Device Modelling of Organic Bulk Heterojunction Solar Cells

Abstract

We review the methods used to simulate the optoelectronic response of organic solar cells and focus on the application of one-dimensional drift-diffusion simulations. We discuss how the important physical processes are treated and review some of the experiments necessary to determine the input parameters for device simulations. To illustrate the usefulness of drift-diffusion simulations, we discuss several case studies, addressing the influence of charged defects on transport in bipolar and unipolar devices, the influence of defects on recombination, device performance and ideality factors. To illustrate frequency domain simulations, we show how to determine the validity range of Mott–Schottky plots for thin devices. Finally, we discuss an example where optical simulations are used to calculate the parasitic absorption in contact layers.

Appendices

Appendix 1: Transfer Matrix Formalism

There have been plenty of descriptions in the literature of the transfer matrix formalism for computing the position dependent generation rate in a multilayer stack with flat interfaces [43–45, 115, 231]. Thus, we will not repeat the whole derivation, but instead give a short summary of the underlying idea. The situation most relevant for organic solar cells is that of normal incidence of light on a glass substrate with a thickness much larger than the wavelength of light followed by several layers with thicknesses comparable or smaller than the wavelength of light. Thus, the glass substrate is first treated incoherently with Lambert–Beer and then the following multi-layer stack is treated with the transfer matrix formalism. Figure 9 shows the general layout of the problem. Light is incident from the left, perpendicular to the surface of the solar cell. To calculate the left and right going electric fields in any layer as a function of the electric fields at the interface between substrate and first layer, we need to define matrices for each interface and each layer. The interface between layers j and k is represented by one interface matrix [43, 44]

Fig. 9

Definition of the electric fields in a multilayer stack consisting of m layers. The matrix transfer formalism uses a matrix for every interface and every layer to express the in- and outgoing electric field on the left of the stack to the in- and outgoing electric field on the right of the stack. With this formalism we can calculate the electric field everywhere in the device and finally the position dependent generation rate

$$ {M}_{\mathrm{jk}}^{\mathrm{I}}=\frac{1}{t_{\mathrm{jk}}}\left(\begin{array}{cc}\hfill 1\hfill & \hfill {r}_{\mathrm{jk}}\hfill \\ {}\hfill {r}_{\mathrm{jk}}\hfill & \hfill 1\hfill \end{array}\right). $$

(24)

where

$$ {r}_{\mathrm{j}\mathrm{k}}=\frac{{\tilde{n}}_{\mathrm{j}}-{\tilde{n}}_{\mathrm{k}}}{{\tilde{n}}_{\mathrm{j}}+{\tilde{n}}_{\mathrm{k}}} $$

(25)

and

$$ {t}_{\mathrm{j}\mathrm{k}}=\frac{2{\tilde{n}}_{\mathrm{j}}}{{\tilde{n}}_{\mathrm{j}}+{\tilde{n}}_{\mathrm{k}}} $$

(26)

are the Fresnel reflection and transmission coefficients for the special case of normal incidence. The layers are characterized by their complex refractive index ${\tilde{n}}_{\mathrm{j}}={n}_{\mathrm{j}}+ i{k}_{\mathrm{j}}$ and their thickness d _j. The thickness becomes relevant for the calculation of the layer matrix [43, 44]

$$ {M}_{\mathrm{j}}^{\mathrm{L}}=\left(\begin{array}{cc}\hfill \exp \left(- i\frac{2\pi {\tilde{n}}_{\mathrm{j}}{d}_{\mathrm{j}}}{\lambda}\right)\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \exp \left( i\frac{2\pi {\tilde{n}}_{\mathrm{j}}{d}_{\mathrm{j}}}{\lambda}\right)\hfill \end{array}\right). $$

(27)

Using the interface and layer matrices, we can express the electric field everywhere in the device as a function of the incoming and outgoing electric field E ₀ at the interface between substrate and first layer via

$$ \left(\begin{array}{c}\hfill {E}_0^{+}\hfill \\ {}\hfill {E}_0^{-}\hfill \end{array}\right)={M}_{01}^{\mathrm{I}}{M}_1^{\mathrm{L}}{M}_{12}^{\mathrm{I}}\dots {M}_m^{\mathrm{L}}{M}_{m m+1}^{\mathrm{I}}\left(\begin{array}{c}\hfill {E}_{m+1}^{+}\hfill \\ {}\hfill {E}_{m+1}^{-}\hfill \end{array}\right). $$

(28)

Using this concept the electric field everywhere in the device can be calculated which we can use for the calculation of the generation rate at any position x.

