Research Article | Open Access
Shosuke Sasaki, "Plateaus in the Hall Resistance Curve at Filling Factors ", International Scholarly Research Notices, vol. 2014, Article ID 468130, 16 pages, 2014. https://doi.org/10.1155/2014/468130
Plateaus in the Hall Resistance Curve at Filling Factors
The fractional quantum Hall (FQH) states with higher Landau levels have new characters different from those with . The FQH states at are examined by developing the Tao-Thouless theory. We can find a unique configuration of electrons with the minimum Coulomb energy in the Landau orbitals. Therein the electron (or hole) pairs placed in the first and second nearest Landau orbitals can transfer to all the empty (or filled) orbitals at , 14/5, 7/3, 11/5, and 5/2 via the Coulomb interaction. More distant electron (or hole) pairs with the same centre position have the same total momentum. Therefore, these pairs can also transfer to all the empty (or filled) orbitals. The sum of the pair energies from these quantum transitions yields a minimum at . The spectrum of the pair energy takes the lowest value at and a higher value with a gap in the neighbourhood of because many transitions are forbidden at a deviated filling factor from . From the theoretical result, the FQH states with are stable and the plateaus appear at the specific filling factors .
The plateau at the filling factor attracts a great deal of attention because of a new fractional quantum Hall (FQH) character. The plateau in the filling factor has characters different from that at . For example, Pan et al.  have found a deep minimum of the diagonal resistance, and , at and . At and the diagonal resistance exhibits a strongly anisotropic behaviour, where has a sharp peak while has a minimum at [1–3]. (The definition of the coordinate axes, , , and , will be shown in Figure 5 of the next section.) Eisenstein et al.  have obtained the plateaus of Hall resistance at and with even denominator. Furthermore, the other plateaus have been discovered at , and with odd denominator as seen in Figure 1.
The plateaus have the precise Hall resistance value. For example, the plateau at has the Hall resistance value within 0.015% as measured in . This accuracy of the Hall resistance indicates that the state has a lower energy than the one at .
Further experimental data are shown in Figure 2 which have been observed by Dean et al.  and Xia et al. . The Hall resistance-curve in the left panel of Figure 2  is different from that in the right panel . This difference means that the shape of the Hall resistance versus magnetic field curve depends on the samples and the experimental conditions (magnetic field strength, etc.). In particular, the difference is large at .
When the magnetic field is tilted from the direction perpendicular to the quasi-2D electron system, the Hall resistance plateau at disappears as seen in Figure 3 which has been found by Csáthy et al. . On the other hand, the plateaus persist with the tilt as in Figure 3.
The temperature dependence of has been measured by many researchers. For example, the temperature dependence of the diagonal resistance curves has been measured by Pan et al. . The diagonal resistance curve at 36 mK is different from that at 6 mK. Furthermore, some local minima in the diagonal resistance curve disappear at 36 mK. Using the temperature dependence of , the Arrhenius plots are drawn to give an energy gap. The energy gap is shown in Figure 4 which is obtained by Choi et al.  and also in the paper .
These experimental findings at have stimulated theoretical studies. Several theories have been proposed to explain the plateaus of the Hall resistance at , especially at . Some of them are briefly reviewed below.
Koulakov et al. have studied the ground state of a partially filled upper Landau level in a weak magnetic field. They have used the effective interaction  which was derived by Aleiner and Glazman in the 2D-electron system with high Landau levels, taking into account the screening effect by the lower fully occupied levels. Then, they have found that the ground state is a charge-density wave (CDW) state with a large period . Moessner and Chalker studied a 2D-electron system with a fermion hardcore interaction and without disorder. They found a transition to both unidirectional and triangular charge-density wave states at finite temperatures . Rezayi et al. numerically studied a 2D-electron system in magnetic field with a high Landau level half filled by electrons. In finite size systems with up to 12 electrons and torus geometry, they found a charge-density wave ordering in the ground state. Their results show that the highest weight single Slater determinant has the occupation pattern 11111000001111100000, where 1 and 0 stand, respectively, for an occupied orbital and an empty orbital .
