The Quantum Hall Effect: Helicity, Graphite and Graphene

Abstract

Abstract-TheHall resistivity is explained to arise from the spin and angular momemtum which leads to fractional charge in a natural way. There are helical orbits for the electrons in a magnetic field so that the + sign in spin gives one helicity while the -sign gives another, which is also related to the sign of the electron velocity. The fractions which arise in the graphite resistivity are explained in terms of principal fractions, resonances, sum processes and clusters. The graphene Hall effect data is explained by using the Landau levels with special treatment of spin. The two layer graphene is well explained. A small number of fractions are explained in terms of sample-dependent clusters which are found in some of the samples. While most of the fractions are a property of pure samples, some of the fractions are induced by the clusters.Index Terms-Graphene, graphite, helicity, quantum hall effect, special spin. I. INTRODUCTIONRecently, we have described the correct theory of the quantum Hall effect as well as explained the origin of 101 fractions found in the experimental data [1]. The cyclotron frequency which appears in the Landau levels is replaced by double valued helical values. The Lande's formula is corrected to include the time-reversed conjugate states. Only the l =0 states are used in the original Landau theory which requires to be modified for finite l value. Since j=l ±s is the total angular momentum keeping only l = 0 leaves out the important effects at high fields. The energy levels which result by these corrections lead to the fractions in the flux-quantized resistivity measured in the Hall effect. The concept of helicity is introduced to understand the right and left moving electrons. Usually the velocity is single valued except that in complex clusters there are rotations which allow two signs of the velocity. Hence, as long as charge is conserved particles can occur with reversed helicity. In a single particle this will not happen but in a cluster it is possible.In this paper, we describe the concept of helicity as applied to the quantum Hall effect of electrons in a high magnetic field and low temperatures and also explain the experimental data of fractions in the Hall effect of graphite and graphene. We look for the evidence of the mixed helicity as well as "reversed helicity". We analyze the experimental data of the quantum Hall effect measurements of noise power to look for the effect of helicity. We find that noise power depends on the helicity. It is possible to heat the samples by a heater wire and monitor the motion of the particles. Hence we explain that while one particle moves to the right, the conjugate will move to the left. This motion of the particles is controlled by the helicity. Hence all particles occur in pairs. The original observation of the quantum Hall effect by von Klitzing et al [2] and Tsui et al. [3] had no theory and the efforts of Laughlin [4] did not yield an interpretation of the data and comparisons with the one-component plasma did not find the ground state...

The Theory of Von Klitzing’s Constant and Phases

“…A "single electron" type theory is sufficient to obtain the value of e and there are no many-body effects which can is not more than  0.2  in 25812.8 , which means that we can get the values correct upto ppm [2]. This point is to be noted so that we are not carried away from the "single particle theory" [3]- [6] of the observed plateau in the Hall resistivity.…”

“…How the accuracy has become so high is another question but in 1965, the value was 137.0388 (6). The gyromagnetic ratio of the electron is given by [2], g/2= 1.001 159 652 180 73(28) (6) This value is related to the fine structure constant,…”

“…In their paper, the value given is 25813 . The calculation of h/e 2 does not require that there should be two dimensionality or there should be Landau levels. However, the experimental value requires the Hall geometry.…”

“…The fine structure constant is defined in such a way that, . At the present time, the value of the inverse fine structure constant [2] is, 1/ = 137.035 999 084(51) which is another way of writing the value of h/e 2 . These are one and the same and not two different quantities.…”