Magneto-absorption spectra[@hl; @ul; @g1] of some bulk semiconductors displays strongly asymmetric Fano-like resonances. In a recent paper of Glutsch [*et al.*]{}[@g1] the observed profiles have been explained by coupling between the discrete and continuum states of magneto-excitons. The authors show that the necessary coupling may be induced by Coulomb interaction. Both experimental and theoretical profiles obtained have asymmetric form. However, the higher energy dip of the resonance seems to be much more pronounced in the data than in the results of the calculations. Moreover, the role of Coulomb interaction between electron and hole should be strongly weakened in the high Landau number $N$ range where the magnetic energy $N\hbar\omega_c$ is large compared to the excitonic Rydberg.

We suggest here that the qualitative explanation of the Fano profile in magneto-absorption stems from the quasi-one-dimensional character of the electronic excitations, and the corresponding singularity in the density of states (placed into continuum of lower Landau bands), rather than from the details of the interaction. We show that a very strong asymmetry, qualitatively resembling Fano profiles, can be found in the magneto-absorption by uncorrelated electron-hole pairs when [*strong enough elastic scattering*]{} by impurities or other defects is assumed be the relevant mechanism. An exact analytical solution of the problem seems to be impossible, and even very advanced numerical calculations face serious problems as it was demonstrated in Ref. \[ \] for absorption by magneto-excitons. Our approach is not aimed at a quantitative analysis of the data but rather at remarking [ *qualitatively*]{} the possible effect of scattering on the absorption by quasi-one-dimensional electronic excitations in a high magnetic field. Since free electron-hole pairs are considered below, the results may break down in the energy interval close to the fundamental gap.

Neglecting the electron-hole interaction, the absorption coefficient $\alpha$ can be written in a factorized form $$\begin{aligned}
\label{1}
\alpha &=&{4n(\omega_l)e^2|{\bf e}_l{\bf p}_{cv}|^2\over
V_0m_0^2c\hbar\omega_l} \int d\omega\sum_{N,k_y,k_z}\nonumber\\
&&{\rm Im}G_e(N,k_z;\omega)
{\rm Im}G_h(N,-k_z;\omega_l-\omega)~,\end{aligned}$$ where $\omega_l$ (${\bf e}_l$) is the incident light frequency (polarization), $n$ the refractive index, $e$ the electron charge, ${\bf
p}_{cv}$ is the interband matrix element of the momentum operator, $V_0$ is the normalization volume, and $m_0$ the bare electron mass. $G_e$ ($G_h$) is the electron (hole) Green function which is renormalized, in principle, by all relevant interactions; when taken as a solution of the Dyson equation, it gives $$\begin{aligned}
\label{2}
&&~~~~\alpha={4n(\omega_l)e^2|{\bf e}_l{\bf p}_{cv}|^2\over
V_0m_0^2c\hbar\omega_l}\int d\omega\sum_{N,k_y,k_z}\nonumber\\
&&{\rm
Im}\left[G_{0e}^{-1}(N,k_z;\omega)-\Xi_e(N,k_z;\omega)
\right]^{-1}\\
&\times&~{\rm Im}\left[G_{0h}^{-1}(N,-k_z;\omega_l-\omega)
-\Xi_h(N,-k_z;\omega_l-\omega)\right]^{-1},\nonumber\end{aligned}$$ where $G_0$ is a bare Green function and $\Xi$ the self-energy evaluated below in the lowest order of perturbation theory.

To show how the self-consistent treatment of scattering in the lowest order perturbation theory results in a dramatic change of the absorption profile, we use a simple model of a parabolic band for electrons and [*infinitely heavy holes*]{}. The latter assumption will enable us to integrate analytically Eq. (\[2\]). The scattering of the electrons is modeled by a $\delta$-function interaction with point defects. Evaluation of the lowest order contribution to the self-energy gives immediately $$\label{3}
\Xi_e(\omega)=-i\Lambda\sum_{N^{\prime}}{(\hbar\omega_{ce})^{3/2}
\over\sqrt{\hbar\omega  
-(N^{\prime}+1/2)\hbar\omega_{ce}}}~,$$ where $$\Lambda =v^2n_{imp}\hbar\omega_{ce}{m_e^{3/2}
\over\pi 2^{3/2}\hbar^3}$$ is a dimensionless coupling proportional to the square of the scattering potential strength $v$ and to the impurity concentration $n_{imp}$, $m_e$ is the electron effective mass. Note that the idealized $\delta$-function form of the scattering potential by defects results in the divergence of the real part of self-energy. To calculate the profiles we cut the number of terms in the sum over Landau index to a reasonable value determined by the laser frequency and the magnetic field strength.