Haldane and Rezayi investigated the pair state with spin-singlet . They used a hollow core Hamiltonian. In the Landau level number , the hollow core Hamiltonian has the first pseudopotential although the zeroth Haldane pseudopotential is zero. They found a ground state called HR state. Moore and Read were inspired by the structure of the HR state, and constructed the pair state a -wave polarized state. They have described the FQH state in terms of conformal-field-theory . The state is called the Moore-Read state (MR state). In , Read wrote “the wavefunction represents BCS [17, 18] pairing of composite fermions. One type are the charged vortices discussed above, with charge which according to MR are supposed to obey nonabelian statistics.” Greiter et al. investigated the MR state from the viewpoint of the composite fermion pair [19, 20]. The statistics are an ordinary abelian fractional statistics.
Morf argued the quantum Hall states at by a numerical diagonalization . He studied spin-polarized and -unpolarized states with electrons. His result indicates that the state is expected to be the spin-polarized MR state. Rezayi and Haldane  confirmed Morf’s results. Their results are based on numerical studies for up to 16 electrons in two geometries: sphere and torus. They found a first order phase transition from a striped state to a strongly paired state. They examined 12 electrons in a rectangular unit cell with the aspect ratio 0.5. They found the stripe state, the probability weight of which is 58% for the single Slater determinant state with the occupation pattern 000011110000111100001111. Also, they found an evidence that the state is derived from a paired state which is closely related to the MR polarized state or, more precisely, to the state obtained particle-hole (PH) symmetrisation of the MR state .
Tao and Thouless [23, 24] investigated the FQH states in which the Landau states with the lowest energy are partially filled with electrons. Thus, the theory does not assume any quasiparticle. The present author has developed a theory on the FQH states at [25–32] by extending the Tao-Thouless theory. We will apply the theory to the problem of 5/2 plateau in Section 3. The plateaus at and at are discussed in Sections 5 and 6, respectively. Before examining this problem, the fundamental properties will be shortly summarized for the quasi-2D electron system below.
2. The Properties of a Quasi-2D Electron System
A quantum Hall device is illustrated in Figure 5 where the -axis is the direction of the current and the -axis is the direction of the Hall voltage. Then, the vector potential, , has the components where is the strength of the magnetic field. The Hamiltonian, , of a single electron in the absence of the Coulomb interaction between electrons is given by where and indicate the potentials confining electrons to an ultrathin conducting layer in Figure 5. Therein is an effective mass of electron and is the electron momentum. The Landau wave function of the single electron is given by where is the wave function of the ground state along the -direction, is the Hermite polynomial of th degree, is the normalization constant, and is the length of the quasi-2D electron system as in Figure 5. The integer is called Landau level number hereafter.
Because of the periodic boundary condition, the momentum is given by The momentum is related to the value in the wave function, (3a), as The eigenenergy is given by where is the Landau level number, is the ground state energy along the -direction, and is the potential energy in the -direction.
When there are many electrons, the total Hamiltonian is given by where is the total number of electrons, is the permittivity and is the single particle Hamiltonian of the th electron without the Coulomb interaction as The many-electron state is characterized by a set of Landau level numbers and a set of momenta . The complete set is composed of the Slater determinant as This state is the eigenstate of . The expectation value of the total Hamiltonian is denoted by which is given by where is the expectation value of the Coulomb interaction defined by Hereafter, we call “classical Coulomb energy.” We divide the total Hamiltonian into two parts and as follows: where is composed of the off-diagonal elements only. Accordingly, the total Hamiltonian of the quasi-2D electron system is a sum of and as follows: The Slater determinant composed of the Landau states is the exact eigenstate of . So we will examine the residual part, namely, quantum transitions via the off-diagonal parts of the Coulomb interaction.
Because the Coulomb interaction depends only upon the relative coordinate of electrons, the total momentum along the -direction conserves in the quasi-2D electron system. That is to say, the sum of the initial momenta and is equal to that of the final momenta and :
Next we discuss the configuration of electrons in the Landau orbitals. The previous article  has verified that the most uniform configuration of electrons is uniquely determined for any filling factor except at both ends. The effects of the boundaries may be neglected in a macroscopic system. At the Landau states with the Landau level number are partially occupied by electrons and all the states with are empty. For example, the most uniform configuration for is constructed by repeating of the unit configuration (filled, empty, filled) as shown in Figure 6. This configuration determines the set of the momenta as where is the minimum value of the momentum.
For an arbitrary filling factor, we can also find the most uniform configuration of electrons in the Landau states. Then, the configuration yields the minimum expectation value of , namely, the ground state of .
