NANO LETTERS
Design of Nanostructured Heterojunction Polymer Photovoltaic Devices
2003 Vol. 3, No. 12 1729-1733
Balaji Kannan,† Kenneth Castelino,† and Arun Majumdar*,‡ Department of Mechanical Engineering, UniVersity of California, Berkeley, California 94720, and Materials Science DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720 Received September 22, 2003; Revised Manuscript Received October 16, 2003
ABSTRACT Solar cells made from blends of conjugated polymers and nanostructured inorganic materials are an important class of organic photovoltaic devices. However, there has been no systematic theoretical analysis of their operation and performance. In this paper, we develop a theoretical model to analyze the performance of two classes of heterojunction solar cells composed of ordered nanostructures. Based on the simulations, we conclude that in order to obtain reasonable efficiencies, the size and spacing of the nanostructures must be on the order of the exciton diffusion length scale. Possible quantum and other confinement effects are qualitatively discussed.
Organic photovoltaic devices based on conjugated polymers have generated a lot of attention in the recent past because they offer the possibility of low-cost large-area devices. Donor-acceptor (D/A) bulk-heterojunction cells, which consist of a blend of conjugated polymer and a second inorganic phase, have much better efficiency than cells based on pure polymers. Modified fullerenes1 and inorganic semiconductors2 have been used as the inorganic phase in such cells, which has led to efficiencies of up to 2.5%.3 Such devices are generally limited by inefficient charge transport due to the discontinuous inorganic phase.4 Continuous ordered inorganic nanostructures such as nanowires and nanoporous semiconducting films can potentially address this issue. However, there has been no systematic analysis of the effect of nanostructuring on device performance. In particular, two questions remain with regards to the overall energy conversion efficiency, namely: (i) the effect of nanostructure size and spacing and (ii) the influence of carrier transport. In this paper, we develop a theoretical model to study both these effects, which forms the basis for rational design of such devices. Figure 1 shows a schematic of the band gap arrangement in a typical bulk-heterojunction cell. There are three main steps in device operation. (1) Absorption of photons generates excitons in the polymer. (2) Excitons reach a donor-acceptor interface and dissociate if the band gap of the two materials are staggered, leading to charge separation. While electrons * E-mail:
[email protected]. † University of California. ‡ Lawrence Berkeley National Laboratory. 10.1021/nl034810v CCC: $25.00 Published on Web 11/21/2003
© 2003 American Chemical Society
Figure 1. Staggered band gap arrangement for organic-inorganic heterojunction photovoltaic cell.
are captured by the inorganic phase, which has the higher electron affinity, holes are retained in the polymer. (3) The separated charges are then transported to their respective electrodes, resulting in an external photocurrent. Because the two materials in a bulk heterojunction cell have a very high interface area, the probability of excitons reaching an interface is high, leading to high charge-separation efficiency. However, charge transport through the inorganic phase to the electrodes can be inefficient if the phase is discontinuous, which then requires hopping transport to occur. Using a continuous nanostructured material would improve the charge transport efficiency and, hence, the overall efficiency, since the electrons can now travel through delocalized energy bands. We examine two different device configurations using ordered continuous nanostructures and simulate their per-
Figure 2. (a) Top view of ordered nanostructure cell. (b) Unit cell used for analysis. Table 1. Material Properties of MEH-PPV Used in Simulations parameter
value
diffusion coefficient (D) exciton diffusion length (1/β) zero-field hole mobility8 number of chargeable sites in polymer8 resistivity Richardson constant9 dielectric constant
8 × 10-3 cm2/s 20 nm 10-7 cm2/V‚s 1022/cm3 104 Ω m 0.9903 S. I. 3.0
formance as a function of nanostructure size in order to identify the effect of size on device efficiency. A top view of the general device configuration is shown in Figure 2a. The wires are assumed to be in a hexagonal array. Depending on the type of matrix used, two specific cases considered are: (a) a nanoporous inorganic semiconductor matrix into which a semiconducting polymer is infiltrated to form polymer nanowires,5 and (b) a forest of semiconductor nanowires immersed in a polymer matrix.6 For both types of devices, the unit cell and characteristic dimensions shown in Figure 2b were used as the basis for all calculations. As a model system, all simulations were done using the polymer MEH-PPV (poly(2-methoxy-5-(2′-ethylhexoxy)1,4-phenylvinylene) as the polymer and CdSe as the inorganic phase with an ITO anode and Al cathode. It must be noted, however, that the methodology developed here is applicable to any suitable polymer-inorganic combination. The material properties of MEH-PPV and the main parameters used in the simulations are listed in Table 1 and were obtained from Malliaras et al.7 and Breeze et al.8 The first step is to compute the photocurrent density and external quantum efficiency of the cell under monochromatic illumination using an exciton transport model. Two limiting models of exciton transport to the donor-acceptor interface, namely, the diffusion and the ballistic models, were considered. The diffusion model is valid when the characteristic length scale of the unit cell is much larger than the exciton diffusion length or mean free path (MFP). The ballistic model is useful at length scales smaller than the exciton MFP. Intermediate length scales may be treated as an interpolation between these two limiting cases. The following assumptions are made in this model. (i) Charge separation of excitons takes place only at the donor-acceptor interface. Electronhole pair generation in the bulk of the polymer is neglected. (ii) Charges that are separated at the interface do not 1730
