1. Introduction

Interest in organic photovoltaic (PV) devices stems mainly from their prospect for a cost-effective source of renewable energy [1, 2]. However, the highest power-conversion efficiency (η_p) of organic PV cells reported to date remains below 7%, far inferior to their inorganic counterparts [3, 4]. In general, an efficient PV device must achieve a high external quantum efficiency over the broad solar spectrum, while minimizing the energy loss associated with the photon-to-charge-carrier conversion process for each absorbed photon. The external quantum efficiency (η _ext) is defined as the probability that an incident photon creates a pair of charge carriers that is collected at the electrodes. This can be expressed as a product of the absorption efficiency (η _abs, the probability that an incident photon is absorbed in the active material(s)) and the internal quantum efficiency (η _int, the probability that a photon absorbed in the active material(s) creates a pair of charge carriers at the electrodes): η _ext=η _abs η _int

The limitation in η_p of organic PV devices arises from three major issues. First, there is an inherent trade-off between η _abs and η _int. Absorption of a photon in an organic PV cell creates a tightly bound exciton, which must diffuse to a donor-acceptor (D-A) interface to be dissociated into an electron and a hole [5]. To achieve η _int above 90%, the thickness of the absorption layer must be chosen to be less than the diffusion length (L_d~10^-8 m) of excitons in organic semiconductors to ensure efficient exciton dissociation. This choice, however, would limit η _abs, since the exciton diffusion length is in general smaller than the optical absorption length (1/α~10^-7 m, where α is the absorption coefficient) in these materials [5]. Second, energy losses incurred during exciton dissociation at the D-A interface and subsequent charge transport process reduce the electric potential energy generated by the cell far below the energy of the absorbed photons [5, 6]. Third, when compared with their inorganic counterparts, organic semiconductors have relatively narrow absorption spectra [5, 7]. To maximally utilize the solar spectrum and minimize the energy loss associated with the photon-to-charge-carrier conversion process, it is crucial to develop a PV cell architecture that can easily accommodate multiple active materials optimized for each wavelength region.

Various device architectures have been demonstrated to overcome these limitations [1, 8], and bulk-heterojunction cells arranged in a vertical tandem architecture [1, 4] have been most commonly adopted for high-efficiency organic PV cells [1, 4]. Bulk-heterojunctions (BHJs) refer to an extended region of interpenetrating networks of donor and acceptor materials, which increase the surface area of the D-A interface. Consequently, efficient dissociation of photo-generated excitons can occur throughout the bulk, in contrast to a planar junction device where excitons generated within the diffusion length from the flat D-A interface can contribute to the photocurrent. Vertical tandem architectures refer to an arrangement where multiple sub-cells, each with an active material (or materials) sensitive to a different spectral region, are stacked. This architecture provides a means to utilize energy contained in the broad solar spectrum. Furthermore, the series connection between the sub-cells allows the open-circuit voltage (V _oc) to be given by the sum of V _oc of the sub-cells. Owing to these features, it is theoretically predicted that a maximum power-conversion efficiency close to 15% is achievable for a BHJ-tandem cell with 2 sub-cells [9]. However, the experimentally demonstrated highest η_p of organic tandem BHJ cells remains below 7 % as mentioned above [4], due to complications in device optimization. A BHJ is typically obtained by relying on phase separation of donor and acceptor materials in a mixed layer, where insufficient phase separation often compromises the device performance due to carrier recombination and poor charge transport [10]. Layers connecting the adjacent sub-cells must possess desirable optical and electrical properties, such as high transparency and energy level alignment favoring charge carrier recombination [4, 11]. In addition, the balancing of short-circuit currents, which is required for a series-connected tandem cell [11], becomes increasingly more difficult as the number of sub-cells increases.

In this paper, we propose an organic PV system consisting of a dispersive-focusing element and resonant sub-cells to overcome the efficiency limitation of organic PV devices. Arranged in a lateral-tandem configuration, the proposed system utilizes spectral re-distribution of the solar photons and continuously-tuned, wavelength-matched resonant sub-cells based on planar metal-organic-metal microcavities. In contrast to the vertical-tandem structure [4, 11], the lateral-tandem configuration does not need inter-cell connection layers with delicate balance of electrical and optical properties. Moreover, the balancing of short-circuit currents is readily achieved by varying the widths of the sub-cells, providing a straightforward platform to integrate multiple sub-cells. Although the splitting of the solar spectrum and the employment of matched sub-cells have been previously proposed for general PV devices [12], the system described here is unique in that it employs continuously-tuned resonant sub-cells to dramatically increase η_p of organic PV cells. We perform numerical analysis of the sub-cells, and demonstrate the tunability of the resonance over a broad spectrum. Our analysis of the system performance suggests that η_p exceeding 18% can be obtained when an ideal dispersive-focusing element and the microcavity sub-cells are employed, demonstrating that the proposed system holds promise for high-efficiency organic PV cells.

