Total dielectric function approach to the electron Boltzmann equation for scattering from a two-dimensional coupled mode system

The nonequilibrium total dielectric function lends itself to a simple and general method for calculating the inelastic collision term in the electron Boltzmann equation for scattering from a coupled mode system. Useful applications include scattering from plasmon-polar phonon hybrid modes in modulation doped semiconductor structures. This paper presents numerical methods for including inelastic scattering at momentum-dependent hybrid phonon frequencies in the low-field Boltzmann equation for two-dimensional electrons coupled to bulk phonons. Results for mobility in GaAs show that the influence of mode coupling and dynamical screening on electron scattering from polar optical phonons is stronger for two-dimensional electrons than was previously found for the three dimensional case.


I. INTRODUCTION
The importance of hybridization of collective modes has been pointed out in the context of several semiconductor systems. Examples include hybrid optical modes in thin semiconductor layers [1], coupled intraband-interband excitations of quasi-one-dimensional electron systems [2], and, more commonly, plasmon-phonon coupled modes in doped polar semicon-ductors [3][4][5]. To the extent that electron scattering depends on the energies and interaction strengths of the system modes, a reliable theory of electron transport depends on treating mode-coupling effects accurately.
In the Born approximation, the interaction between a conduction electron and a coupled electron-phonon system can be represented as an effective electron-electron interaction screened by the nonequilibrium total dynamic dielectric function ǫ T (q, ω), which includes contributions from both electrons and phonons. Recent work [4,5] shows that ǫ T (q, ω) provides a systematic way to determine the collision term in the electron Boltzmann equation for scattering against dynamically screened coupled electron-phonon modes. The result is the sum of an electron-electron collision term and an electron-LO phonon collision term that includes plasmon-phonon mode coupling. Both interactions are dynamically screened by only the electronic part of the total dielectric function for the electron-phonon system. In the random-phase approximation (RPA), the electron-phonon part contains a phonon selfenergy that arises from the polarization of the electron gas. [6] The self-energy correction modifies the longitudinal-optical (LO)-phonon dispersion in doped polar semiconductors, producing hybrid normal modes with phonon strength in each. Numerical results [5] for bulk n-type GaAs show that mode-coupling and dynamical screening should significantly influence electron mobilities in modulation doped structures.
The present paper describes numerical methods for exactly solving the low-field Boltzmann equation for two-dimensional electrons coupled to bulk LO phonons, including dynamical screening and mode coupling for arbitrary electron degeneracy and spherical energy surfaces. In particular, the method for including phonon dispersion in the collision integrals is discussed. The phonon distribution is approximated by its equilibrium form, and the plasmon-pole approximation [7] is used in the LO phonon self-energy to determine the hybrid mode frequencies. Results for GaAs show that mode coupling and dynamical screening are more important for two dimensional electrons than for the three dimensional case.

II. INELASTIC SCATTERING FROM HYBRID PHONONS WITH DISPERSION
The collision integrals in the electron Boltzmann equation for doped polar semiconductors contain the momentum-dependent frequencies of the hybrid LO phonon modes [4]. Using the plasmon-pole approximation in the phonon self-energy, the differential scattering rate due to transfer of momentum q = k − p to the hybrid phonons is [4,5] where Here, ω LO and ω T O are the LO and transverse optical phonon frequencies, while ǫ ∞ and ǫ 0 are the high-frequency and static dielectric constants, respectively. In two dimensions, v q = 2πe 2 /qǫ ∞ . The frequencies ω p , ω + , and ω − are given byω where ω p = (2πne 2 q/m * ǫ ∞ ) 1/2 is the plasmon frequency of the two-dimensional electron gas with concentration n and effective mass m * . The weight factors (ω 2 give the phonon strength in each of the hybrid ω ± modes, so that the differential scattering rate W LO is the rate for scattering from only the phonon component of the hybrid modes. The screening function in (1) is the temperature-dependent RPA dielectric function, , for the two-dimensional electron gas. The iterative procedure used here for solving the low-field Boltzmann equation was described in reference [5]. One assumes a linear form for the nonequilibrium electron distribution, f (k) = f 0 k + x k g k , where f 0 k is the equilibrium (Fermi-Dirac) distribution, x k is the cosine of the angle between the electric field F and k, and g k is an unknown function that is linear in F . Neglecting all scattering mechanisms except electron-LO-phonon gives and x kp is the cosine of the angle between k and p.
The form of the phonon Green's function in equation (2) implies that the energy conservation relation between initial and final electron states, E k − E k−q = ±hω(q), is dependent on the momentum transfer q through the phonon dispersion. In three dimensions [5], it is convenient to choose q as an integration variable when calculating ν LO and τ −1 LO . In this case, one must determine the integration limits by finding the points where the phonon dispersion curve ω(q) intersects the absorption region q 2 + 2kq ≤ 2m * ω(q)/h ≤ q 2 − 2kq and the emission region 2m * ω(q) ≤ −q 2 + 2kq. For the case of two-dimensional electrons, it appears that q is not a preferred choice for integration variable, since the integrand then diverges at the integration limits. Instead one can choose θ kp , the angle between k and p.

III. RESULTS AND CONCLUSIONS
Mobilities are expected to be lower when mode coupling is included, because of increased low-energy electron-phonon scattering due to the ω − mode. Figures 1 and 2 show that this is in fact the case. As in three dimensions [5], mode coupling works to reduce electron mobility especially for low densities at 77K where the thermal occupation of the low-energy hybrid mode is exponentially larger than the high-energy hybrid mode or uncoupled phonon mode. The effect is more pronounced in two dimensions, consistent with previous conclusions concerning the effect of dimensionality on polaronic damping.