recombine because electron transport through the inorganic phase is through the delocalized electronic band structure. This assumption is supported by the recent model of Arkhipov et al.9 In addition, we neglect quantum mechanical effects in our quantitative calculations of exciton transport in the ballistic limit. However, in the latter part of the paper, we qualitatively assess how the quantum nature of carrier transport and band structure might affect device performance. Polymer Nanowires in An Inorganic Semiconductor Matrix. This device consists of polymer nanowires of radius R and center-center distance a surrounded by the inorganic phase as seen in Figure 2a,b. In the diffusion limit, if I0 is the intensity of light of wavelength λ incident at z ) 0, n(r,z) is the spatial distribution of excitons in the polymer, D is the diffusion coefficient, τ is the characteristic recombination time of excitons, and R is the absorption coefficient of the polymer, then the steady-state equation for the excitons in the cylindrical polymer region is obtained by continuity as ∂2n 1 ∂ ∂n r + 2 - β2n ) -γe-Rz r ∂r ∂r ∂z
( )
(1)
x
2 and γ ) 2I0Rλ/hcD. Here, 1/β is the where β ) Dτ planar diffusion length of the exciton in the polymer. We assume that the diffusion-based model is valid when R . 1/β. The solution of the above diffusion equation with the boundary condition n(R, z) ) 0 gives the exciton concentration:
n(r,z) ) e
-Rz
{ }
ωr B γ 2 12 ωR ω B 2
( ) ( )
(2)
where ∞
B(x) )
() xn
∑ n)0 n!
2
, ω ) xβ2 - R2
Assuming that every dissociated exciton generates an electron-hole pair, the photocurrent in each polymer nanowire is iph(λ) ) -
L ∫z)0 {qD2 ∂n∂r|r)R}2πR dz
(3)
where L ) device thickness. Because each unit cell effectively contains three polymer nanowires, the photocurrent density for the unit cell is
{ }
ωR B′ qI0λ 2πR 2 (1 - e-RL) Jdiffusion(λ) ) hc x3a2ω ωR B 2
( )(
)
( ) ( )
(4)
The external quantum efficiency is the ratio of the total number of charges making up the photocurrent to the total Nano Lett., Vol. 3, No. 12, 2003
number of incident photons, which is then given by
{ }
ωR B′ Jph(λ) 2πR 2 EQEdiffusion(λ) ) ) (1 - e-RL) 2 qI0λ ωR x3a ω B 2 hc (5)
( ) (
( )
) ( )
In the ballistic limit where R e 1/β, the interface captures most of the excitons created. Therefore, β is used as the statistical parameter that describes the probability that an exciton reaches the interface before recombination. Equations 6, 7, and 8 give the photocurrent in a single wire, the photocurrent density over the unit cell, and the external quantum efficiency, respectively: iph(λ) )
I λ(1 - e-RL) -β(R - r) 2πr dr e hc
R 0 q ∫r)0
(6)
Jballistic(λ) ) qI0λ 4π e-βR[1 + (βR - 1)eβR](1 - e-RL) (7) hc x3a2β2
( )(
)
Jph(λ) ) qI0λ hc 4π e-βR[1 + (βR - 1)eβR](1 - e-RL) (8) 2 2 x3a β
EQEballistic(λ) )
( ) ( )
Inorganic Nanowires in a Polymer Matrix. This device consists of inorganic nanowires of radius R and centercenter distance a surrounded by polymer, as seen in Figure 2. The difference from the previous case is that the excitons are now generated in the matrix rather than in the nanowires. The unit cell considered is the same as before, but in this case the boundaries as well as the boundary conditions are more complicated and an analytical solution is not possible. In the diffusion limit, the partial differential equation toolbox10 of MATLAB was used to obtain numerical solutions for the steady-state exciton density, n(r,z). We imposed the following boundary conditions: (i) n ) 0 at the interface of polymer and inorganic material (no excitons accumulate at the interface), and (ii) ∇n ) 0 at all other boundaries of the unit cell (ensures that exciton concentration is continuous across unit cells). Once the exciton density is obtained, integrating the gradient of this density over the interface area gives the photocurrent. In the ballistic limit, β is used as the statistical parameter as before, and the photocurrent is obtained by numerically integrating the concentration of excitons generated weighted with the probability of their reaching the inorganic-organic interface. The photocurrent density and EQE are then computed from the photocurrent by using the same methodology as in the previous case. All of the expressions derived above apply for monochromatic illumination. The efficiencies and photocurrents under terrestrial (AM 1.5) conditions11 are obtained by using the Nano Lett., Vol. 3, No. 12, 2003
Figure 3. External quantum efficiency vs nanowire size: (a) polymer nanowires (b) inorganic nanowires. a/R values are based on typical ratios obtained during fabrication of nanowire arrays.