2. Lateral-tandem organic photovoltaic system: concept and analysis

Figure 1 shows the concept of a lateral-tandem cell (LTC) system [13] considered in this paper. The system consists of a one-dimensional periodic array of dispersive-focusing elements (DFEs) and resonant sub-cells. The DFE spectrally separates the incoming solar photons to map the photon wavelength (λ) onto the position (x) on the sub-cells. A cylindrical lens with a grating on one of its surfaces can function as a DFE. The photons delivered to position x are “energetically matched” to the sub-cell at that location in the following sense: (i) the structure of the device is continuously varied along the x-direction so that the optical field incident on x resonantly excites the cell at that location, thereby overcoming the trade-off between η _abs and η _int, and (ii) organic materials in each sub-cell are chosen to ensure that energy losses associated with exciton dissociation and charge transport processes are minimized at that wavelength. The sub-cells are connected in series, as schematically shown in Fig. 1, so that the open-circuit voltage (V _oc) of the LTC is given by the sum of V _oc of the sub-cells [5].

 

Fig. 1. Schematic diagram of organic photovoltaic cell in lateral-tandem configuration employing planar microcavity sub-cells. Lateral-tandem cell (LTC) system consists of a one-dimensional periodic array of a unit-cell and a dispersive-focusing element (DFE) with a period W. The DFE, which is conceptually represented by a lens with a grating on its surface, spectrally separates incoming solar photons, as schematically shown by different colors on the LTC surface. Each unit cell is partitioned into series-connected sub-cells. The planar microcavity shown on the left consists of an organic multilayer of a thickness t located between two silver electrodes (gray), and the organic multilayer consists of a 10-nm-thick absorption layer (blue), and upper and lower transport layers (white). Continuous resonant matching between the sub-cells and incoming photons is achieved by varying t along the x-direction. Also shown on the right is how the series connection between adjacent sub-cells is implemented.

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Since the LTC shown in Fig. 1 possesses one-dimensional periodicity, we consider its period defined by the horizontal arrow in Fig. 1 (referred to as unit cell) to derive an expression for η_p. Assuming for the moment that the unit cell contains only one sub-cell, a region of the LTC ranging from x=x_i to x_j generates a short-circuit current per unit length in the z-direction:

where e is the electronic charge, η_ext(x) is the external quantum efficiency at x, n_ph(x) is the photon flux density at x on the cell surface, ñ_ph(λ) is the flux density of the incoming solar photons per unit wavelength, λ(x) is the position-to-wavelength mapping, and W is the period of the LTC. For givenη_ext(x) and λ(x), the evaluation of Eq. (1) over the unit cell is a conserved quantity, and is denoted as I_sc,tot. The partitioning of the unit cell into N sub-cells are performed in a straightforward manner by successively requiring that the evaluation of Eq. (1) over each sub-cell yields the balanced short-circuit current, I_sc=I_sc,tot/N. The power-conversion efficiency is then given by

where FF _LTC is the fill factor of the LTC, h is Planck’s constant, and c is the speed of light in vacuum. Here, V̄_oc=∑^N _i=1 V ⁽ⁱ⁾ _oc denotes V _oc of the unit cell, where V ⁽ⁱ⁾ _oc is V _oc of the ith sub-cell. The advantages of the LTC approach are evident from Eq. (2). Condition (i) noted above increases η_abs without compromising η_int, thereby maximizing η_ext over the solar spectrum, while condition (ii), together with series connection among the sub-cells, maximizes V̄_oc.

