Received 23 May 2015; accepted 27 June 2015; published 30 June 2015

ABSTRACT

We have investigated the Fractional Quantum Hall Effect (FQHE) on the fundamental Hamiltonian with the Coulomb interactions between normal electrons without any quasi particle. The electron pairs placed in the Landau orbitals can transfer to many empty orbitals. The number of the quantum transitions decreases discontinuously when the filling factor n deviates from the specific fractional number of n₀. The discontinuous decreasing produces the energy valley at the specific filling factors n₀ = 2/3, 4/5, 3/5, 4/7, 3/7, 2/5, 1/3 and so on. The diagonal elements of the total Hamiltonian and the number of the quantum transitions give the total energy of the FQH states. The energy per electron has the discontinuous spectrum depending on the filling factor n. We obtain the function form of the energy per electron in the quantum Hall system. Then the theoretical Hall resistance curve is calculated near several filling factors. Therein the quantum Hall plateaus are derived from the energy valleys. The depths of the energy valleys are compared with the widths of the quantum Hall plateaus appearing in the experimental data of the Hall resistance. Our theoretical results are in good agreement with the experimental results.

Keywords:

Fractional Quantum Hall Effect, 2D Electron System, Quantum Theory, Hall Resistance

1. Introduction

The quantum Hall effect is derived by the total Hamiltonian of a many-electron system which is composed of the single electron Hamiltonian of the i-th electron and the Coulomb interaction between electrons as follows:

(1)

(2)

where, e, and N are the effective mass, the elementary charge, the momentum and the total number of electrons. Therein indicates the potential of the z-direction which confines the electrons to an ultra- thin conducting layer. Also is the electric potential along the Hall voltage (y-direction). The vector potential, , has the components,

(3)

where B is the strength of the magnetic field. The last term of Equation (2) indicates the Zeeman energy where is the effective g-factor, is the Bohr magneton and is the z-component of the i-th electron spin operator. The details have been explained in the previous papers [1] -[14] . When the Coulomb interaction between electrons is ignored in the quasi-2D electron system, the Hamiltonian of the single electron is exactly diagonalized same as in the Landau solution. At a filling factor, all the electrons are placed in the Landau orbitals with the Landau level number. The residual Landau orbitals () are empty. So there are various electron-configurations in the Landau orbitals. We divide the total Hamiltonian into the diagonal part and the non-diagonal part as follows;

We define two symbols W and C which are the expectation values of and the Coulomb energy, respectively. We call C “classical Coulomb energy” (the expectation value of the Coulomb interaction). Then the classical Coulomb energy becomes a minimum for only one electron configuration in the Landau orbitals. This property has been proven in the previous paper [9] . That is to say W becomes a minimum at the only one electron configuration. The electron configuration gives the single ground state for each value of. The residual Coulomb interaction H_I (non diagonal part of H_T) produces many quantum transitions from the ground state. We have examined the perturbation energy via these quantum transitions in details in the previous papers [1] -[14] .

Then all the electron pairs placed in the nearest Landau orbitals can transfer to all the empty orbitals at the specific filling factors. When the filling factor deviates from the fractional numbers the number of the transitions abruptly decreases. The reason comes from the combined effects of the momentum conservation along the x direction (current direction), the most uniform electron configuration and the Pauli exclusion-principle. The abrupt decreasing of the transition number yields the valley structure in the perturbation energy. That is to say the energy of the nearest electron pair takes a minimum at and the energy

for is higher than. Then gives the energy gap (valley depth) as proven in the previous papers [9] [12] [13] .

The total energy of the quantum Hall system is the sum of W (expectation value of) and all the pair energy of electrons (placed in nearest orbital pairs and more distant orbital pairs), because the Coulomb interaction works between two electrons. We will study the function form of the expectation value W in the next section. Then we get the energy spectrum of the quantum Hall system which is quite different from the Halperin result [15] . The valley depth in the pair energy and the function form of W give the quantum Hall plateaus at the specific filling factors. The theoretical results are in good agreement with the experimental data.

2. Expectation Value of the Total Hamiltonian and Its n-Dependence

We describe the expectation value of the total Hamiltonian by the symbol W which is the sum of the single electron energies and the expectation value of the classical Coulomb energy as follows;

(4)

where is the single electron energy of the i-th electron and is the expectation value of the Coulomb interaction between electrons. The total number of Landau states with is equal to where and d are the length and the width of a quantum Hall device respectively. The total charge of electrons at the filling factor is the product of and the total number of Landau states. The total charge is divided by the area, and then we obtain the charge density as

(5)

Figure 1 shows one of the experimental data [16] . The upper figure indicates the Hall resistance divided by the Klitzing constant. The value of is equal to which is almost proportional to the magnetic field strength B except the Hall plateau regions. This property is easily seen by comparing the data with the red line. That is to say, is nearly equal to the constant value. Equation (5) means the charge density to be proportional to. Accordingly the macroscopic Coulomb energy may be treated to be a constant value in the experiment of Figure 1.