The group of Prof. McGehee in Stanford offers a free transfer matrix modelling code on their homepage. The code is available in Matlab and Python and contains a database with the complex refractive indices of common materials of relevance for photovoltaics. More information is available on http://www.stanford.edu/group/mcgehee/transfermatrix/ (accessed 3/1/2013).

Appendix 2: Shockley–Read–Hall Recombination

Let us assume we are dealing with a trap level in the band gap as shown in Fig. 10. To understand how much recombination would be caused by this trap level we would need to know the occupation of that trap level. Initially we would not know which quasi-Fermi level (electrons or holes) would control the occupation of that trap level. This would also depend on how much interaction that trap level has with the conduction band and the valence band. It would depend on whether the trap level is a trap in the fullerene phase, in the polymer phase or at the interface of both. In addition, especially, if the trap can easily interact with both conduction and valence band, we will see that neither the electron nor hole quasi-Fermi level will be able to control the occupation probability of this trap. Instead, we would have to define a new occupation statistics for the trap that is different from that of both conduction and valence band. This new occupation statistics would not necessarily look like a Fermi–Dirac statistics so we might not be able to define a quasi-Fermi level for a trap at all.

Fig. 10

Definition of the four rates of capture and emission of electrons and holes by a single trap level. These four rate equations are the basis of Shockley–Read–Hall statistics, which defines the occupation probability and the recombination rate via this trap

To find the occupation statistics for a trap – the Shockley–Read–Hall statistics [232, 233] – we need to consider the four processes shown in Fig. 10. A single trap can capture and emit an electron and capture and emit a hole. If the same trap captures a hole and an electron, one recombination event happens. If a trap captures and emits an electron or a hole, the trap will have slowed down transport only. Table 3 summarizes the four rates that we need to consider. However, the four rates are not independent of each other in quasi-equilibrium. Because in equilibrium, detailed balance between inverse processes must be obeyed, the capture and emission processes must be connected. In addition, in thermal equilibrium the occupation function for all charge carriers (free or trapped, electrons or holes) must be the Fermi–Dirac function in thermal equilibrium, i.e.

Table 3 Capture and emission rates of single electron trap states

$$ {f}_{\mathrm{theq}}=\frac{1}{\left[1+ \exp \left(\frac{E-{E}_{\mathrm{F}}}{kT}\right)\right]}. $$

(29)

Thus, we can connect the capture coefficients β _n,p and the emission coefficients e _n,p using the conditions r ₁(f _theq) = r ₂(f _theq) and r ₃(f _theq) = r ₄(f _theq). This leads to the conditions

$$ \begin{array}{l}{e}_{\mathrm{n}}={\beta}_{\mathrm{n}}{N}_{\mathrm{c}} \exp \left(\frac{E_{\mathrm{t}}-{E}_{\mathrm{c}}}{kT}\right)\\ {}{e}_{\mathrm{p}}={\beta}_{\mathrm{p}}{N}_{\mathrm{v}} \exp \left(\frac{E_{\mathrm{v}}-{E}_{\mathrm{t}}}{kT}\right),\end{array} $$

(30)

where we used n = N _c exp(−(E _c − E _F)/kT) and p = N _v exp((E _v − E _F)/kT). Now, we can compute the steady-state but non-equilibrium solution for the occupation probability f. Steady state means that the occupation probability of the trap does not change over time. Therefore, the rates need to obey

$$ {r}_1-{r}_2={r}_3-{r}_4. $$

(31)

Using (30) and (31) and the rates defined in Table 3, we can calculate the occupation probability f _srh of our trap level as

$$ {f}_{\mathrm{srh}}=\frac{n{\beta}_{\mathrm{n}}+{e}_{\mathrm{p}}}{n{\beta}_{\mathrm{n}}+ p{\beta}_{\mathrm{p}}+{e}_{\mathrm{n}}+{e}_{\mathrm{p}}} $$

(32)

Note that this occupation probability has indeed no longer the same shape as a Fermi–Dirac distribution. Instead of one inflection point (the Fermi level in Fermi–Dirac statistics), there are two inflection points that are sometimes called quasi-Fermi levels for trapped charge [234, 235].

