# Effect of Self-Consistent Electric Field on Characteristics of Graphene p-i-n Tunneling Transit-Time Diodes

We develop a device model for p-i-n tunneling transit-time diodes based on single- and multiple graphene layer structures operating at the reverse bias voltages. The model of the graphene tunneling transit-time diode (GTUNNETT) accounts for the features of the interband tunneling generation of electrons and holes and their ballistic transport in the device i-section, as well as the effect of the self-consistent electric field associated with the charges of propagating electrons and holes. Using the developed model, we calculate the dc current-voltage characteristics and the small-signal ac frequency-dependent admittance as functions of the GTUNNETT structural parameters, in particular, the number of graphene layers and the dielectric constant of the surrounding media. It is shown that the admittance real part can be negative in a certain frequency range. As revealed, if the i-section somewhat shorter than one micrometer, this range corresponds to the terahertz frequencies. Due to the effect of the self-consistent electric field, the behavior of the GTUNNETT admittance in the range of its negativity of its real part is rather sensitive to the relation between the number of graphene layers and dielectric constant. The obtained results demonstrate that GTUNNETTs with optimized structure can be used in efficient terahertz oscillators.

## Introduction

Pioneering papers by Shur and Eastman [1, 2] have stimulated extensive studies (which continue already for the fourth decade) of ballistic electron and hole transport (BET and BHT, respectively), i.e., collision free transport in short semiconductor structures. The main incentive is the realization of fastest velocities of electrons/holes and, hence, achievement of the operation of diodes and transistors in terahertz range (THz) of frequencies and low power consumption. Even at the initial stage of the ballistic transport research, several concepts of ballistic THz sources have been put forward (see, for instance, an early review [3]). However, the realization THz generation in different semiconductor devices associated with BET/BHT, in particular, analogous to vacuum devices, meets the problems associated with electron scattering in real semiconductor structures. Creation of heterostructures with selective doping with a two-dimensional electron gas (2DEG) spatially separated from the donors has resulted in achievement of very long mean free path of electrons, at least at low temperatures. Recent discoveries of unique properties of graphene [4, 5], in particular, the demonstration of possibility of very long electron and hole mean free path in graphene layers (GLs) and what is even more interesting in multiple graphene layers (MGLs) [6, 7] add optimism in building graphene based THz devices using BET. The concept of graphene tunneling transittime (GTUNNETT) p-i-n diode, which exhibits a negative dynamic conductivity in the THz range, was proposed and substantiated in Refs. [8,9]. This concept based not only on BET or quasi-BET (as well as BHT or quasi-BHT) in GLs and MGLs but also on a strong interband tunneling under the electric field with a pronounced anisotropy [10, 11] due to the gapless energy spectrum, and constant absolute value of electrons and holes velocities [4]. Due to this, the electrons in the conduction band and the holes in the valence band generated owing to the interband tunneling in the electric field propogate primarily in the electric field direction with the velocity (in this direction) virtually equal to the characteristic velocity t_{v} ^ 10^{8}cm/s. A large value of the directed velocity in GLs and MGLs promotes the device operation at elevated frequencies.

As shown (Refs. [8, 9]), for the self-excitation of THz oscillations in a circuit with GTUNNETT diode, this circuit should serve as a resonator. However, at elevated tunneling currents in GTUNNETTs considered previously [8, 9] the self-consistent charge associated with propagating electron and hole streams can affect the spatial distribution of the self-consistent electric field and the electric potential in the i-section. As a result, the self-consistent electric field near the p-i- and i-n-juctions can be substantially reinforced. This, in turn, influences the tunneling generation of electrons and holes and their transit conditions and, hence, the GTUNNETT dc and ac characteristics.

In this chapter, in contrast to the previous treatment [8, 9], we account for the self-consistent electric field associated with the variations of the electron and hole lateral charges in the i-section and their effect on the injection and the dc and ac characteristics. The effects of the space charge in planar TUNNETTs with the propagation of carriers perpendicular to the structure plane were considered by Gribnikov et al. [12]. The problems of calculation of the dc and ac characteristics of devices based on lateral structures accounting for the self-consistent electric field are substantially complicated by the features of the structure geometry (2D electron and hole channels and blade-like contact regions). In particular, as shown in the following, the related mathematical problems are reduced to a system of rather complex nonlinear integral-differential equations. Using the GTUNNETT device model, we derive these equations, solve them numerically, and find the characteristics.