capacitance at 7 8 훾 = 15 meV reaches to only 85% of the geometric value, while the ideal graphene Ctot is lesser than 9 10 70% of the geometric value. 11 12 In contrast, at 300 K (black and magenta curves in Fig. 4(B)), two extreme 훾 values (0 and 30 meV) do 13 not affect the total capacitance significantly as seen by the overlapping curves. The dramatic reduction 14 15 of greater than 50% of the geometric capacitance is observed even under large disorder conditions at 16 17 low bias potentials at higher temperatures and low bias potentials. The physical reason behind a strong 18 impact of disorder only at low temperatures and low bias potentials can be explained from the metal 19 20 carrier density plot at different temperatures presented in Fig. 2(B). Only under low bias potentials and 21 low temperatures, the metal layer carrier density is comparable with the disorder induced density 22 23 leading to its strong influence on the total capacitance. At higher temperatures, electrons in graphene 24 25 occupy energy levels much higher than the Dirac point and the metal carrier density is much higher 26 than the disorder induced charge density. Hence under these conditions no significant impact of the 27 28 electron puddles is observed on the total quantum capacitance. In addition, practical graphene films are 29 inevitably doped due to contaminations during fabrication or the growth process. Hence, devices are 30 31 typically DC biased to bring the chemical potential close to the charge neutrality or the Dirac point (ED) 32 for maximum response. Correspondingly, here, the highest change in total capacitance is predicted at 33 34 higher temperatures and bias potentials close to the charge neutrality point. 35 36 Material thickness is crucial for the device structure and enforces limitations on the observed 37 38 behavior. Graphene layer is necessarily one atom thick and the growth process is self-limiting [13]. 39 AlGaN layers thicker than 65 nm develop cracks due to growth stresses leading to leakage currents [25] 40 41 . Hence, 20 – 30 nm thick quantum well layers are typically grown for device applications. These 42 43 requirements place stringent restrictions on the range of usable material thickness for the proposed 44 structure. The structure was experimentally fabricated to validate the calculations as given below. An 45 46 AlGaN-GaN heterostructure was grown on a commercially available (111) Si substrate using an Aixtron 47 made metal organic chemical vapor deposition (MOCVD) system. The native oxide on the Si surface 48 49 was removed using HF clean and hydrogen treatment at 1000°C. A 400 nm AlN layer was grown 50 directly on top of Si substrate. This is followed by 1 μm thick AlGaN transition layer. A 25 nm film of 51 52 30 % AlGaN was grown on top of the previously mentioned transition layer. The stack was grown by 53 54 using tri-methyl-gallium as the Ga source, tri-methyl-aluminum as the Al source, and ammonia as the 55 nitrogen source. An unbiased 2D charge density of about 1013 cm-2 typically forms at the AlGaN/GaN 56 57 interface. Further information on growth is given elsewhere [26,27]. The graphene monolayer was 58 grown on Cu foils using a home-made CVD system with methane and hydrogen as source gases. The 59 60 monolayer was then transferred onto the AlGaN-GaN substrate using a commonly used wet chemical

