We developed a quantitative theory of the phonon-drag thermopower for the 2D electron gas in a quantum well with parabolic confinement potential. We consider a simple model for the quantum well, in which a two-dimensional electron gas is confined in the x-direction. The parabolic confining potential can be written as U_x=mω₀ ²x²/2, where m is the effective mass of a conduction electron, and ω₀ is the parameter of the parabolic potential. As the temperature gradient (and therefore the current) are directed across the confinement direction, i.e. along the y-direction in plane of a two-dimensional electron gas, it is sufficient to consider the relaxation time approximation and to use the Boltzmann equation. The scattering mechanisms of electrons explicitly considered in the present paper are the acoustic phonon scattering via deformation (da) and piezoelectric (pa) couplings, and the impurity scattering arising from ionized impurities in the quantum well; the scattering mechanisms of phonons are Herring’s, Simons’s and boundary scattering mechanisms. Thermopower consists of two components, the diffusion and the phonon-drag parts. We apply our theoretical results to the 2D degenerate electron gas in GaAs/Al_xGa_1-xAs quantum wells. Numerical calculations for the thermopower have been performed for sample 1 in [1], with the electron density n=1.78·10¹⁵m^-2, the mobility μ=22.7 m²/Vs, the mean free path of phonons L=3·10^–4 m, the width of the quantum well Lx=10^-8 m [1], the mass density ρ=5.3·10³ kg/m³, the longitudinal sound velocity s=5·103 m/s, the effective mass m=0.067m₀ (m₀ is the free electron mass), the acoustic deformation potential E₁=7.4 eV, and the piezoelectric constant e₁₄=0.16 C/m². Our analysis shows that the situations considered in [1] satisfied the conditions of the quantum limit and the strong degeneracy of electron gas. Estimation shows that the dominant mechanism of scattering of the electrons is the scattering by ionized impurities, and for the phonons the dominant mechanism is the surface scattering. We estimated parameter ω₀=7·10¹³ s^-1 of the parabolic potential using condition R~Lx/2, where R=(ħ/mω0)^1/2 is the “oscillator length”.Fig. 1. Variation of thermopower yy a as a function of temperature T: the diffusion component (1), the phonon-drag thermopower (2: due to pa-coupling, 3: due to da-coupling, 4: the total phonon-drag thermopower), and the total thermopower (5). Open and solid points are the experimental data [1].Our theoretical results for the variation of thermopower a_yy as a function of temperature T are shown in Fig. 1. The results are in a good agreement with experimental data in the range 1-10 K. In this temperature interval the diffusion thermopower is by 3 to 8 times smaller than the phonon-drag thermopower. Thus, for example, at T=5K, a _ph = 215mV/K and 36 _dif a = mV/K . At temperatures below 1.5K the main contribution to the thermopower comes from the piezoelectric interaction, and above 1.5K from the deformation interaction with acoustic phonons. The calculations indicate the importance of screening, so the value of the thermoelectric power in the range 1-10K, without taking into account the screening, is 1.5-2 times higher.