Electronic and Plasmonic Properties of Graphene and Graphene Structures

Plasma Waves in Two-Dimensional Electron-Hole System in Gated Graphene Heterostructures

Plasma waves in the two-dimensional electron-hole system in a graphene-based heterostructure controlled by a highly conducting gate are studied theoretically. The energy spectra of two-dimensional electrons and holes are assumed to be conical (neutrinolike), i.e., corresponding to their zero effective masses. Using the developed model, we calculate the spectrum of plasma waves (spatio-temporal variations of the electron and hole densities and the self-consistent *Reprinted with permission from V. Ryzhii, A. Satou, and T. Otsuji (2007). Plasma waves in two-dimensional electron-hole system in gated graphene heterostructures, J. Appl. Phys., 101, 024509. Copyright © 2007 American Institute of Physics.

Graphene-Based Terahertz Electronics and Plasmonics: Detector and Emitter Concepts Edited by Vladimir Mitin, Taiichi Otsuji, and Victor Ryzhii Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4800-75-4 (Hardcover), 978-0-429-32839-8 (eBook) www.jennystanford.com electric potential). We find that the sufficiently long plasma waves exhibit a linear (soundlike) dispersion, with the wave velocity determined by the gate layer thickness, the gate voltage, and the temperature. The plasma wave velocity in graphene heterostructures can significantly exceed the plasma wave velocity in the commonly employed semiconductor gated heterostructures. The gated graphene heterostructures can be used in different voltage tunable terahertz devices which utilize the plasma waves.

Introduction

Recent progress in fabrication of graphene, a monolayer of carbon atoms forming a dense honeycomb two-dimensional (2D) crystal structure (which can also be considered as an unrolled singlewall carbon nanotube or as a giant flat fullerene molecule), and the demonstration of its exceptional electronic properties have attracted much interest [1]. The possibility of creating novel electronic devices on the basis of graphene heterostructures (see, for instance, Refs. [2] and [3]) has caused a surge of experimental and theoretical publications. The massless neutrinolike energy spectrum of electrons and holes in graphene can lead to specific features of the transport properties [4,5], photon-assisted transport [6], and quantum Hall effect [7-9], as well as the unusual high- frequency properties and collective behavior of the 2D electron or hole (2DEG or 2DHG) systems in graphene-based heterostructures [10-13]. The plasma waves in graphene-based heterostructures, in which the 2DEG or 2DHG system can serve as a resonant cavity or as a voltage-controlled waveguide, can be used in different devices operating in the terahertz (THz) range of frequencies [14,15]. Since the spectrum of plasma waves is sensitive to the electron (hole) mass, the plasma waves in graphene heterostructures can exhibit specific features associated with the zero electron and hole masses. The electron and hole sheet densities in graphene heterostructures with the conducting gate can be effectively varied by the gate voltage [1]. Thus, the massless electron and hole spectra and voltage tunability of the electron and hole densities open up new opportunities to create novel THz plasma wave devices based on graphene heterostructures on silicon substrates.

In this work, the spectrum of plasma waves (spatio-temporal variations of the electron and hole densities and self-consistent electric potential) in a gated graphene heterostructure in wide ranges of the gate voltages and the temperatures is studied. A gated graphene heterostructure (with n⁺-Si substrate serving as the gate and the gate layer made of Si0₂) shown in Fig. 1.1 is considered. A limiting case of relatively large (positive) gate voltages at sufficiently low temperatures was considered recently by one of the present authors [13]. In this case, the plasma waves are associated with a degenerate 2DEG. Here, we generalize the study of plasma waves in gated graphene heterostructures for the situation when the gate voltage and temperature vary in wide ranges, so that the contribution of both electrons and holes and the features of their energy distributions can be essential.

Equations of the Model

We shall consider "classical” plasma waves with the wavelength A markedly exceeding the characteristic length of the electron de Broglie wave A_F. In this case, we can use the following kinetic equations coupled with the equation governing the self-consistent electric potential:

Figure 1.1 Schematic view of a gated graphene heterostructure.

Here, f_e = /_e(p, r, t] and /_h = /j,(p, r, t] are the electron and hole distribution functions, respectively, tp = (p[r, z, t) is the electric potential, p = (p_x, p_y) is the electron (or hole) in-plane momentum, r = (x, y) (x and/ correspond to the directions in the graphene layer plane, while axis z is directed perpendicular to this plane), and e=e is the electron charge. The quantity v_p is the velocity of an electron and a hole with momentum p: v_p = Э£р/Эр, where £p = Пр|р| corresponds to the electron and hole massless spectra and Up is the characteristic (Fermi) velocity. As a result, v_p = v^p/p, where p = |p|. The terms 4 and /_h on the right-hand sides of Eqs. (1.1) and (1.2) govern the processes of electron and hole scattering and recombination. For the sake of simplicity, we disregard the electron and hole scattering processes and their recombination, and therefore set 4 = 0 and 4= 0. This is justified when the characteristic frequency of plasma waves significantly exceeds the pertinent collision frequencies.

The Poisson equation which supplements Eqs. (1.1) and (1.2) is presented as

where ae is the dielectric constant of the surrounding space and c>(z) is the Dirac delta function playing the role of the form factor of the 2DEG and 2DHG localization in the direction perpendicular to the graphene plane. Here, we have neglected the finiteness of the localization region thickness. The electron and hole sheet densities Z_e = E_e(r, t) and E_h = 54(r, t) and the distribution functions/_e =/_e(p, r, t) and/_h=_4(p, r, t) are related by the following equations:

where 7i is the reduced Planck constant.

In equilibrium, the electron and hole distribution functions are given by

where £_F is the Fermi energy, T is the temperature, and k_B is the Boltzmann constant. The Fermi energy is determined by the condition of neutrality seV_g/W_g = 4ке(Х_е0 - X)_l0), where V_g is the gate voltage, W_g is the gate layer thickness, and X_e0 and X_h0 are the equilibrium electron and hole densities, respectively. Considering Eq. (1.4} the latter condition can be presented as

Taking into account Eq. (1.5), Eq. (1.6) can be rewritten as

or, introducing the variable = v_Fp/k_BT, as where

At V_g=0, Eqs. (1.7) and (1.8) naturallyyield e_F= 0 (which corresponds to Z_e0 = X_h0).