Electronic properties of graphene on a piezoelectric substrate

Abstract

In the present work, we investigate theoretically the electronic properties of a graphene layer on a piezoelectric substrate. Specifically, we study the effect of the surface acoustic vibration modes on the graphene electrons. The first chapter introduces a basic survey of the main actors in this work. Graphene is a two-dimensional material that has received increasing worldwide attention since its isolation by Novoselov and Geim in 2004. It is formed by a honeycomb lattice of carbon atoms. The single-atom thickness of graphene, combined with its unique electronic properties stemming from the effectively massless behavior of electrons, converts this material into a special object of high fundamental and applied interest. The extremely high carrier mobility in suspended graphene is enabled by the high frequencies of the optical phonons in the stiff honeycomb lattice. Thus, the effects of electron-phonon scattering on transport are small in comparison with conventional metals. However, in most device architectures, graphene is deposited on a substrate, and all lattice modes of the substrate material that induce an electric field will influence the carriers in the graphene sheet, making the choice of substrate material crucial for the resulting transport characteristics of the device. On the other hand, surface acoustic waves (SAWs) created in piezoelectric materials reside at the surface of a solid or at the interface between two solids. They have for long been used to control the properties of semiconductor materials and structures. SAWs may be used to convert mechanical into electric signals and vice versa. A first basic description of piezoelectricity and these waves in the first chapter, is followed by a quantitative study of their propagation and main characteristics in the second chapter. Apart from the mechanical deformation, the vibration of the ionic lattice in a piezoelectric material produces an electric field travelling along with the SAW. The need to quantify its effects accurately both for macroscopic SAWs and their vibration quanta, the acoustic phonons, leads to the third chapter. There, the interaction between the electrons of a two-dimensional metal and the acoustic phonons of an underlying piezoelectric substrate is investigated. Fundamental inequalities can be obtained from general energy arguments. As a result, phonon-mediated attraction can be proven to never overcome electron Coulomb repulsion, at least for long phonon wavelengths. Therefore, in the fourth chapter, we study the influence of these phonons on the electron-electron interactions and the possible pairing instabilities of a two-dimensional electron gas such as graphene. In the fifth chapter, we investigate the many-body properties of graphene on top of a piezoelectric substrate, focusing on the interaction between the graphene electrons and the piezoelectric acoustic phonons. We calculate the electron and phonon self-energies as well as the electron mobility limited by the substrate phonons. We emphasize the importance of the proper screening of the electron-phonon vertex and discuss the various limiting behaviors as a function of electron energy, temperature, and doping level. The effect on graphene electrons of the piezoelectric acoustic phonons is compared with that of the intrinsic deformation acoustic phonons of graphene. The global conclusions of this work are contained in the last chapter. The numerical results for mean free paths and electron mobilities shown are seen to be applicable to a variety of piezoelectrical materials with different lattice structures and piezoelectric strengths. Our study can be thus relevant for graphene devices operating in the ballistic transport regime and for scenarios where quantum interference induces localization phenomena.