Either from r ₁ to r ₂ or from r ₃ to r ₄, we can now calculate the net recombination rate R via the trap or indeed any distribution N _t(E) of traps via

$$ R={\displaystyle {\int}_{E_{\mathrm{v}}}^{E_{\mathrm{c}}}{N}_{\mathrm{t}}(E){\beta}_{\mathrm{n}}{\beta}_{\mathrm{p}}}\frac{ n p-{n}_{\mathrm{i}}^2}{n{\beta}_{\mathrm{n}}+ p{\beta}_{\mathrm{p}}+{e}_{\mathrm{n}}+{e}_{\mathrm{p}}}\mathrm{d} E. $$

(33)

Thus, in the case of a single trap level with concentration N _t, the recombination rate would be

$$ R={N}_{\mathrm{t}}{\beta}_{\mathrm{n}}{\beta}_{\mathrm{p}}\frac{ n p-{n}_{\mathrm{i}}^2}{n{\beta}_{\mathrm{n}}+ p{\beta}_{\mathrm{p}}+{e}_{\mathrm{n}}+{e}_{\mathrm{p}}}. $$

(34)

which is the result often found in textbooks.

Appendix 3: Software

In the following, we will briefly discuss some programs the authors have used and found helpful for doing simulations of thin-film solar cells. The first two programs (ASA, SCAPS) are developed for inorganic solar cells but provide the basic functionality necessary for one-dimensional effective medium simulations. In all cases, detailed information is available on the respective homepages, so the main aim of this section is to make the reader aware of the existence of certain tools rather than to give a detailed assessment of the capabilities of the programs.

3.1 ASA (Zeman Group, Delft)

ASA (Advanced Semiconductor Analysis) has been developed by the group of Prof. Miro Zeman at the Technical University of Delft in the Netherlands as simulation software optimized for amorphous silicon solar cells [236]. Because of this focus on amorphous silicon, the software has several features that make it advantageous for one-dimensional drift-diffusion simulations of organic semiconductors as well. ASA can simulate both the spatially resolved generation rate based on a transfer matrix formalism and the electrical transport and recombination of charges in a multilayer system with a distribution of subgap defects. ASA allows the use of two exponential band tails and one amphoteric Gaussian defect. The inclusion of optical models means that electro-optical simulations are very simple. ASA works by reading in scripts containing the input parameters. It is therefore possible to change variables in these scripts using external programming languages or software like Matlab and therefore control the whole software via external scripts. Figures for this review have been made mostly using ASA and loops to change variables were written in Matlab. This flexibility allows users to use ASA in innovative ways that have nothing to do with the original application of amorphous Si solar cells. One option is to include field dependent photogeneration [75]. This can be done by running ASA once using field independent photogeneration, then reading in the optical generation and the electric field calculated by ASA and finally repeating the electrical calculation until the field no longer changes. Using this technique it is possible to use a commercially available drift-diffusion simulator and concentrate on adding extra features without needing to access the source code of ASA.

Because of the focus of thin-film silicon research on light trapping schemes to optimize light absorption, ASA also contains models to deal with light trapping and to allow the calculation of photogeneration rates with rough scattering surfaces. This may be advantageous for future work on light trapping in organic solar cells as well [237]. In addition, ASA is well suited to model tandem solar cells, also a typical application for thin-film silicon solar cells [230] and likely to be of increasing relevance for organic photovoltaics [15, 238, 239].

ASA is sold by the TU Delft. More information on ASA can be found under the following URL: http://www.ewi.tudelft.nl/en/the-faculty/departments/electrical-sustainable-energy/photovoltaic-materials-and-devices/asa-software/ (accessed 3/1/2013)

3.2 SCAPS (Burgelman Group, Ghent)

SCAPS (Solar Cell Capacitance Simulator) is software developed by the group of Prof. Marc Burgelman at the University of Ghent in Belgium. It was originally developed for use with compound semiconductor thin-film photovoltaics, i.e. for devices based on Cu(In,Ga)Se₂ or CdTe absorbers [228, 229, 240, 241]. However, the electrical models provide a basic functionality similar to that of ASA without having sophisticated optical models. Generation rates calculated with a transfer matrix formalism (which can be done with freeware as shown below) can be imported. The main advantage of SCAPS is that not only steady-state simulations but also frequency domain simulations can be performed. This option allows one to model the capacitance as a function of voltage, frequency and temperature and to use SCAPS for interpretation of impedance spectra [136] and Mott–Schottky plots [49]. In addition, SCAPS is comparatively easy to learn and intuitive to control. More information on SCAPS can be found under the following URL: http://users.elis.ugent.be/ELISgroups/solar/projects/scaps/