Thermodynamic efficiency limit of excitonic solar cells

Excitonic solar cells, comprised of materials such as organic semiconductors, inorganic colloidal quantum dots, and carbon nanotubes, are fundamentally different than crystalline, inorganic solar cells in that photogeneration of free charge occurs through intermediate, bound exciton states. Here, we show that the Second Law of Thermodynamics limits the maximum efficiency of excitonic solar cells below the maximum of 31% established by Shockley and Queisser [J. Appl. Phys. 32, 510 (1961)] for inorganic solar cells (whose exciton-binding energy is small). In the case of ideal heterojunction excitonic cells, the free energy for charge transfer at the interface, Δ$G$, places an additional constraint on the limiting efficiency due to a fundamental increase in the recombination rate, with typical −Δ$G$ in the range 0.3 to 0.5 eV decreasing the maximum efficiency to 27% and 22%, respectively.

Figure 1

(a) Schematic diagrams showing the photovoltaic process for a generalized excitonic solar cell in the radiative limit. The incident solar photon flux (${\mathrm{Ṅ}}_s$) is fully absorbed by the cell, forming bound exciton states with binding energy $E_B$. Radiative exciton recombination produces the luminescent photon flux (${\mathrm{Ṅ}}_r$) emitted from the cell. Alternatively, excitons may dissociate to form free carriers in a reaction that requires heat input (${\mathrm{Q̇}}_{\mathrm{diss}}$) to overcome the binding-energy barrier, detracting from the heat produced in the photovoltaic process, ${\mathrm{Q̇}}_{\mathrm{pv}}$. The current from the liberated electrons (e${}^{-}$) and holes (h${}^{+}$) is given by $I$. (b) The donor-acceptor heterojunction cell operates similarly, except that the free energy change, Δ$G_{\mathrm{CT}}$, at the interface induces charge transfer to form lower energy bound polaron pair states that have a binding energy, $E_{\mathrm{B},\mathrm{CT}}$. Radiative recombination only occurs from polaron pair states since all excitons reach the heterojunction in this ideal cell.

Figure 2

(a) Entropy generation calculated for an excitonic solar cell with optical energy gap $E_x$ $=$ 2 eV and binding energy $E_B$ $=$ 1.2 eV. The black ($V$-shaped) curve represents the heat generated in the photovoltaic process $({\mathrm{Q̇}}_{\mathrm{pv}}=T_a{\mathrm{Ṡ}}_{\mathrm{pv}})$, and the solid blue curve is the heat input required for dissociation $[(I/q)E_B]$ as calculated from the Shockley-Queisser theory, which neglects the thermodynamics of dissociation and leads to a region of negative total entropy generation for 1.4 < $V$ < 1.6 V, in violation of the Second Law of Thermodyamics. In contrast, the excitonic theory maintains thermodynamic consistency by increasing the exciton chemical potential in this region (upper portion of the plot), which leads to a reduced fill-factor in the current-voltage characteristics shown in (b), and hence to a lower maximum power conversion efficiency (${\eta }_{\mathrm{EX}}$) than that calculated from the SQ approach (${\eta }_{\mathrm{SQ}}$).

Figure 3

Maximum efficiencies predicted for excitonic solar cells as a function of optical energy gap for several different binding energies ($E_B$) under 1 sun, assuming the sun is a black body with surface temperature $T_s$ $=$ 5778 K. The excitonic limit falls below the SQ limit for $E_B$ > 1 eV.

Figure 4

Maximum efficiency predicted for an ideal donor-acceptor solar cell as a function of exciton optical energy gap and several values of the free energy for charge transfer at the heterojunction, −Δ$G_{\mathit{CT}}$. Increased recombination from the lower energy bound-pair states leads to reduced efficiency due to the decrease in open-circuit voltage, as shown in the inset for a cell with $E_x$ $=$ 1.8 eV.