We next count the number of the Coulomb transitions via . When the filling factor deviates a little from the specific filling factor, the number of quantum transitions decreases abruptly because of the Fermi-Dirac statistics and the momentum conservation. That is to say, the number of the Coulomb transitions at the specific filling factors takes the largest among those of the neighbouring filling factors. This property produces the minimum energy at the specific filling factors and yields the precise confinement of the Hall resistance. This mechanism can explain the phenomena of the FQHE at without introducing any quasiparticles [25–32].
Here, we remark the edge current in the FQH states. Büttiker  investigated the current distribution in a 2D-electron system and found the edge current. Both total current and Hall voltage are affected by the edge current in the IQHE but the Hall resistance remains to be the original value. The mechanism has been studied for the FQH states under the existence of the edge current in the article . The precise confinement of the Hall resistance is derived from the momentum conservation along the current direction.
3. Explanation for the Appearance of 5/2 Plateau
The energy gap for is shown in Figure 7 which is obtained in . The value of the energy gap changes from sample to sample as in Table 1. The energy gap at is about 4.3 K and that of is about 0.272~0.544 K as measured in  and . Thus, the energy gap in the region is about 1/10 times that in . Therefore, we cannot ignore the small terms of various Coulomb transitions in studying the energy spectrum for .
In the region of , all the Landau states with are filled with electrons with up and down spins, and the Landau states with are partially occupied by electrons. The interactions between electrons depend on the shape of the Landau wave function in the and directions, the wave function along the -direction, the screening effect of electrons in the lowest Landau level, and so on. The theories [10–22] reviewed in Section 1 have employed various types of interactions. For example, the first Haldane pseudopotential for is positive although for the lowest Landau level () is zero. Thus, the wave function and the interaction in higher Landau levels are different from those in the lowest Landau level.
In the previous articles [25–32], we have ignored the energy from the pairs placed in the second nearest neighboring Landau orbitals, because it is expected to be smaller than that in the first nearest Landau orbitals. For the FQH states we have to include the contribution from the electron pairs placed in the second neighboring Landau orbitals because of the small energy-gap as in Table 1.
We first study the FQH state using the method of the previous papers [25–32]. At , the most uniform configuration of electrons in the level is illustrated in Figure 8 where the fully occupied orbitals with are not drawn for simplicity.
We examine the quantum transitions via the Coulomb interaction . All the Coulomb transitions satisfy the momentum conservation along the -axis. Figure 8 shows schematically the quantum transitions from the electron pair CD as an example. The momenta of electrons at C and D are described by and , respectively. These momenta change to and after the transition. The momentum conservation gives the following relation: where is the momentum transfer. The quantum transition is allowed to empty orbitals only. As seen in Figure 8, the empty orbital exists in the odd numbered orbitals from the left of the orbital C. Therefore, the transfer momentum takes the following value derived from (4) and (5): where is eliminated because the transition is forbidden by the Pauli exclusion principle. All the allowed transitions are illustrated by the green arrow pairs in Figure 8. Thus, any electron pair placed in the second neighboring orbitals can also transfer to all the empty orbitals (except ) at .
In order to calculate the pair energies, the following summation is introduced for the Landau level : The summation is carried out for all the momentum changes except and . The elimination comes from disappearance of the diagonal matrix element of . The summation is positive, because the denominator in (18a) is negative. The perturbation energy of the pair CD is expressed by the summation as follows: because the function in (18a) is continuous for the argument and also the momentum change is extremely small for a macroscopic size of the device. Therein the factor 1/2 comes from the fact that the number of allowed transitions is equal to the number of the empty orbitals which is half of the total Landau orbitals with .
There are many electron pairs like CD. The total number of the pairs like CD is equal to which indicates the total number of electrons placed in the Landau orbitals with . Accordingly, the perturbation energy of all the second nearest electron pairs is given by The pair energy per electron is The summation depends on the thickness, size, and material of the quasi-2D electron system. The reasons are as follows. The wave function along the -axis depends on both thickness and potential shape along the -axis. The wave function length of the -direction depends on the device size. The effective mass of electron and the permittivity depend on the material of the device. Therefore, the classical Coulomb energy and the transition matrix element vary with changing the quantum Hall device. Accordingly, the value of varies from sample to sample.
