Threshold of hierarchical percolating systems

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  Random vs regularized OPV: Limits of performance gain of organic bulkheterojunction solar cells by morphology engineering Biswajit Ray n , Muhammad A. Alam School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, USA a r t i c l e i n f o  Article history: Received 18 September 2011Received in revised form22 November 2011Accepted 24 November 2011Available online 4 January 2012 Keywords: Organic bulk heterojunction solar cellSolution processingPlanar heterojunctionOptimum morphology a b s t r a c t Inexpensive solution processing of bulk heterojunction (BHJ) type organic photovoltaic (OPV) cellsoffers an attractive option for the low cost solar energy conversion. Solution processing creates adisordered morphology consisting of two organic semiconductors, intermixed randomly within thelight-absorbing layer of the cell. In this paper, we use a detailed three-dimensional process-device co-modeling framework to show that in spite of the inherent structural randomness of the morphology, the efficiency of solution-processed BHJ cells is nearly optimal – close to those of the perfectly orderedstructures . In addition, we show that the morphological randomness by itself does not increase theperformance variability of large-area cells. Both the results indicate that the inexpensive solutionprocessing of BHJ cells imposes no inherent limitation on the performance/variability and the ultimateefficiency of such solution-processed films should compare favorably to the other ordered OPV cellsfabricated by more expensive techniques. Finally, we explore the theoretical optimum morphology forBHJ cells and find that fill factor is the only parameter through which efficiency can be enhanced bymorphology engineering. We conclude by exploring the performance gains/limits of organic solar cellswith the improvement in transport parameters.Published by Elsevier B.V. 1. Introduction and background Since the invention ofplanar heterojunction (PHJ) based organicsolar cells [1], the efficiency of organic photovoltaic (OPV) technol-ogy has improved continuously, and currently it exceeds 8 %  forbulk heterojunction (BHJ) OPV[2]. The PHJ cells consist of a simpletwo layer stacked structure of donor (D) and acceptor (A) typeorganic semiconductors (Fig. 1a). The innovation of such stackedstructure (or heterojunction) facilitated the efficient dissociation of photo-generated excitons at the planar D–A interface. However,the short circuit current density  ð  J  SC Þ  of PHJ cell is low, primarilybecause most of the photo-generated excitons self-recombinebefore being harvested by the D/A heterointerface. In fact, onlythose excitons generated within a diffusion length  ð L ex  10 nm Þ from the planar HJ can contribute to photo-current (see Fig. 1a).This problem of poor exciton collection (or low  J  SC ) was latersolved by the bulk heterojunction (BHJ) morphology [3], where the junction between the donor and acceptor materials is distributedrandomly throughout the volume of the cell, see Fig. 1b. Regardlessthe point of photo-excitation, this distributed D/A junction canharvest excitons efficiently, and hence BHJ–OPV cells have high  J  SC .Unfortunately, the BHJ–OPV suffers from higher recombination [4]of the photo-generated carriers at the increased donor/acceptorinterfacial area. Thus, even though BHJ solar cells have achievedclose to 100% internal quantum efficiency for  J  SC  [5], the interfacialrecombination lossreduces its open circuit voltage ð V  oc Þ and the fillfactor ( FF  ) significantly.Many groups [6–10] have explored in great detail the trans- port and recombination of excitons, electrons and holes in BHJsolar cells by numerical simulations. These studies confirm thatthe performance of the BHJ–OPV is strongly correlated withthe underlying morphology. However, a statistical analysis of the performance of BHJ cells as a function of the degree of randomness of its morphology has not been reported in theliterature. Thus, despite many recent reports of performancegains of ordered heterojunction OPV (OHJ-OPV, see Fig. 1c) cellsfabricated by various top–down approaches (e.g., nano-imprint,templating, etc.) [11–13], it is fair to ask if one could achieve similar performance gain by optimizing the inexpensive solutionbased