solar spectrum I(λ) and the monochromatic EQE values over the range [λmin, λmax] of wavelengths where the polymer absorbs light (for MEH-PPV,8 the range of interest is from 350 nm-600 nm). Jph )
∫λλ
qλI(λ)EQE(λ) dλ hc
max
min
∫λ
λI(λ)EQE(λ) dλ hc λmax λI(λ) dλ λmin hc
(9)
λmax
EQEAM1.5 )
min
∫
(10)
The external quantum efficiency in eq 10 is plotted in Figures 3a and 3b as a function of nanowire radius R for a specific a/R ratio for each type of solar cell. There is a gap between the ballistic limit and the diffusion limit of exciton transport, which we fill by a simple interpolation formula. For both types of devices, the EQE (and hence the photocurrent) falls off rapidly with increasing size. Even for the smallest devices, the EQE is limited by the fraction of area that the polymer occupies in the unit cell. Hence, to obtain the best performance, the exciton transport has to be in the ballistic regime 1731
and most of the area of the device must be the active material (which is the semiconducting polymer). It should be noted that in the plots of the ballistic limit, the nanowire radii go down to almost 1 nm. For such small length scales, quantum mechanics should be used for predicting the transport of excitons, electrons, and holes. Our equations, which are based on simple kinetic theory, will not hold in this range, but we have nevertheless extended our plots to this regime for the sake of continuity. The next step is to compute the J-V curves from which all the important operating parameters such as the open circuit voltage, short circuit current, fill factor and energy conversion efficiency (ECE) can be determined. The assumptions made in this calculation are listed below and are based on various hybrid solar cells previously fabricated using the same materials:8 (i) A Schottky-type of barrier to hole injection exists from the anode side of the cell. Devices using ITO as anode, P3HT/MEH-PPV as the semiconducting polymer, and TiO2/CdSe/C60 as the inorganic material generally show such Schottky barriers between the anode and the inorganic material.1-3 (ii)Normally, a shunt resistance is used to account for leakage currents through the device, and a series resistance is used to lump together the bulk resistance offered by the various components in the device. Since the shunt resistance is difficult to evaluate theoretically and is usually an order of magnitude greater than the series resistance in many cases,12 it is neglected in the following analysis. We used the equivalent circuit proposed by Petrisch12 for solar cells that incorporate these assumptions and directly give the final form of the J-V relation below.
[ (
J(V) ) Jph - J0 exp
) ]
q(J(V)FL + V) -1 kBT
(11)
where J0 ) reverse saturation current of the Schottky diode given by Richardson’s thermionic emission equation:
( )
J0 ) A*T2 exp -
Vbi kBT
(12)
Here, A* is the effective Richardson’s constant for polymer7 and Vbi is the absolute value of built-in voltage across the polymer due to work function difference between the anode and the inorganic semiconductor. Also, F is the bulk resistivity of the polymer. Because the mobility of charges in the polymer is a few orders of magnitude lower than in inorganic semiconductors, we may assume that all internal resistance arises only from the polymer. The Jph used above is calculated using eq 9. Figures 4a and 4b show the J-V curves for both kinds of devices under AM 1.5 illumination conditions. As the nanowire size increases, the short-circuit current decreases substantially. The values in the ballistic regime compare well with experimental results for bulk heterojunction cells.2 Figures 5a and 5b show a plot of the energy conversion efficiency (ECE) of the devices as a function of nanowire 1732
Figure 4. J-V curves: (a) polymer nanowires (b) inorganic nanowires. The a/R ratios for both plots are the same as those in Figure 3. Arrows indicate increasing values of nanowire radius R.
