PERSPECTIVE pubs.acs.org/JPCL
Entropy of Charge Separation in Organic Photovoltaic Cells: The Benefit of Higher Dimensionality Brian A. Gregg* National Renewable Energy Laboratory, 1617 Cole Boulevard, Golden, Colorado 80401, United States ABSTRACT: The role of entropy in charge separation processes is discussed with respect to the dimensionality of the organic semiconductor. In 1-D materials, the change in entropy, ΔS, plays no role, but at higher dimensions, it leads to a substantial decrease in the Coulomb barrier for charge separation. The effects of ΔS are highest in equilibrium systems but decrease and become time-dependent in illuminated organic photovoltaic (OPV) cells. Higher-dimensional semiconductors have inherent advantages for charge separation, and this may be one reason that C60 and its derivatives, the only truly three-dimensional organic semiconductors yet known, play such an important role in OPV cells.
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unlight has always been the major source of energy on this planet, either employed directly via heat or photosynthesis or indirectly through the products of photosynthesis such as biomass, coal, oil, and natural gas. Wind energy also originates from solar heating of the atmosphere. Major research efforts are presently devoted to converting sunlight directly into electricity (photovoltaics) or fuels (artificial photosynthesis, photoelectrochemistry, etc.). A key step in these processes is the separation of photogenerated charge pairs into free electrons (for producing a photocurrent or for use as reducing agents) and free holes (for a photocurrent or oxidizing agents). A new look at the energetics of this separation process, particularly in organic photovoltaic (OPV) cells, is the focus of this Perspective.
In 1-D materials, the change in entropy plays no role, but at higher dimensions, it leads to a substantial decrease in the Coulomb barrier for charge separation. In a recent paper, Clarke and Durrant1 discussed qualitatively how entropy affects the charge separation process in OPV cells. This was apparently the first report of what seems to be an important and heretofor neglected aspect of the organic photoconversion process. They pointed out that the number of electronic states available to a photogenerated electron, say, escaping from its conjugate hole may be greater than one, and therefore, entropy effects should be taken into account when assessing the energetics of this process. In other words, the enthalpy, ΔH, and the free energy, ΔG, of charge separation must r 2011 American Chemical Society
be distinguished. Here, we expand upon Clarke and Durrant’s notion and provide a more quantitative analysis of entropy effects in organic semiconductors (OSCs). We also distinguish between equilibrium and nonequilibrium systems and discuss the importance of dimensionality, that is, the 1-, 2-, or 3-D nature of the conduction pathways in the semiconductor. Finally, we discuss how these factors affect the separation of photogenerated charges in OPV cells and how they favor higher-dimensional materials. Most inorganic semiconductors are inherently three-dimensional; therefore, these considerations arise only when considering the relatively few lower-dimensional materials such as MoS2 or WSe2. At equilibrium, ΔG = ΔH TΔS, where ΔS is the change in entropy and T is the absolute temperature. The entropy is given by ΔS = k ln W, where k is Boltzmann’s constant and W is the number of available states (degeneracy) at a given energy. State functions such as entropy cannot be directly employed to describe nonequilibrium processes such as the separation of photogenerated charge pairs. Nevertheless, if a suitable equilibrium analogue to the process can be found, it can be treated exactly and then its solution modified to take the time-dependent aspects into account. That is the approach followed here where we first analyze the efficiency of charge separation in an equilibrium n-type doping process in a low dielectric medium. This will then serve as a model for photoinduced charge separation in an OPV cell. We consider first equilibrium charge separation in a material of one, two, or three dimensions with a dielectric constant ε = 4 (Figure 1). The enthalpy of charge separation is given by Coulomb’s Law ΔH ¼ ðq2 =4πεε0 Þð1=rÞ
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Received: September 12, 2011 Accepted: November 16, 2011 Published: November 16, 2011 3013
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The Journal of Physical Chemistry Letters
Figure 1. Calculated equilibrium free energy of charge separation versus distance at room temperature of an electron escaping from an immobile hole at the origin in a dielectric constant of 4. The equations for the entropy in different dimensions are given; a is the lattice constant of the assumed cubic lattice and was set to 1 nm and Ea is the activation energy for charge separation calculated from an assumed ground state of 0.27 eV. Expansion into a circle or sphere was assumed in the 2-D and 3-D cases, respectively.