3. Analysis of planar microcavity sub-cells

We exploit optical resonance occurring in a planar metal-dielectric-metal cavity, where one metal layer is optically thick and the other is semi-transparent [14, 15]. When such a structure is excited by a plane-wave through the semi-transparent metal layer, an optical resonance occurs if the waves inside the cavity resulting from multiple reflections at the metal layers constructively interfere. On resonance, the intensity of the electric field inside the cavity is enhanced as compared with that of the incident plane-wave, enhancing photon absorption by organic molecules (or polymers) strategically located near an anti-node of the electric field.

Figure 1 schematically shows a layer structure of a model device that we analyze, comprising: substrate/150-nm-thick Ag electrode/organic multilayer/10-nm-thick Ag electrode. The organic multilayer with a total thickness of t consists of a 10-nm-thick absorption layer located between optically inactive layers (referred to as ‘upper transport layer’ and ‘lower transport layer’) with equal thickness. The thicknesses of the transport layers are continuously adjusted to tune the wavelength of optical resonance. To estimate η_ext, we perform the electromagnetic and excitonic analyses [5] of the model device illuminated by a monochromatic plane-wave with a wavelength of λ, with the following assumptions: for the absorption layer, the real part of the index of refraction (n) is 1.75, while the absorption coefficient is α=1.5×10 ⁵ cm^-1 throughout the spectral region of interest (400 nm~1000 nm); L_d of the excitons in the absorption layer is 20 nm; n=1.75 for the transport layers; for n of the Ag layers, we use the values reported in literature [16]; the exciton dissociation velocity at the interface between the absorption layer and upper transport layer is infinite, while that at the interface between the absorption layer and lower transport layer is zero [5]; under short-circuit conditions, charge carriers generated by exciton dissociation are collected at the electrodes with unity probability [5]. For the last assumption to hold, a relevant charge transport level of the lower transport layer should be aligned with the corresponding charge transport level of the absorption layer. The constant value assumed for α represents the minimum value required at each wavelength, which can be achieved by appropriately choosing materials for the absorption layer. If active materials withα larger than this value are employed, η_ext will improve over the values presented in this analysis.

Figure 2(a) shows the calculated η_ext for the case of normal incidence as a function of (λ,t), demonstrating that the optical resonance of the microcavity sub-cell can be tuned to achieve high η_ext (>65%) over a wide spectral region (524 nm to 1000 nm). The high-efficiency linear region containing point A (B) corresponds to the first-order (third-order) resonance, where the organic multilayer accommodates approximately half (three halves) the wavelength. The solid line in Fig. 2(a) represents the optimal path in (λ,t) space, along which η_ext is maximized. For some λ, η_ext for the third-order resonance is larger than that for the first-order resonance. We force the optimal path to be continuous by limiting it to the first-order resonance, since the difference in η_ext is negligible. The intensity profile of the electric field corresponding to point A is shown in Fig. 2(b), along with the steady-state exciton density in the absorption layer. The optical resonance dramatically enhances absorption to achieve η_abs=83 % with the absorption layer whose thickness is only 15% of the absorption length. In addition, since the thickness of the absorption layer is L_d/2, 92% of the photo-generated excitons contribute to the photocurrent (η_int=92 %), consistent with the analytical solution obtained assuming that the exciton generation rate is constant throughout the absorption layer [5].

 

Fig. 2. (a) External quantum efficiency as a function of photon wavelength and thickness of the organic multilayer. (b) Electric field profile corresponding to point A in (a). Vertical lines represent the boundaries between the layers, with the leftmost layer corresponding to the 150-nm-thick Ag electrode. E denotes the complex amplitude of the electric field when the amplitude of the incident plane-wave is E _inc. Inset: steady-state exciton density (n_exc) in the absorption layer.

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4. Performance estimation of a lateral-tandem cell employing microcavity sub-cells

