Analytical model for the open-circuit voltage and its associated resistance in organic planar heterojunction solar cells

We derive an analytical formula for the open-circuit voltage $\left(V_{\text{oc}}\right)$ of organic planar heterojunction solar cells under standard operating conditions. We find that the type of free carrier recombination at the interface between the donor and acceptor materials controls the slope of $V_{\text{oc}}$ vs incident light intensity. By using the same derivation, an equation for the resistance around $V_{\text{oc}}$ is obtained. From this, we investigate two parameters in more detail and compare them to experiments. The first is the work function of the cathode metal. We show that, within our model, $V_{\text{oc}}$ does not depend on this work function, while the cell resistance around $V_{\text{oc}}$ is strongly dependent on it. Second, we find that the asymptotic resistance around $V_{\text{oc}}$ is a third-order power function of the thickness of the organic layers (acceptor or donor). The model provides insights to achieve low-resistivity high open-circuit voltage organic solar cells.

Figure 1

Schematic energy diagram of a planar heterojunction solar cell. $\left|{\text{HOMO}}_D\right|-\left|{\text{LUMO}}_A\right|$ is the energy difference between the HOMO of the donor and the LUMO of the acceptor, ${\text{BB}}_A$ and ${\text{BB}}_D$ are the net band bendings in the electrostatic potential, and $\Delta {\varphi }_A$ and $\Delta {\varphi }_D$ are the energy offsets at the contact. The values of ${\text{BB}}_A$ and ${\text{BB}}_D$ are negative if a carrier moving from the interface to the contact loses energy and are positive if the carrier gains energy. As the acceptor layer is an electron region, and the donor layer a hole region, ${\text{BB}}_A$ and ${\text{BB}}_D$ ($\Delta {\varphi }_A$ and $\Delta {\varphi }_D$), show an opposite bending for the same sign. At the bottom, the coordinate system is drawn, with $d_D$ and $d_A$ as the thickness of the donor and acceptor layers, respectively.

Figure 2

Modeled carrier density versus position, where $x=0\text{nm}$ is equal to the contact and $x=50\text{nm}$ is the location of the heterojunction. The different graphs correspond to different carrier densities at the contact [(a) ${10}^{20}$, (b) ${10}^{22}$, and (c) ${10}^{24}\text{carriers}/{\text{m}}^3$], while the different curves on each graph correspond to different carrier densities at the interface ranging from ${10}^{19}$ to ${10}^{26}\text{carriers}/{\text{m}}^3$.

Figure 3

Effect of field dependent barrier lowering $\left({\Delta }_{\text{low}}\right)$. Simulations are carried out by using an injection barrier $\Delta {\varphi }_D=0.42\text{eV}$, giving a contact carrier density $\left(p_c\right)$ of $1\times 20{\text{m}}^{-3}$ without barrier lowering. The lower graph plots the field at the contact $\left(F_c\right)$ needed to reach a certain carrier density at the interface. When a positive field is present, barrier lowering occurs and $p_c$ increases (upper panel). Note that it only increases by a factor of 2 for an increase in $p_i$ of 6 orders of magnitude.

Figure 4

Schematic energy diagrams (upper graphs) and corresponding carrier profile in the organic layers (lower graphs) when using (a) low and (b) high injection barriers at the contacts, $\Delta {\varphi }_D$ and $\Delta {\varphi }_A$. The same incident light intensity is used for the two situations, creating the same density of carriers at the donor-acceptor interface $\left(y_i\right)$. The values for the band bendings, ${\text{BB}}_A$ and ${\text{BB}}_D$, are negative for (a) and positive for (b). The resulting open-circuit voltage $V_{\text{oc}}$ is the same.

Figure 5

Calculated resistance at $V_{\text{oc}}$ as a function of the carrier density at the interface $p_i$ by using different injection barriers $\Delta {\varphi }_D$, and this for a 50 nm thick layer with a mobility of $1\times {10}^{-7}{\text{m}}^2/\text{V}\text{s}$.

Figure 6

Extracted light dependent parameters from planar heterojunction solar cells, consisting of 50 nm CuPc, 50 nm ${\text{PTCDI-C}}_{13}{\text{H}}_{27}$, and a metal contact. The top contact is Yb or Al.

Figure 7

The measured resistance at $V_{\text{oc}}$ versus incident light power $\left(P_0\right)$ for planar solar cells with Yb and Al as contact. As a guide for the eye, the dotted and the dashed lines represent the fitted dark series resistance $R_S$ for cells with Al and Yb respectively.

Figure 8

Upper panel: calculated carrier profile inside layers with different thickness, from 25 nm to 500 nm, and with a mobility of $1\times {10}^{-7}{\text{m}}^2/\text{V}\text{s}$. The carrier density at the contact and the interface is kept constant at $1\times {10}^{24}\text{carriers}/{\text{m}}^3$. Lower panel: the resulting resistance at $V_{\text{oc}}$ for the different thicknesses.

Figure 9

The dark $I\text{−}V$ curves for the solar cells with different ${\text{PTCDI-C}}_{13}{\text{H}}_{27}$ thicknesses and the arrow points to thicker ${\text{PTCDI-C}}_{13}{\text{H}}_{27}$ layers. Inset: series resistance dependence on thickness.