Efficiency Potential of Photovoltaic Materials and Devices Unveiled by Detailed-Balance Analysis

A consistent mathematical approach is presented that connects the Shockley-Queisser (SQ) theory to the analysis of real-world devices. We demonstrate that the external photovoltaic quantum efficiency $Q_e^{\mathrm{PV}}$ of a solar cell results from a distribution of SQ-type band-gap energies and how this distribution is derived from experimental data. This leads us to the definition of a photovoltaic band-gap energy $E_g^{\mathrm{PV}}$ as a reference value for the analysis of the device performance. For a variety of solar-cell devices, we show that the combination of $Q_e^{\mathrm{PV}}$ and electroluminescence measurements allows for a detailed loss analysis that is fully compatible with the principle of detailed balance.

Figure 1

The proposed analysis of practical devices uses the measured external quantum efficiency $Q_e^{\mathrm{PV}}(E)$ [blue curve in (a)] and interprets its derivative [blue curve in (b)] as a distribution of SQ-type band gaps. From the distribution $P(E)$, the photovoltaic gap $E_g^{\mathrm{PV}}$ is determined. The steplike quantum efficiency [red curve in (a)] then defines the reference values $J_0^{\mathrm{SQ}}$ and $J_{\mathrm{SC}}^{\mathrm{SQ}}$ for SQ-type open-circuit voltage $V_{\mathrm{OC}}^{\mathrm{SQ}}$.

Figure 2

Photovoltaic band gaps of ten organic and inorganic, direct and indirect semiconductors as defined by Eq. (7) (blue half-filled circles), the maximum of the derivative of the $Q_e^{\mathrm{PV}}$ (red diamonds), a fit to the $Q_e^{\mathrm{PV}}$ as introduced by Helmers et al. [35] (green squares), and the Tauc plot [33] (yellow triangle).

Figure 3

External quantum efficiency $Q_e$ (black) and the respective distribution of SQ band gaps (blue) and EL emission (red) of six different technologies. The distribution of the SQ band gaps as well as the EL emission is broader for $c\text{−}\mathrm{Si}$ and CIGS than for CdTe and MAPIC. The broadest distributions are found for $a\text{−}\mathrm{Si}:\mathrm{H}$ and $\mathrm{PTB}7:{\mathrm{PC}}_{71}\mathrm{BM}$ with a significant peak shift between SQ band-gap distribution and EL emission.

Figure 4

Open-circuit voltage losses $\mathrm{\Delta }{V}_{\mathrm{OC}}^{\mathrm{SC}}$ due to a nonideal short-circuit current density (green), $\mathrm{\Delta }{V}_{\mathrm{OC}}^{\mathrm{rad}}$ due to radiative recombination (red), and $\mathrm{\Delta }{V}_{\mathrm{OC}}^{\text{nonrad}}$ due to nonradiative recombination for a variety of different solar-cell technologies. The SQ band gap is indicated by a black line and the measured $V_{\mathrm{OC}}$ as a blue bar. *Voltage loss data of GaAs are taken from Ref. [22].

Figure 5

Determination of the radiative ideality factor of $a\text{−}\mathrm{Si}:\mathrm{H}$ by voltage-dependent electroluminescence.