where q is the electronic charge, ε0 is the permittivity of free space, and r is the distance between the two charges. To calculate the entropy in a rudimentary model, we assume a cubic lattice with lattice constant a = 1 nm and calculate the number of available adjacent states at a given distance from the origin using the equations for a line, the circumference of a circle, and the surface area of a sphere in units of a. This number is taken as W. The equations are shown in Figure 1 along with the calculated values of the activation energy, Ea, for escape into an uncorrelated charge pair. A value for the ground-state energy of the contact ion pair was assumed to be 0.27 eV, corresponding to observed values.2 This value depends on the delocalization of the electron and hole wave functions. A different assumed value would shift Ea linearly without qualitatively changing the results. For charge separation in one-dimensional systems, entropy plays no role, and the free energy of charge separation is given simply by Coulomb’s Law (eq 1 and Figure 1, 1-D). For a chemical example of a 1-D semiconductor, consider the discotic liquid-crystal porphyrins3 in which the eight alkyl side chains surrounding the discotic core effectively prevent charge transfer in both dimensions perpendicular to the ππ stacking axis (Figure 2, ZnOOEP).4 Thus, these are effectively 1-D conductors in which the charge carrier mobility is many orders of magnitude higher along one axis than that along the other two. In this case, an electron attempting to escape from a positive charge at the origin has only a single isoenergetic state available to it at each distance. That is, the degeneracy of the states is unity, and ΔS = 0. Along this pathway, an electron must climb a barrier of Ea = 0.27 eV in order to escape the attraction of the hole. The equilibrium charge separation efficiency, known as the doping efficiency Φ, the number of free carriers produced per added dopant, can be approximated by Boltzmann statistics, Φ ≈ exp(Ea/kT). In this case, Φ ≈ 3 105 at room temperature, that is, only about 30 free electrons would be produced per million added dopants. In 1-D materials, therefore, doping probably does more harm than good because dopants quench
PERSPECTIVE
Figure 2. Examples of OSCs that in their solid form have different dimensionalities. Crystalline ZnOOEP conducts almost exclusively along the ππ stacking axis; poly(3-hexylthiophene), P3HT, can conduct along the ππ stacking axis as well as along the π-conjugated backbone; C60 can conduct equally well in all three dimensions.
excitons and decrease carrier mobility even when they do not efficiently produce free carriers.5
In illuminated materials, ΔS should always be less than that in the equilibrium case. The two-dimensional case (Figure 1, 2-D) is qualitatively different. Here, the electron has an increasing number of states available to it as it gets further from the bound positive charge. In the model, W increases as the circumference of a circle in units of a around the origin. A charge in the 2-D system would only have to go Ea = 0.13 eV uphill to escape the attraction of the positive charge. The doping efficiency, Φ ≈ 7 103, is still low but orders of magnitude higher than that in the 1-D case. In 3-D, where W increases as the surface area of a sphere in units of a, the electron would only have to surmount a barrier of Ea = 0.054 eV to escape, and the doping efficiency would increase to Φ ≈ 0.12. Many organic materials are not readily classified as being integral dimensional conductors. The 1-D and 3-D limits are easily recognized, but many materials are simply somewhere in between. Nevertheless, the trend is clear; equilibrium charge separation becomes more efficient in higher-dimensional materials because entropy decreases the barrier to charge separation. In a nonequilibrium system such as an illuminated OPV cell, the magnitude of the entropic effects on charge separation should be diminished from the equilibrium case (Figure 1) by two factors. One is purely geometrical; charges are photogenerated by exciton dissociation at interfaces. At a planar bilayer interface, a half-space compared to the calculations in Figure 1, ΔS will be decreased by a factor of ln 2, assuming that the hole is immobile. In a bulk heterojunction, the most efficient sort of OPV cells at the moment,1,6,7 it will be further decreased by the smaller domain size relative to a planar interface. The other factor is the limited lifetime of photogenerated carriers before they recombine. This limits the number of available states that are accessible during a lifetime. Thus, the effective ΔS will depend on the carrier mobility and the recombination rate. In a high-mobility, low recombination rate material, the 3014
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The Journal of Physical Chemistry Letters effect of ΔS may be approximately as shown in Figure 1 but scaled by the geometrical factors. However, the effective ΔS will diminish as the mobility decreases and the recombination rate increases because fewer states can be sampled before recombination. In short, the results in Figure 1 are still qualitatively correct for separation of photogenerated charges, but ΔS should always be less than that in the equilibrium case. If this model is correct in essence, higher-dimensional OSCs have an inherent advantage over lower-dimensional materials in the initial charge separation process that is so crucial to photoconversion. Most small-molecule OSCs crystallize in lowsymmetry lattices and thus are intrinsically low-dimensional conductors, with a dimensionality probably between 1-D and 2-D. Amorphous films of small molecules are effectively higherdimensional than crystals of the same material, but they suffer from lower exciton diffusion lengths and carrier mobilities. On the other hand, most π-conjugated polymers conduct both along the polymer backbone and along the ππ stacking axis and thus have quasi-two-dimensional character (Figure 2, P3HT). It is natural to note that the best electron acceptor for OPV cells is also one of the only truly 3-D conductors known among organic molecules. The C60 family of acceptors (Figure 2, C60), including its derivatives such as the ubiquitous PCBM, seem to be far superior to any other acceptors even after years of searching for alternatives.8 Perhaps their 3-D nature, and its associated entropy in solid state systems, is a key attribute.Higher-dimensional semiconductors have inherent advantages for charge separation, and this may be one reason that C60 and its derivatives, the only truly three-dimensional organic semiconductors yet known, play such an important role in OPV cells.