Under the assumptions made, the absorption coefficient reduces to $$\begin{aligned}
\label{7}
&&~~~\alpha=-{2m_en(\omega_l)e^2|{\bf e}_l{\bf p}_{cv}|^2\omega_{ce}\over
\pi m_0^2c\hbar\omega_l}\sum_N\int dk_z\nonumber\\
&&{\rm
Im}\left[\hbar\omega_l-\hbar\omega_g-{\hbar^2k_z^2\over 2m_e}-\left(N
+{1\over 2}\right)\hbar\omega_{ce}\right.\nonumber\\
&+&i\Lambda\left.
\sum_{N^{\prime}}{(\hbar\omega_{ce})^{3/2}\over\sqrt{
\hbar\omega_l-\hbar\omega_g-(N^{\prime}+1/2)\hbar
\omega_{ce}}}\right]^{-1}~.\end{aligned}$$ Equation (\[7\]) can be easily integrated analytically to yield: $$\begin{aligned}
\label{4}
&&\alpha ={2\sqrt{2}n(\omega_l)e^2m_e^{3/2}
|{\bf e}_l{\bf p}_{cv}|^2\omega_{ce}\over
m_0^2c\hbar^2\omega_l}\nonumber\\
&&\times{\rm Re}\sum_N\left[
\hbar\omega_l-\hbar\omega_g-\left(N+{1\over 2}\right)
\hbar\omega_{ce}\right.\nonumber\\
&&\left.+i\Lambda\sum_{N^{\prime}}{(\hbar\omega_{ce})^{3/2}
\over\sqrt{\hbar\omega_l
-\hbar\omega_g-(N^{\prime}+1/2)\hbar\omega_{ce}}}\right]^{-1/2}. \end{aligned}$$ The results of calculation according to Eq. (\[4\]) and for coupling $\Lambda=0.2$ are shown in Fig. \[2f\]. We want to emphasize that, due to the resonant character of the renormalization (see Eq. (\[3\])), the minimum point on the higher energy side of each resonance [*corresponds to the singularity in the bare density of states for Landau level $N$*]{}. However, every individual resonance in Eq. (\[4\]) is composed of contributions from different Landau levels lying [*both above and below*]{} the bare level $N$. The resulting shape of the resonance is very different from the bare density of states and magneto-absorption profile acquires the Fano-type form.

Our model involves essentially only the electron density of states since the infinite effective mass of the hole have been assumed in calculations. A more elaborate theory is needed for quantitative analysis of the data.

The authors thank Prof. M. Cardona and Dr. T. Ruf for indicating the problem and stimulating discussions, and Prof. R. G. Ulbrich for useful comments. V. I. B. and S. T. P. thank the European Union, Ministerio de Educacion y Ciencia de España (DGICYT) and the Russian Fundamental Investigation Fund (93-02-2362) for financial support and the University of Valencia for its hospitality. This work has been partially supported by Grant PB93-0687 (DGICYT).

on leave from the P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow, Russia H. Lage, [*Magneto-optic of GaAs*]{}, Diplomarbeit, Institut für Physik der Universität Dortmund, 1989, unpublished. W. Becker, B. Gerlach, T. Hornung, and R. G. Ulbrich, in [*Proceedings of the 18th International Conference on the Physics of Semiconductors, Stokholm, Sweden, 1986*]{}, edited by O. Engström (Wold Scientific, Singapore, 1987), p. 1713. S. Glutsch, U. Siegner, M.-A. Mycek, and D. S. Chemla, Phys. Rev. [**B 50**]{}, 17009 (1994).
