Higher-order composite fermion states are correlated with many quasiparticles. The energy calculations are very complicated. We develop the theory of Tao and Thouless to explain them. The total Hamiltonian is

Precise experiments on ultra-high-mobility samples revealed many local minima of diagonal resistivity

We study the other description in order to remove these ambiguities. Tao and Thouless [

Quantum hall device.

That is to say the potential well in the

Next we consider the many electron states

The electrons should most uniformly occupy the lowest Landau levels so that the classical Coulomb energy has the lowest value. The electron momenta

In the next section we will schematically draw the most uniform electron configuration at several filling factors. Then it is clarified that the most uniform electron configuration is unique. The most uniformity yields the minimum eigen energy of

We can estimate the perturbation energy via the Coulomb transitions by using the usual perturbation method of nondegenerate case. Electron (or hole) pairs in the nearest Landau orbitals are most affected by the Coulomb interaction. The perturbation energies of the nearest electron (or hole) pairs are sensitively dependent upon each electron configurations. The sensitivity is caused by Fermi-Dirac statistics and the momentum conservation of the

All the

At the filling factors of

We examine another type of gap which indicates the excitation energy gap from the ground state to excited states. This new gap is highly correlated with the gap in the spectrum of energy versus filling factor. The mechanism is studied in Section

We can find out the many-electron states with the minimum energy of

(a) Electron configuration at

Therein the bold line indicates a Landau state filled with electron, and the dashed line means an empty state. It is noteworthy that the current direction is described by the

We explain the searching method to find the electron configuration with the minimum Coulomb energy, because it is nontrivial to find the filling way for any fractional filling factor

In the whole region, three electrons exist inside every 5 sequential landau states. Then the filling factor becomes 3/5.

Two electrons exist in 5 sequential Landau states for some parts, and four electrons exist in 5 sequential Landau states for some other parts. And the average filling factor is equal to 3/5.

The Coulomb energy of Case 1 is smaller than one of Case 2 because the filling way of Case 1 is more uniform than one of Case 2. Therefore, it is sufficient to consider all the filling ways inside 5 sequential states. They are 10 filling ways as shown in Figure

All unit arrangements of electron configurations for

The five filling ways (a-1, a-2, a-3, a-4, and a-5) give the same electron configuration A by numerous repeating of themselves except both end parts. The electron configuration A is shown in Figure

(a) Configuration A at

Similarly the five filling ways (b-1, b-2, b-3, b-4, and b-5) in Figure

It is noteworthy to examine the connections between different arrangements in Figure

Connections between different arrangements.

Therefore, these connections belong to Case 2 and then have a classical Coulomb energy larger than one of configuration A in Figure

For any filling factor

Most uniform unit arrangements of electron configuration.

When we repeat the unit arrangement in Figure

We draw some examples with higher Coulomb energy for the denominator

Filling ways with higher classical Coulomb energy.

Thus only one configuration has the minimum classical Coulomb energy among the enormous many configurations. The whole-electron configuration is created by repeating of only one unit arrangement of electron at any fractional number of

The shape of Landau wave function with

Allowed transitions of

The electron configurations in Figure

Coulomb transitions from nearest electron pair AB at

The momenta of the

Similarly the

We can systematically describe the perturbation energies at any filling factors by using the value of

The number of

Therein,

The number of

It has been clarified in the previous subsection that all the nearest electron pairs can transfer to all empty Landau orbitals at the filling factor of

Transitions from nearest electron pairs for

It is noteworthy that some nearest electron pairs cannot transfer to some empty states because of momentum conservation and Pauli’s exclusion principle. We take this prohibition of the transitions into consideration and obtain the perturbation energy of nearest electron pairs as follows:

Next we examine the neighbourhood of

The gap structure is produced from the property that all the

Table

Energy gaps of

2/3 | −(1/6) | −(1/12) | −(1/12) | |

3/5 | −(2/15) | −(3/30) | −(1/30) | |

4/7 | −(3/28) | −(5/56) | −(1/56) | |

5/9 | −(4/45) | −(7/90) | −(1/90) | |

6/11 | −(5/66) | −(9/132) | −(1/132) | |

7/13 | −(6/91) | −(11/182) | −(1/182) | |

8/15 | −(7/120) | −(13/240) | −(1/240) |

We draw the most uniform electron configuration at

Coulomb transitions from nearest hole pair AB.

Therein, solid lines indicate the Landau orbitals filled with electron, and dashed lines indicate the vacant Landau orbitals.

Energy gaps of

1/3 | −(1/6) | −(1/3) | −(1/6) | −(1/6) | |

2/5 | −(2/15) | −(1/5) | −(3/20) | −(1/20) | |

3/7 | −(3/28) | −(1/7) | −(5/42) | −(1/42) | |

4/9 | −(4/45) | −(1/9) | −(7/72) | −(1/72) | |

5/11 | −(5/66) | −(1/11) | −(9/110) | −(1/110) | |

6/13 | −(6/91) | −(1/13) | −(11/156) | −(1/156) | |

7/15 | −(7/120) | −(1/15) | −(13/210) | −(1/210) |

Tables

Many electrons in the electric current are scattered by impurities and thermal vibrations. These scatterings yield the diagonal resistance of the

At

The excitation-energy gaps suppress the electron scatterings. Then the diagonal resistance becomes small. This mechanism produces local minima in the diagonal resistance curve. The theoretical results in Tables

Many local minima of the diagonal resistance in the region of

The local minima appear at the filling factors of

The most uniform electron configurations are illustrated in Figure

All Coulomb transitions from nearest electron pairs are forbidden.

Next we draw the most uniform electron configuration for

All Coulomb transitions from nearest hole pairs are forbidden.

That is to say we obtain the following relations:

1/2 | 0 | ||

3/4 | 0 | 1/4 | 0 |

5/6 | 0 | 1/6 | 0 |

7/8 | 0 | 1/8 | 0 |

Consequently the states with

In this section we examine the states with the filling factors

Coulomb transitions of nearest electron pairs at

The

Coulomb transitions of nearest hole pairs at

Then the perturbation energy of the nearest hole pairs is obtained as

Similar configurations are schematically drawn in Figures

Coulomb transitions of nearest hole pair at

Coulomb transitions of nearest hole pair at

Coulomb transitions of nearest hole pair at

Coulomb transitions of nearest hole pair at

All the red hole pairs can transfer to all electron states as in Figures

We have used the term “energy gap” for the gap in the spectrum of energy versus filling factor in the previous sections. This gap produces the plateau in the Hall resistance curve [

We first consider one of excited states at

Electron configuration in excited state #1 at

The perturbation energies of the pairs CE and DC are described by the symbols

The spreading width of Landau state is denoted by

The second example is shown in Figure

Electron configuration in excited state #2 at

Electron configuration in excited state #3 at

Thus the excitation-energy gap has a correlation with the energy gap in the spectrum.

When the device size is very small, the ratio

We have developed the theory of Tao and Thouless. Then we have found the momentum conservation of the

The author would like to acknowledge Professor H. Hori and Professor K. Oto for useful discussions. The author expresses his heartfelt appreciation for the comments of the referee and the editor. This paper has been improved and revised by the comments.