Higher-dimensional semiconductors have inherent advantages for charge separation, and this may be one reason that C60 and its derivatives, the only truly three-dimensional organic semiconductors yet known, play such an important role in OPV cells.
PERSPECTIVE
’ BIOGRAPHY Brian A. Gregg obtained a Ph.D. in physical chemistry from the University of Texas at Austin in 1988. After a 2 year postdoctoral appointment developing enzyme electrodes, he joined NREL. His research interests include the design and synthesis of materials for solar energy conversion, organic semiconductors, photoelectrochemistry, and photoelectrocatalysis. http://www.nrel.gov/basic_ sciences/technology_staff.cfm/tech=14/ID=15
’ ACKNOWLEDGMENT We thank Ross Larsen and Marshall Newton for helpful discussions. This work was funded by the U.S. Department of Energy, Office of Science, Basic Energy Science, Division of Chemical Sciences, Geosciences, and Biosciences, under Contract No. DE-AC36-08GO28308 to NREL. ’ REFERENCES (1) Clarke, T. M.; Durrant, J. R. Charge Photogeneration in Organic Solar Cells. Chem. Rev. 2010, 110, 6736–6767. (2) Gregg, B. A.; Chen, S.-G.; Cormier, R. A. Coulomb Forces and Doping in Organic Semiconductors. Chem. Mater. 2004, 16, 4586–4599. (3) Gregg, B. A.; Fox, M. A.; Bard, A. J. 2,3,7,8,12,13,17,18-Octakis(beta-hydroxyethyl)porphyrin (Octaethanolporphyrin) and Its Liquid Crystalline Derivatives: Synthesis and Characterization. J. Am. Chem. Soc. 1989, 111, 3024–3029. (4) Liu, C.-Y.; Pan, H.-L.; Fox, M. A.; Bard, A. J. High-Density Nanosecond Charge Trapping in Thin Films of the Photoconductor ZnODEP. Science 1993, 261, 897–899. (5) Hains, A. W.; Liang, Z.; Woodhouse, M. A.; Gregg, B. A. Molecular Semiconductors in Organic Photovoltaic Cells. Chem. Rev. 2010, 110, 6689–6735. (6) Chen, H.-Y.; Hou, J.; Zhang, S.; Liang, Y.; Yang, G.; Yang, Y.; Yu, L.; Wu, Y.; Li, G. Polymer Solar Cells with Enhanced Open-Circuit Voltage and Efficiency. Nat. Photonics 2009, 3, 649–653. (7) Kippelen, B.; Bredas, J.-L. Organic Photovoltaics. Energy Environ. Sci. 2009, 2, 251–261. (8) Brabec, C. J.; Gowrisanker, S.; Halls, J. J. M.; Laird, D.; Jia, S.; Williams, S. P. PolymerFullerene Bulk-Heterojunction Solar Cells. Adv. Mater. 2010, 22, 3839–3856.
In summary, we show that entropy effects in higher-dimensional organic semiconductors give them a major advantage in equilibrium doping efficiency over lower-dimensional systems. Likewise, the process of photoexcited charge separation is more facile in higher-dimensional materials, although, in this case, ΔS becomes time-dependent, and its magnitude decreases relative to the equilibrium case. These results suggest that synthesizing new three-dimensional organic semiconductors, besides the C60 family, could be a promising new research direction.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 3015
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