fabrication process [14–17]. Indeed, there is no convincing theoretical/numerical argument to show that the intrinsic ran-dom morphology of BHJ–OPV must necessarily lead to inferiorperformance compared to OHJ–OPV created by sophisticatedfabrication methods. There is also very little discussion regardingthe performance variability of the solution-processed BHJ cells Contents lists available at SciVerse ScienceDirectjournal homepage: Solar Energy Materials & Solar Cells 0927-0248/$-see front matter Published by Elsevier B.V.doi:10.1016/j.solmat.2011.11.042 n Corresponding author. E-mail addresses: (B. Ray), (M.A. Alam).Solar Energy Materials & Solar Cells 99 (2012) 204–212  srcinating from the inherent structural randomness and theimplications of such variability for large-area modules. A sys-tematic understanding and statistical analysis of such variabilityare important to predict/improve the panel efficiency of series-connected cells [18]. Finally, even though there are several recentworks on the optimum geometry for OPVs [19–21], it remains unclear if such structures provide significant efficiency gain overthat of random BJH-OPV (for comparable transport/recombina-tion parameters). In the absence of such explicit comparison, it isdifficult to predict the circumstances for which engineeringordered morphology may be appropriate.In this paper, we address the above mentioned issues througha computational framework (based on transport simulation of excitons, electrons and holes) which connects the morphology of an OPV to its efficiency. We have four major conclusions: (1) Wefind that the regularization of BHJ morphology does not improvethe OPV efficiency significantly. Instead, we show that efficiencyof random BHJ cells, fabricated by inexpensive solution-basedprocessing (with optimum mixing ratio and anneal duration), isclose to that of the perfectly ordered structures. (2) We alsodemonstrate that even though the morphology of BHJ cells isinherently random, this  intrinsic   structural randomness does notaffect the performance of large-area cells. However, precisecontrol of various process parameters related to  extrinsic   varia-bility remains a significant concern. (3) Next, we show thatinstead of the fully ordered structure (Fig.1c), a fin-like geometry(Fig. 4a), whose dimensions have been optimized for a givencombination of material parameters, offers the highest efficiency.(4) Finally, we explain how the transport parameters affect OPVefficiency, and we find the limits of efficiency enhancement byimproving the transport parameters like mobility, exciton diffu-sion length, etc.The paper is organized as follows: We first describe theprocess and device modeling approach used in this analysis. Thenwe present a detailed comparison between the performance of ordered and random BHJ cells. Next we discuss the performancevariability of BHJ cells due to its morphological randomness.Finally, we formulate the design rules for the optimal morphologyand discuss the performance gain/limit with the improvement of various transport parameters like exciton diffusion length, chargecarrier mobility, etc. 2. Process device co-modeling of BH–OPV  In order to explore the effect of bulk heterojunction morphol-ogy on the device performance, we first simulate the random BHJmorphology by the phase field approach. Specifically, we use theFlory–Huggins free-energy formulation within the Cahn–Hilliardtransport model [22] to describe the spinodal decompositionof the respective donor–acceptor organic semiconductors. Thedetails of the process model equations and the model parameters[23,24] are summarized in Tables 1 and 2. More sophisticated phase-field models and kinetic Monte Carlo approaches [8] havealso been used by many groups to describe the finer features of the morphology. Since our goal is to explore the generic impact of morphology on efficiency of OPV cell, a simpler description of phase segregation based on the Cahn–Hilliard model, analogousto those used in Ref. [6,7], is adopted. Once the morphology is simulated, it is characterized by an average domain size [25],  W  ,as shown in Fig. 2a. Unlike the BHJ morphology, the planarheterojunction (PHJ) and the ordered heterojunction (OHJ) struc-tures are simulated not from a process model, but with fixedgeometrical dimensions associated with the top–down fabrica-tion process.The