Reconciling edge states with compressible stripes in a ballistic mesoscopic conductor

Abstract

The well-known Landauer-Buttiker (LB) picture used to explain the quantum Hall effect uses the concept of (chiral) edge states that carry the current. In their seminal 1992 article, Chklovskii, Shklovskii and Glazman (CSG) showed that the LB picture does not account for some very basic properties of the gas, such as its density profile, as it lacks a proper treatment of the electrostatic energy. They showed that, instead, one should consider alternated stripes of compressible and incompressible phases. In this letter, we revisit this issue using a full solution of the quantum-electrostatic problem of a narrow ballistic conductor, beyond the CSG approach. We recover the LB channels at low field and the CSG compressible/incompressible stripes at high field. Our calculations reveal the existence of a third 'hybrid' phase at intermediate field. This hybrid phase has well defined LB type edge states, yet possesses a Landau level pinned at the Fermi energy as in the CSG picture. We calculate the magnetoconductance which reveals the interplay between the LB and CSG regimes. Our results have important implications for the propagation of edge magneto-plasmons.Electrostatic energy is very often the largest energy scale in a physical situation. Yet, the electrostatic landscape is equally often taken for granted as an external potential, which may result in a wrong physical picture. A well known example is the quantum Hall effect [1] (QHE) that has been largely discussed using the concept of edge states in a non-interacting Landauer-Buttiker (LB) picture [2, 3]. Despite being very successful for the understanding of e.g. the quantification of the Hall resistance and the vanishing longitudinal resistance, the LB picture also fails spectacularly to describe basic physics such as the density profile of the electron gas. In a series of articles [4-6] that culminated with the work of Chklovskii, Shklovskii and Glazman (CSG) [7-9], it was recognized that the LB picture should be revisited. It was shown that the interplay between quantum mechanics and electrostatics leads to the emergence of compressible and incompressible stripes, a concept related, yet somehow different, to the original edge states. An important effort has been devoted to the experimental observation of these stripes [10][11][12][13][14][15][16][17][18][19][20]. CSG work, as well as a large fraction of the subsequent litterature [21][22][23][24][25][26][27][28][29][30] was based on the Thomas-Fermi approximation which is suitable at high magnetic field but inadequate at low field where the LB approach is expected to work well. The self-consistent problem can also be tackled directly in presence of a finite temperature [31]. More recent works improved on Thomas-Fermi by incorporating a Gaussian broadening of the Landau levels [32][33][34]. Solving the full self-consistent electrostatic-quantum problem, beyond the above approximations and at low temperature, is a difficult task however, as the presence of the Landau levels (and the associated Dirac comb for the den...

The self-consistent quantum-electrostatic problem in strongly non-linear regime

“…In this section, we consider the geometry of Fig.1b in presence of a perpendicular magnetic field. The physics contained here has been discussed in a separate paper [34]. The results shown below aim at illustrating the algorithm as well as providing additional data that were not shown in [34].…”

How do edge states position themselves in a quantum Hall graphene pn junction?

“…The assumption that the electrostatic potential is unaffected by the magnetic field is in fact not valid [9]. In contrast, it is the electronic density n s (x) shown on the right panel of Fig.…”

“…The purpose of this article is to show that this paradox is due to a breakdown of the LB picture in the QHE regime. The LB approach does not account for the (dominating) electrostatic energy [9] and must be replaced by the more elaborate Chklovski-Shklovskii-Glazman (CSG) model [10]. Performing the CSG construction of the compressible and incompressible stripes in a graphene pn junctions, our main result is a simple explanation to the above mentioned paradox.…”

“…In these regions, the electrostatic potential U (x) remains constant. We refer to [10,15] for the details of the original construction and to [9,14] for a more recent version compatible with numerical calculations. In this picture, the size of each incompressible stripe is proportional to √ E ex [14].…”

Coulomb interactions and effective quantum inertia of charge carriers in a macroscopic conductor

“…[31]. Going beyond this phenomenological but practical model would involve a multiscale treatment combining our approach to interchannel interactions with a self-consistent microscopic approach solving the problem of electrons in the presence of intrachannel Coulomb interactions [65][66][67].…”