The term $H^e\gamma =\u2212(e/m0)A^(r,t)\xb7p^$ is used to describe the interaction of charge carriers with photons that are required for radiative generation and recombination processes via corresponding nonlocal self-energies, e.g., for electrons in the conduction band, Display Formula
$\Sigma cc<(gen)(k\u2225,z,z\u2032,E)=(em0)2\u2211\mu pcv\mu (z)pcv\mu *(z\u2032)\u222bdE\gamma Gvv<(k\u2225,z,z\u2032,E\u2212E\gamma )\u2062A\u22121\u2211q\u2225A\mu (q\u2225,z,E\gamma )A\mu *(q\u2225,z\u2032,E\gamma ),$(7)
Display Formula$\Sigma cc>(rec)(k\u2225,z,z\u2032,E)=(em0)2\u2211\mu ,\nu pcv\mu (z)pcv\nu *(z\u2032)\u222bdE\gamma 2\pi \u210fGvv>(k\u2225,z,z\u2032,E\u2212E\gamma )\u2062A\u22121\u2211q\u2225i\u210f\mu 0D\mu \nu >(q\u2225,z,z\u2032,E\gamma ),$(8)
Display Formula$\u2248n036\pi 2\u210fc03\epsilon 0(em0)2\u2211\mu pcv\mu (z)pcv\mu *(z\u2032)\u222bdE\gamma E\gamma Gvv>(k\u2225,z,z\u2032,E\u2212E\gamma ),$(9)
where the local approximation of the momentum-averaged GF of free field photon modes, Display Formula$A\u22121\u2211q\u2225D\mu \nu ,0>(q\u2225,z,z\u2032,E\gamma )\u2248\u2212in03E\gamma 3\pi \u210fc0\delta \mu \nu ,(z\u2248z\u2032),$(10)
was used in the last line. This corresponds to emission into an optically homogeneous medium. The classical electromagnetic vector potential $A$ in the multilayer device is obtained from the conventional transfer matrix method (TMM). The local extinction coefficient used in the TMM is related to the local absorption coefficient as obtained from the microscopic interband polarization in terms of the charge carrier GFs.^{8} At this point, it should be noted that for detailed balance to hold, both self-energies should be expressed in terms of the same photon GF, which can be obtained from an additional set of Dyson and Keldysh equations similar to Eqs. (3) and (4).^{9}