Reservoir hypothesis for the quantum hall effect

Abstract

It is assumed that in modulation-doped heterojunctions the 2D electron gas is in thermodynamical equilibrium with an external reservoir. In a quantizing magnetic field energy gaps appear in the density of states of the 2D system. This produces, when the magnetic field B increases, oscillations of the density of the free 2D electrons, because of the transfer of electrons into (and out of) the reservoir. It is shown that these oscillations of N , induce the formation of plateaus of the Hall resistivity exy with the exact quantization in h/ie2 and vanishing values of the resistivity exx, when the Fermi energy is between Landau levels. The model developed in the triangular well approximation accounts well for the width and shape of the Hall steps and for the eXx behaviour in a magnetic field, in a large range of temperatures.Nous supposons que, dans les heterojonctions a modulation de dopage, le gaz d'electron bidimensionnel est a l'equilibre thermodynamique avec un reservoir exterieur. En presence d'un champ magnetique quantifiant, des "gaps" d'energie apparaissent dans la densite d'etats de ce systeme 2D. Ceci provoque, lorsque le champ magnetique B croit, des oscillations de la densite N , d'electrons libres 2D, ii cause du transfert d'electron dans (et hors de) ce reservoir. Nous montrons que ces oscillations de N , induisent, lorsque I'energie de Fermi est entre deux niveaux de Landau, dune part la formation de plateaux de la resistivite de Hall exy presentant la quantification exacte en h/ie2 et d'autre part des zones dans lequelles la resistivite exx tend vers zero. Le modele developpe dans I'approximation du puits triangulaire rend bien compte, dans une large gamme de temperatures, de la largeur et de la forme des "marches" de Hall ainsi que de la variation de ex, en fonction du champ magnetique.

Nonlinear dependence of the magnetophotoluminescence energies of asymmetricGaAs∕Ga0.67Al0.33Asquantum wells on an external magnetic field

“…16,17 Also, this hypothesis had been proposed in the explanation of the quantum Hall effect. [25][26][27][28][29][30] We do not go here into the interpretation of the QHE which is a much more complicated problem than the interpretation of MPL. The reason is that the dc transport involves localized electron states, mobility edges, sample edges, etc.…”

Reservoir model for two-dimensional electron gases in quantizing magnetic fields: A review

“…However, the above formalism does not include the localisation of electron states. Thus, in order to describe the plateaus of ρ xy and zeros of ρ xx we use the reservoir hypothesis following the work of Raymond and Sibari [10]. This treatment uses the triangular well approximation.…”

“…[6,7]. Still, the reservoir model has kept appearing in the literature under different names in order to explain various observations on the 2D electron gases: quantum transport [8][9][10], Fermi energy behaviour [11], cyclotron resonance [12,13], interband photo-magneto-luminescence [14], magnetic susceptibilities [15], magneto-plasmon dispersion [16,17], etc. In his well known book, Mahan [18] treats the localization and reservoir interpretations of QHE on equal footing.…”

Self-consistent equilibrium of a two-dimensional electron system with a reservoir in a quantizing magnetic field: Analytical approach

“…At B = B 0 the magnetic length ℓ is about 250 nm. Thus d = 25 nm will be sufficient to assure homogeneous broadening of the LLs up to B = 10 T. At the last step one need to justify inequality (2). This means that QW width should be about 3 nm.…”

“…A model of grand canonical equilibrium of a 2D system with a reservoir was proposed to explain the integer quantum Hall effect (IQHE) more than twenty years ago. 1 This so-called reservoir hypothesis can describe main features accompanied to the IQHE such as plateaus in a Hall resistance, 1 the Shubnikov-de Hass effect 2 and oscillations of a magnetization and a thermoelectric power. 3 Key points of the reservoir model are following: a 2D chemical potential is fixed by the reservoir and it doesn't depend on the magnetic field, a carriers concentration and 2D subband energies oscillate in the magnetic field.…”