transport of carriers (excitons, electrons, and holes) on thesimulated morphology is modeled by the drift–diffusion formal-ism (see Eqs. (3–10) in Table 1). Optical absorption is consideredonly in the donor material, as is typical for P3HT:PCBM basedsystem [26]. However, the key conclusions will not change if both donor and acceptor absorb photons. The absorption profile isassumed uniform with an effective exciton generation rate in thedonor material (details on absorption profile are described in[26,27]). We then solve the steady state exciton diffusion equa- tion in the distributed donor regions. We assume that theefficiency of exciton dissociation at the donor–acceptor hetero-interface is very high and is independent of field at the interface[28,29], so that the exciton concentration at the D–A boundary can be set to zero regardless the operating conditions. In recentyears, the model for exciton transport has been generalized toinclude hopping transport and more complex dissociationdynamics [30], but we adopt a simpler semi-classical approach for this first discussion regarding the efficiency/variability of BHJ–OPV due to the randomness in the morphology.The solution of exciton transport equation gives the excitondissociation flux at the D–A interface, which we use as the chargecarrier generation term in the electron and hole continuityequations [6,7]. The recombination term in the continuity equa- tions is implemented to reflect a bi-molecular recombinationprocess at the D/A interface [31]. Even though there is a debate in the literature regarding the dominant recombination mechan-ism at the hetero-interface [32], the key conclusions in the paperdo not depend sensitively on the details of the recombi-nation mechanism. Since the D–A interface is the only regionwhere free carriers can be generated from excitons and the free Fig. 1.  Structures of various organic solar cells. (a) Planar heterojunction (PHJ),(b) Bulk heterojunction (BHJ), and (c) ordered heterojunction (OHJ) OPV cell in thechronological order of development. (d) Comparison of typical  J  – V   characteristicsfor various OPV geometries. B. Ray, M.A. Alam / Solar Energy Materials & Solar Cells 99 (2012) 204–212  205  electron/holes can recombine with each other, the generation andrecombination terms in the e–h continuity equations are non-zero only at the D–A interfacial nodes. Electron and hole transportare modeled by the drift-diffusion formalism. While the drift–diffusion equation cannot explicitly account for complex chargetransport in polymeric crystals [33], most groups adopt this approach to explore the broad aspects of carrier transport withinthe OPV structures.The boundary conditions for carrier densities at the metal–semiconductor contact are determined by the equilibrium carrierdensities (defined by the metal work function, the HOMO andLUMO energy levels, and the corresponding effective density of states). Numerical values of all these parameters are tabulated inTable 5. Finally, we assume that the charge-density is low enoughso that the Poisson equation need not be solved explicitly but canbe approximated by a constant field defined by the built-involtage ( V  bi ), applied voltage ( V  ), and device thickness ( T  film ),i.e.,  E  ¼ð V  bi  V  Þ = T  film . While the validity of the approximation canbe easily established for relatively high mobility OPV cells,its application to very low mobility OPV devices requires addi-tional care. We wish to mention explicitly that the transportmodel used in this paper follows Refs. [6,7]. We claim no new contribution to model development; rather, we use this well-known, well-calibrated, and well-tested process-device model toexplore the implications of morphology on the efficiency of organic solar cells.For a given OPV structure (Fig. 1(a–c)), we solve the coupledtransport equations for excitons, electrons, and holes self-consis-tently at each bias condition to construct the current-voltagecharacteristics of the solar cell, see Fig. 1d. Based on the  J  – V  characteristics so generated, we calculate efficiency as well asother solar cell parameters such as short circuit current density,open circuit voltage, and FF. The short circuit current in the  J  – V  curve is defined as the current at the zero terminal voltage, i.e.,  J  SC   J  ð V  ¼ 0 Þ , the open circuit voltage is defined as the terminalvoltage for which current from the cell is zero, i.e.,  V  OC  V  ð  J  ¼ 0 Þ ,and the FF is calculated by the formula  FF  ¼  J  m V  m = ð  J  SC V  OC Þ , where  J  m  and  V  m  are the current density and voltage at the maximumpower point of the  J  – V   curve, respectively. 