Based on the result in Sec. 3, we can estimate η_p of the LTC system employing the planar microcavity sub-cells. A monochromatic optical beam incident on the grating surface of the DFE experiences a wavelength-dependent tilt in the propagation direction before it gets focused on the LTC surface. We assume that an ideal DFE, capable of establishing the wavelength-toposition mapping without loss in the optical power, is available. The wavelength-to-position mapping is assumed to be linear from 400 nm to 1000 nm across the unit cell of the LTC (top axis, Fig. 3). Then, t must be continuously varied following the solid line in Fig. 2(a) to obtain the optimized external quantum efficiency [η_ext,opt(x)] over the entire LTC surface [first and second panels in Fig. 3(a)]. The third panel shows η_ext,opt(x)n_ph(x), when ñ_ph(λ) corresponds to AM1.5 solar illumination with an integrated intensity of 100mW/cm 2. The area under this curve in a sub-cell is proportional to its short-circuit current. The vertical dotted lines in the lower two panels represent the partitioning of the LTC into 7 sub-cells, each generating identical short-circuit current. The bottom panel shows V ⁽ⁱ⁾ _oc, which are assumed to be V ⁽ⁱ⁾ _oc=V ⁽ⁱ⁾ _ph-0.8 V [17]. Here, V ⁽ⁱ⁾ _ph=(hc)/[eλ(x(i))] is the minimum of the photovoltages of photons incident on the ith sub-cell, where x(i) is the position of the right boundary of the i th sub-cell. Under these assumptions, V̄_c=6.6 V and I_sc=3.7 mA/cm². Finally, assuming that FF _LTC=0.7, η_p is estimated to be 18.2%. Figure 3(b) shows the dependence of η_p on the number of sub-cells N, along with η_p for the ideal sub-cells with η_ext=1 for all x. We note that ηp reaches 86% of the asymptotic value when N=5.

For each λ, the DFE transforms an incoming solar beam with a width W into a focused beam, which increases the spread in solid angle of the outgoing beam incident on a sub-cell, consistent with the constant-radiance theorem [18]. Figure 4 shows the dependence of η_ext on incident angle for unpolarized illumination. For an incident angle smaller than 30°, η_ext is more than 89 % of the normal incidence case for all λ, and the focal length of the DFE can be chosen to limit the angular spread below this value. The effect of daily movement of the sun on η_p can be minimized by aligning the z-axis of the LTC system to the projection of the sun’s trajectory on the x-z plane. Seasonal re-aligning or tracking may be necessary to further preserve the LTC performance.

 

Fig. 3. Performance analysis of the LTC employing microcavity resonant sub-cells. Top panel: the thickness of the organic multilayer that optimizes the external quantum efficiency throughout the LTC surface. Second panel: the optimized external quantum efficiency. Third panel: the optimized external quantum efficiency multiplied by the photon flux density on the LTC surface. Fourth panel: open-circuit voltages of the sub-cells. Dashed line represents the photovoltage at each location x. Vertical dotted lines denote the partitioning of the LTC into seven sub-cells. (b) The power-conversion efficiency as a function of the number of sub-cells employed. Square symbols are for the LTC employing the microcavity sub-cells. Also shown is the power-conversion efficiency of the LTC employing the ideal sub-cells whose external quantum efficiency is unity over the entire LTC (dot symbols).

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5. Conclusion

We proposed a systematic solution to the efficiency limitation of organic PV cells by employing a DFE and continuously-tuned, series-connected sub-cells arranged in lateral-tandem configuration. Our electromagnetic and excitonic analyses show that the planar, metal-organic-metal microcavity device with a 10-nm-thick active layer can be tuned to achieve η_ext larger than 65% over a wide spectral region (524 nm to 1000 nm), assuming that the optical absorption and exciton diffusion lengths in the absorption layer are 67 nm and 20 nm, respectively. We showed that η_p of the lateral tandem cell system employing an ideal DFE and the microcavity sub-cells can exceed 18%.

The concept of LTC — spatial and spectral separation of incoming solar photons and their delivery to continuously optimized sub-cells — can be widely applied to photovoltaic devices whose performance is limited by insufficient absorption of solar energy and mismatch between photovoltage and optical band gap of the active materials. We note that it is particularly well-suited for organic PV cells. Unconventional deposition and patterning processes, such as ink-jet printing [19, 20], layer transfer [21, 22] and organic vapor-phase deposition (OVPD) [23], may be adopted in achieving continuous variation in device geometry over large areas at low cost. Furthermore, a diverse set of organic semiconductors available, together with the tunability of their optical and electronic properties [24], provides a greater flexibility in the sub-cell optimization over a wide spectral range, as compared with conventional inorganic devices.

 

Fig. 4. Dependence of the external quantum efficiency on the incident angle. If the model device shown in Fig. 1 with t _opt(λ) shown in Fig. 3(a) is excited by an obliquely incident unpolarized plane-wave, the resonance condition is not maintained, decreasing the external quantum efficiency (η_ext). η⁰ _ext,opt is the external quantum efficiency spectrum optimized for the case of normal incidence.

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