Universal Introduction to the Quantum Hall Effect

Shrivastava, Keshav N.

doi:10.7763/ijapm.2013.v3.207

Cited by 4 publications

(7 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Usually, this will split E 1 , E 2 and E 3 into doublets. However, in the present case, we consider the spin in a more intricate manner so that we get the correct plateaus in the quantum Hall effect [4,[11][12][13][14][15][16][17][18][19][20][21][22]. This theory explains the Stormer's data [2], Pan's data [4] and Tsui's data [21] correctly and all fractions observed [22,23] in the experiments are well explained.…”

Section: Spin Hubbard Modelmentioning

confidence: 75%

“…If the experimental data does not give the square root dependent shift, then it means that there is no Coulomb interaction as such except that which might be absorbed in the unperturbed Hamiltonian by a transformation. In the case of special properties of spin with both signs in the total angular momentum j = l ± s, there is no interaction and all the interpretation is based on the unperturbed Hamiltonian [12][13][14][15][16][17][18][19][20][21]. The understanding of the experimental data [22,23] is fully achieved by the theory of ref.…”

Section: Search For An Interactionmentioning

confidence: 99%

See 1 more Smart Citation

Theory of Nano-Carbon Based Materials: Cyclotron Resonance, Kohn's Theorem and Hubbard Model

Shrivastava

2013

AMR

View full text Add to dashboard Cite

A monolayer of carbon is called graphene. It exhibits unusual properties in the Hall effect and in the cyclotron resonance. It is found that it exhibits fractional charge in the Hall effect. The interactions amongst electrons almost become constant at low temperatures. Hence, the Kohn's theorem, which shows that the interactions do not play much role in determining the cyclotron resonance, becomes operative at low temperatures. The experiments on graphene do not depend on the wave vector dependence of the frequency. Hence whether the dispersion depends on k2 or on k does not matter. The Hubbard model has been very successful in explaining the ground state of several electron systems. We consider a triangle with three vortices. Each vortex can be occupied by two electrons. By using the spin in a particular way, we can obtain new features in the Hubbard model. There is a doubling in the Peierls-Luttinger phase factor and eigen values acquire higher multiplicities than are known for the usual treatment of spin. The flux is distributed on the area of the triangle. The graphene consists of hexagons of carbon atoms but the Hall effect shows that there are defects on which electrons form clusters so that there is spin wave type behaviour. A cluster of electrons shows spin-waves leading to "spin deviation" of several per cent.

show abstract

Section: Spin Hubbard Modelmentioning

confidence: 75%

Section: Search For An Interactionmentioning

confidence: 99%

Theory of Nano-Carbon Based Materials: Cyclotron Resonance, Kohn's Theorem and Hubbard Model

Shrivastava

2013

AMR

View full text Add to dashboard Cite

show abstract

“…For this case the resistivity (30) is the same as von Klitzing's value. In cases of finite l and s, the physics of the problem is different from that of von Klitzing et al, so that von Klitzing's constant becomes a special case of "spin-dependent" phenomenon [3]. The are given in Table I as a function of l is given in Fig.…”

Section: The Theorymentioning

confidence: 96%

“…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. Another observation is that plateaus are formed and destroyed as the magnetic field is varied.…”

Section: Turning Pointsmentioning

confidence: 99%

The Theory of Von Klitzing’s Constant and Phases

Shrivastava¹

2013

IJAPM

Self Cite

View full text Add to dashboard Cite

Abstract-TheHall resistivity is found to become a function of spin. For positive spin, one value is found but for negative sign in the spin, another value occurs. In this way, there is never only one value of the resistivity but there is doubling of values. The value of the von Klitzing's constant is a special case of more general dependence of resistivity on the spin. We investigate the effect of Landau levels. For extreme quantum limit, n=0, the effective charge of the electron becomes (1/2)ge. The fractional charge arises for finite value of the angular momentum. There is a formation of spin clusters. As the field increases, there is a phase transition from spin ½ to spin 3/2 so that g value becomes 4 and various values of n in Landau levels, g(n+1/2), form plateaus in the Hall resistivity.

show abstract

“…Recently, in three papers, we have laid down the basically correct theory of the quantum Hall effect [1]- [3]. It has been found [4], [5] in the Hall effect that the transverse conductivity as a function of magnetic field shows plateaus at certain fractions of e 2 /h.…”

Section: Introductionmentioning

confidence: 99%

Hierarchy of States and the Dirac Equation in the Quantum Hall Effect

Shrivastava¹

2013

IJAPM

Self Cite

View full text Add to dashboard Cite

Abstract-The hierarchy of states is developed with increasing the value of l, the orbital angular momentum, is now called "Shrivastava's hierarchy of fractional states". We have examined the odd denominator states as extended to a hierarchy of continued fraction which describes a very large abundance of fractional states. The heirarchy developed by random numbers m and p i (i=1,2,…,4) by using the continued fraction is known as the Haldane's hierarchy. We find that the predictions of the Dirac equation agree with the idea of fractional charges. We have introduced the combination of spin and orbital quantum numbers, including the negative sign for spin, in such a way that there occur fractional charges through the Bohr magneton. This leads to doubling of eigen values so that we define an additional matrix the properties of which are important when magnetic field is present. There is a spin-charge coupling so that spin ½ particle can have the zero or one charge. The Dirac equation can accommodate not only charges of 0 and ±e but also fractional values such as 1/3 and 2/3. Index Terms-Dirac equation, fractional charges, hierarchy, angular momentum. I. INTRODUCTIONRecently, in three papers, we have laid down the basically correct theory of the quantum Hall effect [1]- [3]. It has been found [4], [5] in the Hall effect that the transverse conductivity as a function of magnetic field shows plateaus at certain fractions of e 2 /h. The measurement of the resistivity in the plateau region can be performed with very high precision. In the absence of scattering processes, at low temperatures and high fields, the classical Hall current is described by the fraction, I x = B eV n y s( 1) where V y is the applied voltage, e the charge of the electron, B the magnetic field and n s the surface carrier density,where  is a filling factor andis the degeneracy factor per unit area obtained from the shift Manuscript received March 30, 2013; revised May 30, 2013. Keshav N. Shrivastava is with the University of Malaya, Kuala Lumpur, and with the University of Hyderabad, Hyderabad 500046, India (e-mail: keshav1001@yahoo.com).of the oscillator wave function upon the application of magnetic field. Substituting (2) and (3) in (1) we find that,which means that the conductivity is given by, From (2) and (3) we find that,which means that the carrier density and the field are adjusted in such a way that the filling factor  of the energy levels is an integer. From the occurrence of plateau in the Hall the value of h/e 2 . The plateaus in the Hall resistivity as a function of magnetic field were observed at many different fractions in addition to integer values. II. HALDANE'S HIERARCHYHaldane [6] suggested that the fractional filling factors occur according to a hierarchy which is built from numbers m and p i (i=1, 2 or 3) determined from a continued fraction in which the number of terms determine the number of fractions. These numbers m and p i are not derived from any physical properties but are integers. Since m and p i are not related to t...

show abstract

Universal Introduction to the Quantum Hall Effect

Cited by 4 publications

References 27 publications

Theory of Nano-Carbon Based Materials: Cyclotron Resonance, Kohn's Theorem and Hubbard Model

Theory of Nano-Carbon Based Materials: Cyclotron Resonance, Kohn's Theorem and Hubbard Model

The Theory of Von Klitzing’s Constant and Phases

Hierarchy of States and the Dirac Equation in the Quantum Hall Effect

Contact Info

Product

Resources

About