3. Results and discussion With the model system described in the previous section, wenow explore how the efficiency of a cell depends on its morphol-ogy. Although we choose typical values for the model parameters(see Tables 1 and 2) based on P3HT:PCBM system to illustrate ourapproach, the conclusions are general and should apply to a broadrange of polymeric OPV.  3.1. Comparison of ordered and random BHJ cells Let us first consider how the randomness of the OPV morphol-ogy affects the key device parameters such as efficiency,  J  SC , V  OC ,and  FF  . To quantify the structural randomness of OPV, wegenerate a series of three dimensional (3D) BHJ morphology fordifferent anneal duration ( t  a ) and temperature ( T  a ); the details of process simulation are described in Ref. [6,7]. The initial condition ( t  a ¼ 0) for simulating the morphology is a structure with randomcomposition fluctuation around a mean composition value givenby the mixing ratio of the D–A molecules. This random initialcondition implies that even if the anneal duration and tempera-ture are identical, the morphologies generated from differentinitial conditions will be structurally different. Thus, the random-ness in OPV morphologies has two distinct sources: one comesfrom the initial/starting random composition and the other isfrom process variation ( t  a ,  T  a ).In Fig. 2(a), we show the time evolution of BHJ morphology,simulated on a 3D domain of 100  100  100 grid points withuniform grid spacing of 1 nm. Once the morphology is simulated,we characterize it by the average cluster size,  W   (see Fig. 2a). Thetime evolution of   W   is plotted in Fig. 2b. We also plot in Fig. 2c the evolution of the total D/A interfacial area of the simulatedmorphology. Each point in Figs. 2b and 2c represents a distinct 10 0 10 1 10 2 51015Anneal Time (min)    I  n   t  e  r   f  a  c   i  a   l   A  r  e  a   (  a .  u   ) 10 0 10 1 10 2 102030Anneal Time (min)    <   W    >   (  n  m   ) <W>Anneal timeDonor Acceptor  Fig. 2.  Morphology evolution with anneal duration. (a) Three distinct morphologies of the active layer are plotted with increasing anneal duration. For clarity, the metalelectrodes and the blocking layers of the complete cell have not been shown. (b) The average cluster size ð W  Þ of the simulated morphologies are plotted with anneal time(log–log plot). (c) Interfacial area of the 3D morphologies are plotted with anneal duration. B. Ray, M.A. Alam / Solar Energy Materials & Solar Cells 99 (2012) 204–212 206  morphology corresponding to the anneal duration shown inx-axis. The spread in the domain size (or interfacial area) for agiven anneal duration reflects the randomness in the OPV mor-phology. For each of the simulated morphology, characterized by itsdomain width  W  , we calculate the  J  – V   characteristics by followingthe procedure described in section 2. From the  J  – V   characteristics,we compute all the solar cell performance metrics and plot them asa function of morphology-specific parameter  W   as open circles inFig. 3(a–d). The scatter of the points reflects randomness of themorphology resulting from initial conditions as well as fluctuationin anneal temperature, anneal time, etc. Generally speaking, we findthat an optimized BHJ–OPV cell with the specified parameters(Tables 2–5) can achieve average efficiency of   Z  5 : 5 %  with  J  SC  14 mA cm  2 ,  V  OC  0 : 62  V  ,  FF   0 : 63.Next, we define a series of ordered morphology (OHJ) with thedomain width ð W  Þ and domain height equal to the film thickness(see Fig. 1c). Unlike the C–H process model for BHJ–OPV, theseordered OPV structures are created mechanically with fixedgeometric dimensions. In practice, these structures could befabricated by stamping or lithography [12,13]. We assume min- ority carrier blocking at both contacts. We calculate the  J  – V  characteristics of this series of ordered structures using exactlythe same transport equations and boundary conditions as we didfor the BHJ–OPV. Finally, we plot the corresponding quantities of  V  OC ,  FF  ,  J  SC ,  and efficiency as a function of   W   as solid blue lines inFig. 