Toward High-Peak-to-Valley-Ratio Graphene Resonant Tunneling Diodes

The resonant tunneling diode (RTD) is one of the very few room-temperature-operating quantum devices to date that is able to exhibit negative differential resistance. However, the reported key figure of merit, the current peak-to-valley ratio (PVR), of graphene RTDs has been up to only 3.9 at room temperature thus far. This remains very puzzling, given the atomically flat interfaces of the 2D materials. By varying the active area and perimeter of RTDs based on a graphene/hexagonal boron nitride/graphene heterostructure, we discovered that the edge doping can play a dominant role in determining the resonant tunneling, and a large area-to-perimeter ratio is necessary to obtain a high PVR. The understanding enables establishing a novel design rule and results in a PVR of 14.9, which is at least a factor of 3.8 higher than previously reported graphene RTDs. Furthermore, a theory is developed allowing extraction of the edge doping depth for the first time.


Methods
Graphene and h-BN flakes were mechanically exfoliated from graphite crystals (NGS Naturgraphit) and h-BN crystals (2D Semiconductors) using adhesive tape (Nitto ELP BT-130E-SL).The top and bottom h-BN flakes were both tens of nanometres thick, whereas their surfaces were atomically flat.In the dry transfer process, the 'pick up' and 'tear and stack' techniques 1 were adopted to guarantee the momentum matching.The polydimethylsiloxane (PDMS)/polymethyl methacrylate (PMMA) film successively picked up h-BN, graphene, h-BN, graphene, and h-BN.
The top and bottom graphene flakes were torn from one graphene flake.The exfoliation and transfer processes were performed under ambient conditions.The source and drain electrodes were defined by electron-beam lithography (Raith e-LiNE Plus).The h-BN flakes were etched using inductively coupled plasma (ICP; Oxford Instruments PlasmaPro 100 Cobra) to expose the graphene.The one-dimensional contacts 2 were formed by 3-nm Cr/70-nm Au, which was deposited by electron-beam evaporation (HHV auto 500).
The device was placed inside a cryogenic probe station (Lakeshore CRX-VF) for roomtemperature and low-temperature measurements.The electrical measurements were performed using a Keysight B2902A instrument.

Numerical simulation
The numerical simulations in this study were based on the following principles.The drain current Id of the GRTD is an integral of the tunnelling current density Jtun throughout the overlapping region of the two graphene flakes in real space (x, y) and is expressed as For each coordinate pair (x, y), Jtun is an integral of the tunnelling throughout two Dirac cones in reciprocal space (kx, ky).Without considering a constant factor, it can be expressed as 3 Here, vF is the Fermi velocity of graphene; γ is the broadening of electronic states; sT/B, which has a value of 1 for the conduction band and −1 for the valence band, is the band index of the top or bottom graphene flake.k is the wave vector; kT/B is the wave vector starting at the Dirac point of the top or bottom graphene flake; KT/B is the wave vector of the Dirac point of the top or bottom graphene flake; thus, k is expressed as For a small twist angle θ, the wave vector difference of two Dirac points is approximately Here, lx/y/z is the unit vector of each direction in reciprocal space; Δφ = φT − φB is the energy difference of two Dirac points.Electrostatic doping nT/B,ele and edge doping nT/B,dop are considered to contribute to the sheet carrier density of the top or bottom graphene flake nT/B, which is counted positively for positive charge and is expressed as Considering charge conservation, the sheet carrier density of the gate electrode ng satisfies n T,ele + n B,ele + n g = 0. (S6) The chemical potential of the top or bottom graphene flake μT/B is given by Considering a conservative field, without considering other resistances in the circuit, the drain voltage Vd satisfies The gate charge corresponds to the electric field in the gate dielectric, and the gate voltage Vg, thickness of SiO2 d SiO 2 , and relative dielectric constant of SiO2 ε SiO 2 satisfy The electrostatic doping of the top graphene flake corresponds to the electric field in the interlayer h-BN flake, and the thickness of interlayer h-BN flake dBN, number of interlayer h-BN layers lBN, and relative dielectric constant of h-BN flake εBN satisfy Equations ( S3) and (S4) involve momentum matching, where θ is an important factor.Equations (S5)-(S11) involve energy matching.All unknowns can be obtained by solving the equations, where nT/B,dop passes the inhomogeneous distribution in real space (x, y) to Jtun, thus leading to the broadening and suppression of the resonant tunnelling peak, which is the focus of this study.
Moreover, the tunnelling process is determined by the difference in the number of quantum states occupied by the carriers between the top and bottom graphene flakes.fT/B satisfies the Fermi-Dirac distribution and is given by Here, kBol is the Boltzmann constant, and T is the temperature.The numerical simulation in Figure 3d

Other factors that influence the PVR
In addition to the APR metric emphasized in this paper, traditional factors reported in past literature, 1,[3][4][5][6] such as the broadening of electronic states γ, twist angle θ, and number of interlayer h-BN layers lBN, should also not be neglected.In the literature, the PVR was considered to decrease with increasing γ and θ, and lBN influenced the tunnelling current density.Simulation results of the PVR as a function of γ, θ, and lBN are shown in Figure S4.After considering various APRs, PVR still decreases with γ and θ; however, lBN between 3 and 5 influences the PVR insignificantly.
Therefore, in this study, γ was considered as an intrinsic property of materials that cannot be controlled, θ was minimized with the help of the 'tear and stack' technique, and lBN was maintained

Figure S1 .
Figure S1.Output characteristics and extracted PVR values of 13 devices.(a-z) Output
Figure S3.The two treatments agree qualitatively, and thus do not affect the conclusion of this

Figure S3 .
Figure S3.Comparison between double-and single-flake doping.Simulation results of PVR as a

Figure S4 . 5 .
Figure S4.Influence of the broadening of electronic states γ, twist angle θ, and number of

Figure S5 .
Figure S5.Electronic characteristics of the high-PVR device (Device D) from another

Table S1 .
Size information on 14 devices.