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1 2 3 process. The details of the growth condition and the wet chemical transfer are provided elsewhere 4 5 [14,28,15]. An array of graphene-2DEG capacitors was fabricated using photo-lithographic processes 6 on top of AlGaN/GaN substrates [29]. Capacitance value was measured at different temperatures 7 8 ranging from 77K to 300 K with the graphene electrode as the forcing terminal, the 2DEG and the 9 10 substrate bottom were grounded. The capacitance measurements obtained two different applied DC bias 11 are plotted in Fig. 5 for two different devices of same dimensions (squares and stars are values from 12 13 two different devices respectively). Calculated total capacitance using the description above is shown 14 in the same figure with blue solid line faithfully following the temperature trend. The overall reduction 15 16 in capacitance due to the quantum capacitance effect is also seen to be more prominent at lower bias 17 potentials as expected from the calculations. 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Figure 5 Measured total capacitance (normalized with geometric capacitance) of graphene-GaN-AlGaN capacitor at various 39 temperatures for two devices are shown in squares and stars respectively at two different potentials color coded as shown. 40 The calculated value, shown in continuous line, is seen to faithfully describe the experimental measurements. 41 42 43 44 Most of the results reported here stems from the temperature dependent variations in the 45 electron distribution and the chemical potential of the electronic subsystems. Increase in temperature 46 47 permits population of electrons at higher energies leading to higher total energy in the electronic 48 distribution (∫퐸퐹(퐸)퐷(퐸)푑퐸). The increasing share of metal charge density at higher temperatures for 49 50 any imposed graphene charge density, as seen in Fig. 2 (A and B) is a direct consequence of the above 51 52 as the total energy minimization criteria, enforced in Eq. 8, balances the energy per added charge on 53 the 2DEG and the electric field energy in the substrate dielectric due to metal charge density. In addition, 54 55 the relative change in chemical potential per increase in charge density exponentially drops with 56 temperature. This is again due to the wider Fermi-function tail at higher temperatures explaining stark 57 58 reduction in slope of lines in Fig. 2(B) at higher temperature. 59 60

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1 2 3 Capacitance reflects inverse of the change in chemical potential on the electron distribution per 4 5 change in the imposed charge density. The linear density of states in graphene with vanishingly low 6 state density at low carrier densities causes wide differential change of chemical potential at low bias. 7 8 This results in significant reduction in overall capacitance at low bias as witnessed in Fig. 3A. However, 9 10 a much stronger signature of the linear density of states is observed as temperature is increased. The 11 increase in chemical potential at higher temperatures at constant carrier density is more significant 12 13 causing the overall capacitance to drop to half its geometric value. The theoretical calculations were 14 performed by assuming an ideal graphene scenario with the Fermi level at the Dirac point. However, 15 16 practical sheets are unintentionally doped by defects, adsorbents and fabrication induced contaminants 17 leading to raised chemical potentials. This is accommodated by comparing the experimental results 18 19 offset by the Dirac potential (푉퐺 ― 푉퐷 ) to the calculated results as shown in Fig. 4. The total capacitance 20 21 drops significantly at lower bias potentials as expected by the theory. 22 23 In summary, we introduce a unique quantum capacitor with quantum electronic states on both 24 25 its plates and describe the charge density distribution between the plates at various temperatures. The 26 quantum capacitance of the structure is potential dependent with a prominent drop in the capacitance at 27 28 low biases due to linear density of states of graphene. The total capacitance also displays significant 29 drop in value at higher temperatures and drops to 50% of its geometric capacitance at room temperature. 30 31 The structure was fabricated using 2DEG in AlGaN/GaN heterostructure and graphene monolayer and 32 the measured capacitance corroborate the theoretical results. 33 34 35 36 Data Availability 37 The data that support the findings of this study are available from the corresponding author upon 38 39 reasonable request. 40 41 42 Acknowledgments 43 44 The submitted research was funded through the Initiation grant, Indian Institute of Technology 45 – Kanpur. The author thanks Dr. Hareesh Chandrashekar for the valuable discussions and comments on 46 47 the manuscript. The quantum well stack was grown by Dr. Nagaboopathy Mohan and Prof. Srinivasan 48 49 Raghavan at Centre for Nanoscience and Engineering, Indian Institute of Science, Bangalore. Electrical 50 measurements and nanofabrication were conducted at the National Nanofabrication centre and Micro- 51 52 Nano characterization centre at Indian Institute of Science, Bangalore. 53 54 55 Bibliography 56 57 [1] T. P. Smith, B. B. Goldberg, P. J. Stiles, and M. Heiblum, 1985 Direct measurement of the density 58 of states of a two-dimensional electron gas, Phys. Rev. B, 32 , 2696 59 60

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