3(a–d). For completeness, we also plot the solar cell perfor-mance metrics for PHJ cell (dashed red lines) in Fig. 3(a–d).How does the random OPV compare with the ordered OPV?First, among the PV parameters, the open circuit voltage is foundleast sensitive to morphology and is mainly determined by thematerial constants (see Fig. 3a). In Fig. 3a we plot  V  OC  variation forrandom BHJ, OHJ, and PHJ cells, which shows that  V  OC  value isalmost same for all these morphologies. This insensitivity of   V  OC with geometry can be understood from the fact that both  J  SC  andrecombination current are proportional to the interfacial area[34]. Since  V  OC  is determined by the ratio of these two currents,the interfacial area dependence cancels out and  V  OC  becomesinsensitive to morphology (detailed discussion on the insensitiv-ity of   V  OC  is given in Ref.[7]). Second, the fill-factors calculated from numerical simulationfor both ordered and random morphologies are close and vary inthe range of 0.55–0.65. The experimentally reported  FF  BHJ valuesfor P3HT:PCBM based BHJ cells are slightly lower compared to thesimulation, typically in the range of    0.5–0.6 [35]. It is interest- ing that these FFs are much lower than that of PHJ cell ( FF  PHJ  : 8in Fig. 3b). There are many experimental evidences for this higherFF of PHJ cells, where typically  FF  PHJ is measured in the range of    (0.6–0.7) [36]. Although specific values are slightly different, our conclusion of higher  FF   for PHJ geometry compared to therandom BHJ/OHJ structures is broadly consistent with experi-ment. We attribute the lower  FF   of BHJ/OHJ cells to the higher intrinsic   series resistance associated with the percolating paths of the device. Among the various OPV structures, the average carrierextraction length in PHJ is the lowest and hence it offers thelowest  intrinsic   series resistance, which makes its  FF   higher thanBHJ/OHJ OPV especially for high-mobility materials.Third, we find that the real advantage of OHJ morphology overthe random BHJ structures lies in the enhancement of shortcircuit current density (see Fig. 3c) for smaller  W   (or early annealphase). More fundamentally, this higher  J  SC  of ordered cells forlower  W   arises from the fact that the ordered morphologyoperates above the percolation threshold for all domain width W   and volume ratio of the polymers, see Fig. 1c. Thus, thegenerated charge carriers always find continuous pathways forcoming out of the cell. For random BHJ cells, at the early phase of annealing (this corresponds to the lower value of   W   in Fig. 3), theactive layer contains large number of floating islands of D/Aphases and the heterointerfaces between the D–A regions are notwell formed. Hence  J  SC  of BHJ cells is lower compared to OHJ cellsfor smaller  W   as shown in Fig. 3c (detailed discussion of anneal-ing effect on the  J  SC  of BHJ cell is given in [7]). With the increase in domain size  W   (late annealing phase), the interfaces form sharpheterojunction and the floating islands in the active layerdisappearbymergingintopercolatingpathsbetweentheelectrodes.Given the optimum 1:1 D-A mixing ratio (by volume), both thedonor and acceptor phase volumes exceed the 3D site percolationthreshold of    31%. It is well known that for such conditions, thedonor and acceptors belong to their respective percolation clustersand there are very few isolated floating islands in the morphology.Thus,  J  SC  of BHJ cells approaches the corresponding value of OHJcells for larger  W   (or late annealing phase).Since  V  OC  and FF of both OHJ and BHJ cells are similar, theefficiency comparison follows the analogous trend of   J  SC , asshown in Fig. 3d. The figure clearly shows that efficiency of random BHJ cells is very close to that of perfectly ordered (ideal)structures. Thus, the plot confirms that there is no fundamentallimitation related to morphology arising from the inexpensivesolution-based processing of BHJ cells, provided that the processvariables such as mixing ratio, anneal temperature, anneal dura-tion, etc. have been chosen optimally.Finally, we analyze the performance variability of random BHJcell for a series of 3D morphologies. Fig. 3d might suggest that theefficiency of the random BHJ cells could vary considerably (  1% inabsolute terms) depending on the cluster size and structuralrandomness. This variability in efficiency is mainly due to thestructural randomness srcinated from the initial composition.However, we find that the scatter in efficiency reflects finite sizeof the simulation domain, i.e., 100nm  100nm  100 nm. In sup-plementary material, we plot the efficiency variation for thestructures with the same thickness but increasing area to show that 10203034567< W > (nm)    E   f   f   i  c   i  e  n  c  y   (   %   ) 10203061015< W > (nm)    J    S   C    (  m   A  c  m   -   2    ) 1020300.