Microscopic Perspective on Photovoltaic Reciprocity in Ultrathin Solar Cells

The photovoltaic reciprocity theory relates the electroluminescence spectrum of a solar cell under applied bias to the external photovoltaic quantum efficiency of the device as measured at short circuit conditions. Its derivation is based on detailed balance relations between local absorption and emission rates in optically isotropic media with nondegenerate quasiequilibrium carrier distributions. In many cases, the dependence of density and spatial variation of electronic and optical device states on the point of operation is modest and the reciprocity relation holds. In nanostructure-based photovoltaic devices exploiting confined modes, however, the underlying assumptions are no longer justifiable. In the case of ultrathin absorber solar cells, the modification of the electronic structure with applied bias is significant due to the large variation of the built-in field. Straightforward use of the external quantum efficiency as measured at short circuit conditions in the photovoltaic reciprocity theory thus fails to reproduce the electroluminescence spectrum at large forward bias voltage. This failure is demonstrated here by numerical simulation of both spectral quantities at normal incidence and emission for an ultrathin GaAs $p\text{−}i\text{−}n$ solar cell using an advanced quantum kinetic formalism based on nonequilibrium Green’s functions of coupled photons and charge carriers. While coinciding with the semiclassical relations under the conditions of their validity, the theory provides a consistent microscopic relationship between absorption, emission, and charge carrier transport in photovoltaic devices at arbitrary operating conditions and for any shape of optical and electronic density of states.

Figure 1

Band profile—conduction band edge ${\mathrm{E}}_C$ and valence band edge ${\mathrm{E}}_V$—and quasi-Fermi levels for electrons (${\mu }_n$) and holes (${\mu }_p$) in a 100 nm thin GaAs $p\text{−}i\text{−}n$—diode at applied bias voltage of $V=1.1\mathrm{V}$. Both quantities are obtained from the solution of the full NEGF-Poisson problem.

Figure 2

Local absorption coefficient $\alpha$ and spectral volume emission rate in the center of the intrinsic region of a 100 nm thick GaAs $p\text{−}i\text{−}n$ solar cell. Dark lines represent the NEGF absorption coefficients at $V=0\mathrm{V}$ (dashed) and $V=1\mathrm{V}$ (solid), respectively. The corresponding generalized Planck (GP) spectra for quasi-Fermi level splitting of $\mathrm{\Delta }\mu /q=1\mathrm{V}$ are given by light solid and dotted lines, respectively. The isotropic NEGF result ${\overline{\mathcal{R}}}_{\mathrm{em}}^{\mathrm{NEGF}}$ for the emission at $V=1\mathrm{V}$ (open symbols) is in excellent agreement with the generalized Planck spectrum ${\overline{\mathcal{R}}}_{\mathrm{em}}^{\mathrm{GP}}$ obtained from the isotropic NEGF absorption coefficient at this bias voltage, but strongly broadened and redshifted as compared to the generalized Planck spectrum obtained from the absorption coefficient at short circuit conditions ($V=0\mathrm{V}$, dotted line).

Figure 3

Total absorptance $a_{\mathrm{NEGF}}$ (open squares) and spectral emission rate ${\mathfrak{R}}_{\mathrm{em}}^{\perp ,\text{Poynt}}$ (open circles) of light with propagation direction normal to the left slab surface, for a bias voltage of $V=1.1\mathrm{V}$ applied at the contacts, as given by Eqs. (18) and (23). The results coincide with the values obtained directly from the transfer matrix method and the generalized Kirchhoff law [Eq. (24)], if the actual absorptance of the biased system is used. For the absorptance at short circuit conditions ($V=0\mathrm{V}$, dashed line), the generalized Kirchhoff emission spectrum (dotted line) deviates significantly from the actual NEGF emission spectrum, exhibiting again a strong redshift and broadening attributed to the field-induced tailing of the joint density of states.

Figure 4

Same as Fig. 3, but now for a system with a gold back reflector attached at the right side. Additionally, the global emission rate ${\mathfrak{R}}_{\mathrm{em}}^{\perp ,\text{glob}}$ obtained from the $z$ integration of the local rate for normal emission is shown (dash-dotted line), revealing the effect of reabsorption on the spectral shape and intensity of the emission at the surface.