Furthermore, the Landau wave function is zero at its center position because of the Hermite polynomial of degree as in (3a). Accordingly, the function form in (18a) is quite different from that for . Additionally, we need to consider the screening effect from the electrons. The effect is also unknown. Therefore, we do not go into detail of the summation and treat as a parameter.
We have ignored the quantum transitions into higher Landau levels with . The contribution is extremely small because the excitation energy is very large as follows: the excitation energy from the Landau level with to that with is given by which is derived from (6). The effective mass differs from material to material and the value in GaAs is about 0.067 times that of free electron. For example, this excitation energy is estimated at the magnetic field strength as In the perturbation calculation, the denominator is the energy difference . When the intermediate state belongs to , the main part of is for . The value is about , and so the absolute of the denominator for the intermediate states with is very large compared with that for the intermediate states with in (18a). Therefore, the contribution from the intermediate states with is extremely small, so we ignore them. We examine now the energy gaps in Table 1 which have the magnitude of about 0.1 K. The absolute value of the denominator for the intermediate state with is about 800 times the energy gaps. Accordingly, the intermediate states with may be neglected.
Next, we study the perturbation energy in the neighborhood of . As an example, the state is examined. The most uniform electron configuration is illustrated in Figure 9 where the Landau orbitals with are not shown, for simplicity. The unit configuration is , where 1 indicates a filled orbital and 0 an empty orbital.
The electron pair CD can transfer to all the empty orbitals as shown by green arrow pairs. On the other hand the pair GA can transfer to only one site per unit configuration as shown by black arrow pairs in Figure 9 where the red symbol indicates the forbidden transition. So the pair energy is Therein the coefficient 1/19 indicates that the electron pair can transfer to one site per 19 Landau orbitals. The energies of the other pairs are calculated, the results of which are We calculate the electron pair AB which is placed in the nearest neighboring Landau orbitals as in Figure 9. In order to calculate the energy of the nearest pair, we introduce the following summation : where which is different from (18b). This summation is positive because the denominator of (26) is negative. The value is also treated as a parameter because varies from sample to sample. The pair AB can transfer to any empty state as shown by the red arrow pairs in Figure 9. The number of the empty states is 9 per unit configuration. Accordingly, the pair energy of AB is given by The total perturbation energy from all the electron pairs placed in the first and second neighboring Landau orbitals with is Substitution of (24), (25), and (28) into (29) yields The pair energy per electron for is One more example will be examined. The most uniform configuration is shown in Figure 10 where the Landau orbitals with are not shown. The electron pair CD can transfer to all the empty orbitals as shown by green arrows. On the other hand, the pair JA can transfer to only one site per unit configuration as illustrated by black arrows in Figure 10.
Accordingly, the pair energy is The perturbation energies of the other second nearest pairs are given by The electron pair AB can transfer to any empty state as shown by the red arrows in Figure 10. The number of the empty states is 15 per unit configuration. Accordingly, the pair energy of AB is given by The total perturbation energy from all the electron pairs placed in the first and second neighboring Landau orbitals with is Substitution of (32), (33), and (34) into (35) yields The pair energy per electron for is This filling factor is close to . The difference between and 78/31 is about 0.6%. We compare the perturbation energy of the first and second nearest pairs per electron between and 78/31: From (38) the pair energy at is lower (or higher) than that at for (or ). We examine the following two Cases 1 and 2.
Case 1 (). In this case, the perturbation energy of the first and second pairs per electron is Accordingly the pair energy at is sufficiently lower than that in the neighborhood of . So the state is very stable and the Hall plateau appears at .
Case 2 (). In this case, the Hall plateau does not appear because the pair energy at is higher than that in its neighborhood.
Thus the FQH state is sensitive to the relative value of and which is dependent on the materials, thickness of the conducting layer, device structure, and so on. In the next section, we discuss the sample dependent phenomena based on the theory obtained above.
4. Sample Dependent Phenomena
For example, the 5/2 and 7/2 Hall plateaus do not exist on the red curve of Hall conductance obtained by Dean et al. in the article  as seen in Figure 11. On the other hand, the experimental results in Figures 1–3 indicate the appearance of the 5/2 and 7/2 Hall plateaus. Thus, the appearance or disappearance of the and 7/2 plateaus seems to depend upon the samples used in the experiments.
According to our theory examined in Section 3, this property comes from the relative value of and .
