Excitonic modes in a Bose-condensed electron-hole gas in the pairing approximation.

Abstract

In studying the low-temperature state of an electron-hole gas in an optically pumped semiconductor there has been increasing realization that the pair approximation (similar to the Gor'kov approximation for BCS superfluids) can deal with both the low-density phase of excitons in a Bose-condensed state as well as the high-density excitonic insulator phase. Using functional differentiation techniques, we use this pair approximation to give a systematic derivation of the two-particle Green's function and the associated collective modes. Our equations of motion give what is often called the generalized random-phase approximation (GRPA). The collective modes are shown to correspond to the solution of the standard electron-hole ladder and bubble diagram sum, but with 2\ifmmode\times\else\texttimes\fi{}2 matrix propagators. Our model calculations are for a simple two-band direct-band-gap semiconductor with parabolic bands and a positive band gap (the semiconductor limit as opposed to the semimetal limit) and thus might be appropriate for optically pumped ${\mathrm{Cu}}_{2}$O. In the appropriate limits, our formalism leads to the phonon modes discussed (in the late 1960s) by Maksimov and Kozlov in the excitonic insulator and by Keldysh and Kozlov in the Bose-condensed state of excitons. In the low-density limit, the excitation of these collective modes is crucial in understanding how the Bose condensate is depleted at higher temperatures. Besides the excitonic modes, our equations of motion allow a systematic study of the complete spectrum of collective modes in the GRPA, including plasmons. Throughout, we emphasize the formal similarity with the theory of collective modes and excited Cooper-pair states in BCS superfluids.