^{1}

^{2}

^{3}

^{*}

Mani observed zero-registance states similar to those quantum-Hall-effect states in GaAs/AlGaAs but without the Hall resistance plateaus upon the application of radiations [R. G. Mani, Physica E 22, 1 (2004)]. An interpretation is presented. The applied radiation excites “holes”. The condensed composite (c)-bosons formed in the excited channel create a superconducting state with an energy gap. The supercondensate suppresses the non-condensed c-bosons at the higher energy, but it cannot suppress the c-fermions in the base channel, and the small normal current accompanied by the Hall field yeilds a B-linear Hall resistivity.

In 2002 Mani et al. [

with ω = radiation frequency, m = effective mass, e = electron charge, indicating the superconducting state with an energy gap in the elementary excitation spectrum. The phenomenon is similar to that observed in the same system in the traditional quantum Hall effect (QHE) regime (T ~ 1.5 K, b ~ 10 T) [

rather than (Mani’s case). They also noted a noticeable side resistivity minimum besides the principal set of the minima.

In finer analysis Mani et al. [

correlates with the resistance such that nearly vanishes when, and (b) is negative, and it is antisymmetric with respect to small B-fields:

The property (b) means that there is a current due to “hole”-like particles having the charge of the opposite sign to that of the majority (“electron”-like) current carrier. In other words the applied radiation generates the “holes”. This may be checked by applying the circularly polarized lasers, which can excite “electrons” or “holes” selectively, depending on the sense of the circular polarization. The small slope means that the “hole” density is considerable higher than the “electron”

density.

Mani et al. [

Earlier Fujita et al. [

In the present work we shall extend our theory to Mani’s phenomena. We show that the most prominent zero-resistance states observed by Mani et al. [

If the magnetic field is applied slowly, the classical electron can continuously change from the straight line motion at zero field to the curved motion at a finite B. Quantum mechanically, the change from the momentum state to the Landau state requires a perturbation. We choose for this perturebation the phonon exchange between the electron and the fluxon. Consider the cparticle with a few fluxons. If the B-field is applied slowly the energy of the electron does not change but the cyclotron motion always acts so as to reduce the magnetic fields. Hence the total energy of the c-particle is less than the electron energy plus the unperturbed field energy. In other words the c-particle is stable against the break-up, and it is in a bound (negative energy) state. In our theory a c-particle is simply a dressed electron carrying fluxons. The c-particle moves as a boson (fermion) depending on the odd (even) number of fluxons in it. At the Landau level (LL) occupation number, odd, the c-bosons with fluxons are generated, and condense below certain critical temperature. The Hall resistivity plateau is due to the Meissner effect, see below.

GaAs forms a zinc blende lattice. We assume that the interface is in the plane (001). The ions form a square lattice with the sides directed in [

where is the phonon momentum (energy); the interaction strength between the electron (fluxon) and the phonon; the Landau quantum number is omitted; the bold denotes the two dimensional (2D) guiding center momentum and the italic the magnitude. If the energies () of the final and initial electron states are equal as in the degenerate LL, the effective interaction is attractive, i.e., .

Following Bardeen, Cooper and Schrieffer (BCS) [

where is the number operator for the “electron” (1) [“hole” (2), fluxon (3)] at momentum and spin with the energy. We represent the “electron” (“hole”) number by ^{†}, where are annihilation (creation) operators satisfying the Fermi anticommutation rules:

We represent the fluxon number by, with, satisfying the anticommutation rules. ,. The prime on the summation means the restriction:, = Debye frequency. If the fluxons are replaced by the conduction electrons (“electrons”, “holes”) our Hamiltonian is reduced to the original BCS Hamiltonian, Equation (2.14) of Ref. [

The c-bosons, each with one fluxon, will be called the fundamental (f) c-bosons. Their energies are obtained from [

where is the reduced wavefunction for the fcboson; we neglected the fluxon energy. The denotes the strength after the ladder diagram binding, see below. For small, we obtain

where is the Fermi velocity and the density of states per spin. Note that the energy depends linearly on the momentum.

The system of free fc-bosons undergoes a Bose-Einstein condensation (BEC) in 2D at the critical temperature [

The interboson distance calculated from this expression is. The boson size calculated from Equation (4), using the uncertainty relation and, is, which is a few times smaller than. Hence, the bosons do not overlap in space, and the model of free bosons is justified. For GaAs/AlGaAs, , m_{e }= electron mass. For the 2D electron density 10^{11} cm^{−2}, we have cm∙s^{−1}. Not all electrons are bound with fluxons since the simultaneous generations of ±fc-bosons is required. The minority carrier (“hole”) density controls the fc-boson density. For cm^{−2}, K, which is reasonable.

In the presence of Bose condensate below the unfluxed electron carries the energy, where the quasi-electron energy gap is the solution of

Note that the gap depends on. At, there is no condensate and hence vanishes.

Now the moving fc-boson below has the energy obtained from

where replaced in Equation (3). We obtain

where is determined from

The energy difference:

represents the T-dependent energy gap. The energy is negative. Otherwise, the fc-boson should break up. This limits to be at 0 K. The declines to zero as the temperature approaches from below.

The fc-boson, having the linear dispersion (12), can move in all directions in the plane with the constant speed. The supercurrent is generated by the fc-bosons condensed monochromatically at the momentum directed along the sample length. The supercurrent density (magnitude), calculated by the rule:, is

The induced Hall field (magnitude) equals. The magnetic flux is quantized, fluxon density. Hence we obtain

If, , we obtain in agreement with the plateau value observed.