5. FQH States at Filling Factors
As will be shown below, in the region of , we find fractional filling factors where both first and second nearest electron pairs can transfer to all the empty Landau orbitals with . These filling factors are 8/3, 14/5, 18/7, and 19/7 for the denominator smaller than nine. The allowed transitions are shown by the arrow pairs in Figures 12, 13, 14, and 15.
Figure 12 shows the most uniform configuration at where two types of electron pairs exist. The pair AB represents the first nearest electron pair and the pair CD the second nearest one. Both the pairs AB and CD can transfer to all the empty orbitals with . The allowed transitions are shown by black and green arrow pairs, respectively, in Figure 12.
The number of empty orbitals with is 1/3 of the Landau orbitals with . Therefore, the pair energies are given by
The total energy of the electron pairs placed in the first and second neighboring Landau orbitals with is where indicates the total number of electrons in the Landau orbitals with . Substitution of (40a) and (40b) into (41a) yields We examine the pair energy in the limit from the right or left to . Using the same method reported in the previous papers [25–32], we obtain the right and left hand limits as Therefore, a valley in the energy spectrum appears as The , and states have the most uniform configuration as shown in Figures 16, 17, and 18, respectively. The allowed transitions are schematically drawn by the black and green arrow pairs for the first and second nearest electron pairs, respectively.
The number of the allowed transitions is 1/5 times the number of Landau orbitals with . Then, the pair energy of AB and CD is given, respectively, by There are many electron pairs represented by AB and CD. The total number of the pairs represented by AB is equal to and also that by CD is equal to . Then, we obtain the total pair energy for the electron pairs placed in the first and second neighboring Landau orbitals with as Figure 14 shows the allowed transitions of the pairs AB and CD at the filling factor . The number of the empty orbitals is 3/7 times that of the Landau orbitals with . Accordingly, the pair energy of AB and CD is given, respectively, by Then we obtain Next we count the number of allowed transitions of the pairs AB and CD at . The electron pairs AB and CD in Figure 15 can transfer to all the empty Landau orbitals with .
Since the number of the allowed transitions for each of the AB and CD pairs is two per unit configuration composed of the seven Landau orbitals, the pair energy of AB and CD is given, respectively, by Then we obtain Thus, the electron pairs AB and CD can transfer to all the empty orbitals at 5/2, 8/3, 14/5, 18/7, and 19/7, and therefore the pair energy becomes very low, resulting in a strong binding energy.
The values of and may vary from sample to sample. We examine the condition that the 5/2 plateau appears. From (38), the state is stable when is sufficiently lager than . In the experiment , the energy gaps have been measured as in Figure 4. In the high mobility sample , the energy gap at is nearly equal to that at . Equations (21) and (41b) give the pair energy per electron as follows: The experimental data  can be explained by the present theory under the following condition: Equations (21), (41b), (45), and (49) give the theoretical ratio of the pair energies at , and 19/7 as follows: When condition (51) is satisfied, the theoretical ratio of the pair energies becomes From Figure 4, the experimental data of the energy gap yield the ratio for the high mobility sample as Thus, the present theory explains reasonably well the experimental data.
6. FQH States at Filling Factors
Next we examine the FQH states with . The most uniform configurations at , and 16/7 are schematically drawn in Figure 16. The hole-pairs AB and CD can transfer to all the electron states in as easily seen in Figure 16. This property produces a strong binding energy between the hole-pairs.
The number of allowed transitions for the hole-pairs at is equal to that for the electron-pairs at . This symmetry between electron and hole is clearly seen by comparing the number of transitions (namely, number of arrows) for 8/3 and 7/3, as easily seen in Figure 17.
From the discussion given in Sections 3, 5, and 6, we find that the state with , and and 19/7 is stable in the region . The left panel of Figure 2 quoted from  shows the plateaus of the Hall resistance at , and 16/7. Similar investigation for can be performed by using the method of this section. The pair energy becomes large at the filling factor , and so on. The Hall plateaus at these filling factors have been found in several experiments.
The number of the allowed transitions via the Coulomb interaction discontinuously varies with changing the filling factor. The discontinuous variation is caused by the combined effect of the momentum conservation along the current, the most uniform configuration of electrons and the Fermi-Dirac statistics. This effect produces the stability of FQH states at the several filling factors.