We next examine the microscopic charge-distribution in more details. Figure 2 shows the electron configuration with a minimum classical-Coulomb-energy at. Therein the bold lines express the occupied orbitals with electron and the dashed lines indicate the empty orbitals. The electron pair located at the orbitals AB is one example of the nearest-electron-pairs. Two electrons placed at B and C show the second nearest pair. The electron pair at A and C is the third nearest pair. The pair at A and D is the fourth nearest pair and so on. The classical Coulomb energy between two electrons is expressed by the symbols and respectively as in Figure 2. Therein is the largest one of the classical Coulomb pair energy, is the second largest, is the third largest, is the fourth largest and so on.

The classical Coulomb energy between the pair (A, C) is weakened by the screening (shielding) effect of electron B. Also the classical Coulomb energy between the pair (A, D) is weakened by the screening effect of electrons B and C. Accordingly the n-dependence of the classical Coulomb energy mainly comes from the first nearest and the second nearest pairs. The number of the more distant pairs (third, fourth, fifth and so on) are enormous many. The total number of electron pairs is. On the other hand the total number of the first and second nearest pairs is N. Accordingly the residual energies (namely the sum of all the more distant pair

energies) may be approximated by the macroscopic Coulomb energy which is a constant value mentioned above. On the other hand the sum of the first and second nearest pair-energies is strongly dependent upon the filling factor. The n-dependence is examined for various filling factors as follows:

(Case of)

We can ignore the boundary effect in both ends for a macroscopic electron-number N. Then the number of the first nearest pairs is equal to and the number of the second nearest pairs is equal to for the configuration of Figure 2. Then the sum of the classical Coulomb energies between the first nearest pairs is equal to and that between the second nearest pairs is equal to. Accordingly, the total classical Coulomb energy is approximated by

(6)

Next we estimate the classical Coulomb energy for n = 3/5, 4/7, 5/7 where the electron-configurations with the minimum classical Coulomb energy are shown in Figures 3-5.

(Case of) The number of the first nearest pairs is equal to and the number of the second nearest pairs is equal to at as seen in Figure 3. Then the total classical Coulomb energy is nearly equal to

(7)

(Case of) Figure 4 shows that the number of the first nearest pairs is equal to and the number of the second nearest pairs is equal to at. The total classical Coulomb energy is given by

(8)

(Case of) The number of the first nearest pairs is equal to and the number of the second nearest pairs is equal to at as easily seen in Figure 5. Then the total classical Coulomb energy is

(9)

(Any case of)

We calculate the classical Coulomb energy for a general case of . As it is proven in Ref. [9] the electron-configuration with the minimum classical Coulomb energy is constructed by repeating the representative unit-configuration where r electrons exist in sequential q Landau orbitals. The number of empty orbitals per unit-configuration is. All the empty orbitals are separated by one or more filled-orbitals at. That is to say, all the empty orbitals are isolated as seen in Figures 2-5. Therefore the second-nearest pairs exist per unit-configuration due to the presence of the empty orbitals. The total numbers of the first and the second nearest pairs is equal to the total number of electrons as easily seen in Figures 2-5. Therefore the number of nearest pairs becomes per unit-configuration. The total number of nearest electron pairs is equal to and the total number of second nearest electron pairs is equal to for the filling factor. Consequently the total classical Coulomb energy is given by

This equation is expressed by using as

(10)

From Equations (4) and (10) the expectation value of the total Hamiltonian is obtained as

(11)

The single electron eigenenergy has been investigated in the previous papers and the result is the following form.

(12)

where expresses the ground state energy along the z direction (direction of the thickness in the thin conducting electron channel). Also is the potential along the y direction (Hall voltage direction). Substitution of Equation (12) into Equation (11) yields the expectation value of the total Hamiltonian as follows:

(13a)

We put together the constant parts as follows;

(13b)

Therein f is the constant value as

(14a)

(14b)

where is the mean value of the potential along the y-direction. Equation (13b) gives the function-form of W, which is illustrated in Figure 6.

The function W depends linearly upon and the proportional coefficient is negative, because the classical Coulomb energy between the first nearest electron pair, , is larger than that between the second nearest pair,. Thus the expectation value of the total Hamiltonian W changes continuously with as in Figure 6. Accordingly the classical Coulomb energy has no energy-gap and so cannot produce the plateaus of Hall resistance. The confinement of the Hall resistance comes from another reason as studied in the previous papers [6] [9] [12] [13] . The allowed transitions of electron-pairs decrease abruptly when the filling factor deviates from the specific filling factor. This structure is named “valley structure”. Summation of the valley energy and W gives the energy spectrum of the quasi 2D-electron system as examined in the next section.

3. n-Dependence of the Total Energy

We already calculated the energy of electron pairs placed in the nearest orbitals by employing the perturbation calculation in the previous papers [1] - [13] . The exact