< W > (nm)    V    O   C    (   V   ) 1020300.< W > (nm)    F   F PHJPHJOHJBHJPHJPHJBHJBHJOHJ OHJOHJBHJ Fig. 3.  Performance comparison of ordered heterojunction (OHJ, solid line), randombulk heterojunction(BHJ, symbols) and planar heterojunction (PHJ, dashed line) solarcell. The performance metrics are (a) open circuit voltage ( V  OC ), (b) fill factor ( FF  ),(c) short circuit current density (  J  SC ), and (d) efficiency ( Z ). Each scattered symbolcorresponds to a particular random morphology (BHJ) with an average domain size o W 4  (x-axis). (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article). B. Ray, M.A. Alam / Solar Energy Materials & Solar Cells 99 (2012) 204–212  207  the spread in efficiency decreases monotonically with larger cellarea. Thus, even for the smallest practical device dimensions (e.g.,1 mm  1 mm  100nm), such spread in efficiency disappears andonly the mean efficiency will be realized.  Hence the main conclusionof this study is that the inherent structural randomness of BHJ cells isnot a concern for the performance variability of BHJ–OPVs.  It isimportant to note that a 2D analysis of such geometrical random-ness overestimates the performance variability of BHJ–OPV [37],because the 2D percolation threshold of 50% incorrectly suggestssignificant number of floating islands even for 1:1 mixing ratio (byvolume). However, for the large area cells (fabricated by roll-to-rollor other technique), there will be extrinsic variation in processparameters such as anneal temperature, mixing ratio, etc., whichwillleadtovariationinaverageclustersize W   (andhenceefficiency)among the cells. This extrinsic process variation might inducesignificant performance variability even for a large area cells. Othersources for variability in performance can arise from metal deposi-tion and shunt formation [38], variability in the thickness of the blocking layers, etc. Thus, performance variability for BHJ cellsremains a significant concern for achieving higher panel/moduleefficiency, even though our analysis shows that  intrinsic   morpholo-gical randomness does not contribute to this variability.  3.2. Design rules for optimum morphology In this section, we explore the theoretical optimum morphol-ogy of OPV cells for the same set of transport parametersdiscussed in the previous section (see Table 2). The discussionin the previous section clearly indicates that it is difficult toimprove  J  SC  and  V  OC  by engineering morphology, because  V  OC  ismorphology insensitive and  J  SC  is almost optimal for averagedomain width  W   L ex . Thus,  FF   is the only parameter throughwhich efficiency can be enhanced by morphology engineering.In Fig. 4a we show an interpenetrating fin like structure (Fin-OPV), which optimizes the  FF   without affecting the  V  OC  and  J  SC and thus maximizes the OPV efficiency. However, the dimensionsof the Fin-OPV need to be optimized as a function of the transportparameters such as exciton diffusion length, mobility and therecombination strength.As shown in the Fig. 4a, the Fin-OPV has four independentgeometrical dimensions, namely, the donor fin width ( W  D ),acceptor fin width ( W   A ), donor offset height ( H  D ) and acceptoroffset height ( H   A ). For simplicity, we assume the structure issymmetric with equal volume of donor and acceptor material, sothat  W   A ¼ W  D ¼ W  fin  and  H   A ¼ H  D ¼ð T  flim  H  fin Þ = 2, with two inde-pendent variables of fin height ( H  fin Þ  and fin width ( W  fin ) avail-able for optimization. The general problem of optimizing partiallyordered OPV with unequal volume fraction is left as an openproblem for future work.The proposed fin-like morphology in Fig. 4a unifies both thePHJ ð H  fin ¼ 0 Þ and the fully ordered ð H  fin ