Bosonization theory of excitons in one-dimensional narrow-gap semiconductors

Abstract

Excitons in one-dimensional narrow gap semiconductors of anti-crossing quantum Hall edge states are investigated using a bosonization method. The excitonic states are studied by mapping the problem into a non-integrable sine-Gordon type model. We also find that many-body interactions lead to a strong enhancement of the band gap. We have estimated when an exciton instability may occur.PACS numbers: 71.27.+a, 71.35.Cc One-dimensional (1D) narrow gap semiconductors can be realized in anti-crossing quantum Hall edge states 1 and nanotubes 2 . The gaps in these systems are single particle gaps and not many body gaps. Theoretically, they provide the unusual condition that the bare band gap t is much smaller than the characteristic Coulomb energy scale E c . Moreover, strong quantum fluctuations are present in these systems, reflecting the 1D character. The ground state of these systems may be unstable against the spontaneous formation of excitons if the exciton binding energy exceeds the band gap 3,4 . In three dimensional semiconductors excitons can be treated successfully by solving the Bethe-Salpeter equation 4 . However, in the strong coupling regime of 1D systems, the perturbative approaches are not expected to be reliable due to large quantum fluctuations. If the Coulomb scale is much larger than the gap one might naively expect that exciton instability would occur. However, this simple picture neglects screening which is expected to be large due to the smallness of the gap. It is unclear whether a bound state of an "electron" and a "hole" can exists in the presence of strong quantum fluctuations. Not much is known about the physics of excitons in 1D narrow gap semiconductors.This problem can be addressed within a bosonization approach, which is applicable as long as the the characteristic energy scale of the problem is smaller than the band width W . Thus bosonization provides a natural framework for studying 1D excitons. In this work we will consider anti-crossing of quantum Hall edge states in the barrier region of between two 2D electron gases. Assume that the applied magnetic field is sufficiently strong that the system is spin-polarized with the filling factor ν = 1 and that the Fermi level is in the gap of anti-crossing edge states. We find that the problem can be mapped into a sine-Gordon (sG) model with an extra term representing the long range Coulomb interaction. Due to this extra term the model becomes non-integrable and an exact solution is unavailable. However, the non-linear cosine term of this sG model can be expanded provided that δ ex ∼ v/v0 ln W/t < 1, where v is the Fermi velocity, v 0 = e 2 /ǫh, W =hv/a, and a is the smallest length scale in the problem 6 . In this regime a perturbative theory may be applied to calculate excitonic energies. Our work also indicates that a large enhancement of band gap occurs. Our perturbative approach suggests an approximate estimate for when an exciton instability may occur: δ ex > 1 . (We cannot exclude the possibility that the higher order corrections n...

Cascade of Quantum Phase Transitions in Tunnel-Coupled Edge States

“…These resonances are particularly enhanced in its width and height at zero-bias crossings, indicating the importance of electron-electron interaction. Proposed explanations include enhanced tunneling driven by electron-electron interaction [16,17,18], mixing of the states with equal transverse momentum from the opposite sides of the barrier [19,20,21], and a coupled Luttinger liquid interacting through a strongly backscattering center in the barrier [22].…”

Tunneling in fractional quantum Hall line junctions

“…They have studied the conductance of this structure as a function of the applied magnetic field and the voltage bias between the two gases. Many theoretical works have tried to explain their results [5,6,7,8,9,10,11,12,13,14].…”

Optical Conductivity of One-Dimensional Narrow-Gap Semiconductors

“…First consider the spinless fermion case realized in the spin-polarized AQHE. 1,5 The system is modeled by the following Hamiltonian…”

“…Bosonization method allows the (almost) exact treatment of strong Coulomb interaction, and it also transforms the bare band gap term into the non-linear cosine potentials leading to a non-integrable sine-Gordon (sG) type model. 5 It was argued that under certain conditions the excitons can exist even for the Coulomb interaction much larger than the bare gap, being accompanied by the enhanced single particle gap. 5 In this paper we study the optical conductivities of AQHE 1,5 and CNT in semiconducting phase, assuming the absence of exciton instability.…”

“…5 It was argued that under certain conditions the excitons can exist even for the Coulomb interaction much larger than the bare gap, being accompanied by the enhanced single particle gap. 5 In this paper we study the optical conductivities of AQHE 1,5 and CNT in semiconducting phase, assuming the absence of exciton instability. A simple lowest order renormalization group analysis shows that the cosine term [bare band gap] can be treated perturbatively at high frequency/temperature.…”