The model can be extended to the integer QHE at. The field magnitude is less. The LL degeneracy is linear in, and hence the lowest LL’s must be considered. The fc-boson density per LL is the electron density over and the fluxon density is the boson density over:

At there are c-bosons, each with two fluxons. The c-fermions have a Fermi energy. The c-fermions have effective masses. The Hall resistivity has a b-linear behavior while the resistivity is finite.

Let us now take a general case, odd. Assume that there are sets of c-fermions with fluxons, which occupy the lowest LL’s. The cfermions subject to the available B-field form c-bosons with fluxons. In this configuration the c-boson density and the fluxon density are given by Equations (18). Using Equations (17) and (18) and assuming the fractional charge [15,16]

we obtain

as observed. In our theory the integer denotes the number of fluxons in the c-boson and the integer the number of the LL’s occupied by the parental c-fermions, each with fluxons.

Our Hamiltonian in Equation (6) can generate and stabilize the c-particles with an arbitrary number of fluxons. For example a c-fermion with two fluxons is generated by two sets of the ladder diagram bindings, each between the electron and the fluxon. The ladder diagram binding arises as follows. Consider a hydrogen atom. The Hamiltonian contains kinetic energies of the electron and the proton, and the attractive Coulomb interaction. If we regard the Coulomb interaction as a perturbation and use a perturbation theory, we can represent the interaction process by an infinite set of ladder diagrams, each ladder step connecting the electron and the proton. The energy eigenvalues of this system is not obtained by using the perturbation theory but they are obtained by solving the Schrödinger equation directly. This example indicates that a two-body bound state is represented by an infinite set of ladder diagrams and that the binding energy (the negative of the ground-state energy) is calculated by a non-perturbative method.

Jain introduced the effective magnetic field [17-19]

relative to the standard field for the composite (c-) fermion at the even-denominator fraction. We extend this to the bosonic (odd-denominator) fraction. This means that the c-particle moves field-free at the exact fraction. The c-particle is viewd as the quasiparticle containing an electron circulating around fluxons. The jumping of the guiding centers (the CM of the c-particle) can occur as if they are subject to no B-field at the exact fraction. The excess (or deficit) of the magnetic field is simply the effective magnetic field B. The plateau in is formed due to the Meissner effect. Consider the case of zero temperature near. Only the energy matters. The fc-bosons are condensed with the ground-state energy, and hence the system energy at is, where is the number of –fc-bosons (or fc-bosons). The factor 2 arises since there are ±fcbosons. Away from we must add the magnetic field energy, so that

When the field is reduced, the system tries to keep the same number by sucking in the flux lines. Thus the magnetic field becomes inhomogeneous outside the sample, generating the magnetic field energy . If the field is raised, the system tries to keep the same number by expeling out the flux lines. The inhomogeneous fields outside raise the field energy as well. There is a critical field. Beyond this value, the superconducting state is destroyed, generating a symmetric exponential rise in. In our discussion of the Hall resistivity plateau we used the fact that the ground-state energy of the fc-boson is negative, that is, the c-boson is bound. Only then the critical field can be defined. Here the phonon exchange attraction played an important role. The repulsive Coulomb interaction, which is the departure point of the prevalent theories [

In the presence of the supercondensate the noncondensed c-boson has an energy gap. Hence the noncondensed c-boson density has the activation energy type exponential temperature-dependence:

In the prevalent theories the energy gap for the fractional QHE is identified as the sum of the creation energies of a quasi-electron and a quasi-hole [20-23]. With this view it is difficult to explain why the activationenergy type temperature dependence shows up in the steady-state quantum transport. Some authors argue that the energy gap for the integer QHE is due to the LL separation =. But the separation is much greater than the observed. Besides from this view one cannot obtain the activation-type energy dependence.

The BEC occurs at each LL, and therefore the c-boson density is less for high, see Equation (18), and the strengths become weaker as increases.

The experiments by Mani et al. [

In the neighbarhood of the principal QHE at the carriers in the base and excited channels are respectively c-fermions and c-bosons condensed. The currents are additive. We write down the total current density as the sum of the fermionic current density and the bosonic:

where and are the drift velocities of the fermions and bosons. The Hall fields are additive, too. Hence we have

We note that the Hall effect condition () applies for the fermions and bosons. We therefore obtain

Far away from the midpoint of the zero-resistance stretch the c-bosons are absent and hence the Hall resistivity becomes after the cancellation of. At the midpoint the c-bosons are dominant. Then, the Hall resistivity is approximately since

where we used the flux quantization [], and the fact that the flux density equals the c-boson density. The Hall resistivity is not exactly equal to since the c-fermion current density is much smaller than the supercurrent density, but it does not vanish. In the horizontal stretch the system is superconducting, and hence the supercurrent dominates the normal current:. The deviation is, using Equation (26),

If the field is raised (or lowered) a little from the midpoint, is a constant () due to the Meissner effect. If the field is raised high enough, the superconducting state is destroyed and the normal current sets in, generating a finite resistance and a vanishing. Hence the deviation and the diagonal resistance are closely correlated.

In

Mani et al.’s experiments,

In summary the QHE under radiation is the QHE at the upper channel. The condensed c-bosons generate a superconducting state with a gap in the c-boson energy spectrum. The supercondensate changes the c-fermion energy from to in the base channel. This energy spectrum has no gap, and hence the cfermions cannot be suppressed completely at the lowest temperatures, and generate a finite resistive current accompanied by the Hall field. This explains the b-linear Hall resistivity. Our microscopic theory can be tested experimentally by examining 1) the “hole”-like excitations by a circularly polarized laser; 2) the bosonic state at and; 3) the “hole” band edge.