Influence of hot carriers on parametrically interacting polaron mode in semiconductors

In the present paper effect of hot carriers due to parametrically interacting electron- longitudinal optical phonons in polar semiconductor is analytically investigated. Presence of hot carriers is found to significantly modify the threshold and amplification characteristics in the presence of external magnetic fields. Expressions for threshold pump field required for the onset of polaron induced parametric interaction and amplification characteristics are explicitly derived. It is found that at moderate magnetic field and high carrier concentrations hot carriers affect threshold and amplification characteristics strongly. Resonance between polaron frequency and plasma frequency is found to be favourable for the minimum threshold field. Presence of hot carriers and magnetic field along with mass modulation effects are found to be additive and resulted into increment in the parametric gain. Typical dependence of parametric gain on magnetic field and carrier concentration could be utilized for the construction of optical switches.


Introduction
Parametric interactions offer an efficient way to heat semiconductor plasma, since the incoming photons participating in the process are completely converted into plasmons. The parametric amplification of collective-waves in polar semiconductors such as phonons and polarons at the expense of the pump wave has already aroused much attention for modern optoelectronic device applications [1] e.g. for the generation of tunable laser with high conversion efficiency, photovoltaic effect, far-infrared diagnostic systems and optical parametric amplifiers (OPAs).
Large changes at room temperature and below have been shown to be due to impact ionization of the lattice by hot carriers [2,3] in polar semiconductors. History of inclusion of carrier heating effects in semiconductor plasmas dates back to 1973 [4]. Recently authors analytically investigated polaron induced parametric interactions in magnetized semiconductors [5] (here after referred as paper I). Nonlinear electromagnetic effects connected with free carriers in homogeneous semiconductors result from heating up of the carriers and peculiarities in the dynamics of the carriers in the bands or the dependence of their effective mass * m on the field. However which of the mechanisms play the dominant role in the processes of Infrared parametric generation is still a question of important. A more fundamental motivation for such studies is that they help in elucidating the interaction of the carriers with the scattering mechanism: the lattice, impurities, and other defects. Therefore in the present paper emphasis will be on the heating of carriers due to Fröhlich interaction and the underlying theory, significant results will be cited. Several factors, including acceptor concentration, carrier density, temperature, magnetic field, mass modulation, all affect hot carrier scattering, with different mechanisms dominating in various regimes. Here we examine how these external parameters can be used to extract analytically the carrier heating dependence of this interaction in polar semiconductors.

Theoretical Formulation
In this section, the parametric amplification of polaron mode arising due to three-wave interaction induced by   2  in polar semiconductor duly irradiated by a relatively high-power laser with photon energy much below the forbidden energy gap of the crystal has been studied. Well-known hydrodynamic model of homogeneous semiconductor plasma, satisfying the condition kl <<1( k and l being the wave number and electron mean free path, respectively) has been used. We assume that a spatially uniform   Since we are interested in three wave interactions, the phase matching conditions are

Nonlinearity due to mass modulation
The effective mass is taken to be energy dependent based on Kane's model [6] as where,

Nonlinerarity due to electron collision frequency
When the mobility is high, heating of the carriers occurs in the fields as low as a few volts per centimeter. Particularly in high mobility semiconductors like InSb, it is an established fact that due to high intensity pump (which is one of the pre-requisite conditions for the onset of parametric instability) heating of carrier becomes inevitable. This heating of electrons modifies the electron collision frequency (ECF) through the relation [8] where, e T is the effective temperature of electrons, 0 T is the lattice temperature, and 0 e  is the ECF when 0 T T e  . As a result, the momentum-transfer collision frequency (MTCF), mobility of the carriers and conductivity of the medium become function of the pump amplitude and hence produce refinement effects. The momentum transfer of carriers is assumed to be due to acoustical phonon scattering and the energy transfer due to polar optical phonon (POP) scattering mechanisms in an ntype III-V semiconductor. In steady state, the power absorbed per electron from the pump is just equal to the power lost per electron in the POP scattering. Hence for moderate heating, carrier temperature is-

Threshold and amplification characteristics
Using basic equations from paper I [5] second-order nonlinear susceptibility The parametric amplification can be achieved at excitation intensity above this threshold value under favourable conditions and nonlinear absorption coefficient is given by   The nonlinear parametric gain of the signal as well as the idler waves can be possible only if para  obtained from equation (6) is negative.

Results and discussions
Numerical estimations were carried out by using the relevant parameters chosen for InSb medium are listed in [9]. Figure 1 depicts the variation of threshold pump electric field   para oth E (with or without CH effects) with carrier density. The threshold pump electric field achieves its minimum value carrier density when dependent modified plasma frequency becomes smaller and smaller and finally resulting into   in comparison with that in the absence of magnetic field. It is found that presence of magnetic field is favourable for polaron induced parametric gain in heavily doped semiconductor. It is also noticed that the nonlinearity due to mass modulation and electron collision frequency modifies threshold and amplification characteristics strongly. CH effects are found to lower the threshold pump field and effectively raise the parametric gain.