size. As in the case of the EQE plots, we interpolate between the ballistic and diffusion regimes. As expected, the ECE falls rapidly with increasing nanowire size or unit cell dimension. In the case of polymer nanowires (Figure 5a), if R is kept fixed and a increases (thereby increasing a/R), the same current flows through a larger cross-sectional area, reducing the quantum yield and, hence, the ECE. For the case of inorganic nanowires in a polymer matrix (Figure 5b) in the diffusion limit, larger a/R values imply larger crosssectional area (for fixed R) and more loss of excitons (which have to diffuse through a larger region to reach the interface), resulting in a low ECE. In the ballistic regime, the fraction of area occupied by the active material (polymer) is the critical factor that determines the upper limit of efficiency. In this regime, larger a/R values result in a larger filling fraction of polymer, thereby improving the ECE. The opposite trends in both these regimes can be seen in Figure 5b. As mentioned earlier, the above calculations do not account for quantum mechanical effects. Here, we briefly discuss how carrier transport in the ballistic regime might be affected in the quantum limit. If the nanowires are quantum confined, the density of states (DOS) of the carriers would be modified to have states concentrated at certain energy levels. First, this would change the probability of transmission of an electron from the polymer to a nanowire, Nano Lett., Vol. 3, No. 12, 2003
ing interchain hopping, exciton transport may be more efficient, which could potentially increase the efficiency. Hence, confinement could lead to many different effects, which can be qualitatively discussed. However, which of these effects is dominant and how do they interplay to determine the overall device performance can be predicted only through quantitative analysis of the quantum effects, which is beyond the scope of this paper. In summary, the performance of two types of hybrid inorganic-organic photovoltaic devices was simulated as a function of the nanostructure size. The external quantum efficiencies, J-V characteristics, and power conversion efficiency of these devices were calculated based on exciton transport models in different regimes. The simulations clearly show that the advantages of efficient charge transport are realized only when the dimensions of the inorganic phase are on the order of the exciton diffusion length or less, which in the case of MEH-PPV is about 20 nm. For larger structures, the improvement in charge transport in the inorganic phase is offset by the reduced contact area between the two phases. Thus, ordered structures on the order of 10 nm or less would be required in order to outperform bulk heterojunction devices.
Figure 5. Energy conversion efficiency: (a) polymer nanowires (b) inorganic nanowires. Plots for different a/R ratios are shown.
since that depends on the availability of the electronic states. In this paper, we have assumed the transmission to be 100 percent, but that could be reduced due to quantum confinement, which could reduce the overall device efficiency. Second, quantum confinement in the nanowires could increase electron mobility due to reduced phonon scattering and thereby reduce the resistance. Conversely, boundary scattering of electrons with the nanowire walls could increase its resistance. However, since the main electrical resistance is not in the nanowire, this would have marginal effect on the overall efficiency. Now consider confinement in the organic phase. In this paper, we started with diffusive transport of excitons and considered the ballistic limit of diffusion. This assumes that diffusion contains only one length scale-the mean free path. One must note that exciton diffusion in a polymer consists of both intrachain transport and interchain hopping. These two processes have their respective characteristic length scales. Hence, when a polymer is confined such that one encounters only single chains between nanowires, using the limit of macroscopic diffusive transport may not be accurate. In fact, by eliminat-
Nano Lett., Vol. 3, No. 12, 2003
Acknowledgment. The authors thank Rohit Karnik and Pramod Reddy for helpful discussions regarding the theoretical formulation of the problem. B.K. was supported through funding from the College of Engineering, UC Berkeley, for research on nanoengineering. K.C. was supported through a grant from DARPA. A.M. acknowledges the support of Basic Energy Sciences, Department of Energy and the National Science Foundation through a NIRT project. A.M. alsothanks the Miller Institute of UC Berkeley for supporting him through a Miller Professorship. References (1) Yu, G.; Gao, J.; Hummelen, J. C.; Wudl, F.; Heeger, A. J. Science 1995, 270, 1789. (2) Huynh, W. U.; Dittmer, J. J.; Alivisatos, A. P. Science 2002, 295, 2425. (3) Shaheen, S. E.; et al. Appl. Phys. Lett. 2001, 78(6), 841. (4) Greenham, N. C.; Peng, X.; Alivisatos, A. P. Phys. ReV. B 1996, 54(24), 17628. (5) Coakley, K. M.; Stucky, G. D.; et al. AdV. Funct. Mater. 2003, 13(4), 301. (6) Wu, Y.; Yan, H.; Yang, P. Top. Catal. 19(2), 197, 2002. (7) Scott, J. P.; Malliaras, G. G. Chem. Phys. Lett. 1999, 299, 115. (8) Breeze, A. J.; Schlesinger, Z.; Carter, S. A.; Brock, P. J. Phys. ReV. B 1998, 64, 125205. (9) Arkhipov, V. I.; Heremans, P.; Baessler, H. Appl. Phys. Lett. 2003, 82(25), 4605. (10) MATLAB, v. 6.5; Mathworks Inc., 2002. (11) Standard Tables for Reference Solar Spectral Irradiance at Air Mass 1.5: Direct Normal and Hemispherical for a 37 Degree Tilted Surface, ISO 9845-1. (12) Petritsch, K., Ph.D. Thesis, Technischen Universitaet Graz, Austria, 2000.
NL034810V
1733