As described in Section 1, different states have been proposed by different authors. For example, the FQH state is explained by the stripe HR or MR states and so on. The FQH state is said to be composed of the composite fermions where each electron binds to two flax quanta and the FQH state is explained to be composed of the composite fermions where each electron binds to four flax quanta. The theory presented here explains the FQH phenomena occurring at and those at in a coherent way without assuming any quasi-particles.
7. Further Investigation of the Pair Energy for
We examine the exact energy of any FQH state. The total energy of the quasi-2D electron system is the sum of the eigenenergy of and the pair energy via the interaction as follows: where has been already given by (10) as follows: Equations (5) and (6) yield the following equation: The energy difference between different Landau levels is extremely large as shown in (22) and (23). The interval of Landau energies has been estimated for GaAs The experimental values of the energy gaps at are smaller than 1 as in Table 1 and therefore higher Landau levels can be ignored at low temperatures. So, the ground state at is obtained by superposing many-electron states that all the Landau states with are occupied by electrons with up and down spins and the Landau states with are partially occupied by electrons. We express the number of electrons in the Landau level by and the number of Landau orbitals by , respectively. The ratio is described as follows: where depends on the sample and the magnetic field strength but is independent of . The total number of electrons is Substitution of (59a), (59b), (59c), (59d), and (60) into (57) gives the eigenenergy of as follows: Next, we investigate the pair energy which is caused by the quantum transitions via . The electron pairs in the ground state with have been classified into the following three types: First type: both electrons in the pair are placed in the orbitals with only. Second type: one electron is placed in and the other in . Third type: both electrons in the pair are placed in only.
These pair energies are described by the symbols , , and , respectively. The total energy of all the electron pairs is Therein the pair energies and are negligibly small because of the following reason. Any order of the perturbation energy is obtained by a summation of the functions with the denominator containing the energy difference of between the ground and intermediate states. Any electron pair belonging to the first or second types can transfer only to the intermediate states with a higher Landau level because all the Landau orbitals with are already occupied by electrons with up and down spins. Therefore, the energy difference between the ground and intermediate states is very large as in (58). Then we may ignore the pair energy belonging to the first and second types: On the other hand, the electron pairs in can transfer to empty orbitals with for . Then, the energy difference of between the ground and the intermediate states comes from the difference in the classical Coulomb energies and so the difference is very small. We will examine any electron (or hole) pair placed in any Landau orbitals with . As an example, we discuss the case of . Figure 18 schematically shows the electron pairs at . The electron pairs IL, HM, and GN possess the total momentum same as that of the pair JK. These pairs can transfer to all the empty orbitals as easily seen in Figure 18.
The momenta of the electrons at G, H, I, J, K, L, M, and N are described by the symbols , and , respectively. Then, the total momenta of the electron pairs take the same value because of (4) and (5): The energies of the pairs GN, HM, IL, and JK, , and , can be reexpressed systematically by using a symbol , where and indicate the total momentum and the distance between the pair as follows: Therein the momentum of each electron is given as Thus, any momentum-pair is related to and as Because both momenta and should be equal to , the values of and are classified to the following two cases: We have already examined the case of odd integer in Figure 18.
Next, we examine the case of even integer . Figure 19 shows quantum transitions with even integers given by (68b). All the electron pairs possessing the total momentum same as that of the pair KL can transfer to all the empty orbitals as in Figure 19. The electron pairs KL, JM, IN, and HO indicate the cases of , and 10, respectively.
The pair energies are described as The total energy of all the electron pairs is described by the symbol defined by (55). Use of (62) and (63) gives This energy at is the sum of all the pair energies with and : Equations (55), (70), and (71) yield the total energy of the quasi-2D electron system as follows: Substitution of (61) into (72) gives We express the pair energy per electron by the symbol which is defined by The exact pair energy is the sum of all order terms in the perturbation calculation as follows: where indicates the th order of the perturbation energy. Substitution of (74) and (75) into (73) yields Therein the function form of is continuous with the change in . On the other hand, the pair energy has a discontinuous form for the argument , because the number of the allowed transitions depends discontinuously upon . This discontinuous property produces the plateaus of the Hall resistance at specific filling factors. We have already calculated the second order perturbation energies for and 2 as in (21), (31), (37), (41b), (45), (47), and (49). We list the results in Tables 2 and 3.