Relationship between energetic disorder and open-circuit voltage in bulk heterojunction organic solar cells

We simulate organic bulk heterojunction solar cells. The effects of energetic disorder are incorporated through a Gaussian or exponential model of density of states. Analytical models of open-circuit voltage ($V_{\mathrm{OC}}$) are derived from the splitting of quasi-Fermi potentials. Their predictions are backed up by more complex numerical device simulations including effects such as carrier-density–dependent charge-carrier mobilities. It is predicted that the V${}_{\mathrm{OC}}$ depends on: (1) the donor-acceptor energy gap; (2) charge-carrier recombination rates; (3) illumination intensity; (4) the contact work functions (if not in the pinning regime); and (5) the amount of energetic disorder. A large degree of energetic disorder, or a high density of traps, is found to cause significant reductions in $V_{\mathrm{OC}}$. This can explain why $V_{\mathrm{OC}}$ is often less than expected in real devices. Energetic disorder also explains the nonideal temperature and intensity dependence of $V_{\mathrm{OC}}$ and the superbimolecular recombination rates observed in many real bulk heterojunction solar cells.

Figure 1

Energy-level diagram showing an OPV device at open-circuit voltage. $E_{\mathrm{vac}}$ is the vacuum level, and all other symbols are as defined in the text. All energy intervals are defined as positive as shown in this diagram.

Figure 2

Schematic density of states (DOS) for (a) classical crystalline inorganic semiconductor and (b) disordered organic semiconductor showing tail states extending into the band gap. (c) and (d) A Gaussian or exponential DOS respectively (red lines) used to approximate the tail of the valence states. The widths of the model DOS are σ and $E_{\mathrm{t}}$, and the areas enclosed are $N_{\mathrm{h}}$ and $N_{\mathrm{t},\mathit{h}}$, respectively.

Figure 3

Schematic diagram illustrating different recombination modes in mobility-edge model. (t-t) trapped-carrier to trapped-carrier recombination; (f-t) free-carrier to trapped-carrier recombination; and (f-f) free-carrier to free-carrier recombination.

Figure 4

Simulations of BHJ devices with Gaussian disorder. (a) and (b) Simulated current-voltage curves for varying degrees of disorder (σ = 0, 25, 50, 75, 100, and 125 meV) at low ($G$ = 10${}^{20}$ cm${}^{-3}$s${}^{-1}$) and high ($G$ = 10${}^{22}$ cm${}^{-3}$s${}^{-1}$) intensities, respectively. (Dashed red lines are dark currents.) (c) and (d) Same as (a) and (b) except without field-dependent generation and recombination rates. (e) $V_{\mathrm{OC}}$ vs. disorder at low intensity. Filled symbols: using field-dependent continuity equation [Eq. (16)]. Open symbols: using field-independent continuity equation. Solid line: Eq. (7a) with effective recombination coefficient $\mathit{\gamma }={\mathit{\gamma }}_L/\mathit{\beta }\left({\mathbf{\mu }}_{\mathbf{h}}+{\mathbf{\mu }}_{\mathbf{e}}\right)$. Dashed line: Eq. (7a) with $\mathit{\gamma }={\mathbf{\gamma }}_{\mathbf{L}}$. (f) $V_{\mathrm{OC}}$ vs. intensity with varying disorder and field-dependent generation and recombination. Symbols: numerical simulation; solid lines: Eq. (7a).

Figure 5

Simulations of BHJ devices with exponential density of trap states. (a) Simulated current-voltage curves with $E_{\mathrm{t}}$ = 100 meV and varying trap densities ($N_{\mathrm{t}}$ = 10${}^{16}$, 10${}^{17}$, 10${}^{18}$, and 10${}^{19}$ cm${}^{-3}$) at low ($G$ = 10${}^{20}$ cm${}^{-3}$s${}^{-1}$) intensity. (b) $V_{\mathrm{OC}}$ vs. trap density for the same conditions. Filled symbols: including trap-to-trap recombination (t-t); open symbols: excluding trapped-to-trapped recombination, but including free-to-trapped-carrier recombination (f-t). (c) $V_{\mathrm{OC}}$ vs. intensity excluding trapped-to-trapped recombination for different trap distributions. (d) $V_{\mathrm{OC}}$ vs. intensity including trapped-to-trapped recombination for different trap distributions. Lines: analytical solutions from text.

Figure 6

Simulated energy diagrams in open-circuit conditions with $G$ = 10${}^{20}$ cm${}^{-3}$s${}^{-1}$ and (a) no disorder. (b) σ = 150 meV Gaussian disorder.

Figure 7

Simulations of BHJ devices with Gaussian disorder and varying electron injection barriers. (a) Current-voltage characteristics for devices with σ = 100 meV and $G$ = 10${}^{20}$ cm${}^{-3}$s${}^{-1}$. (b) $V_{\mathrm{OC}}$ vs. electron injection barrier under the same conditions with and without disorder.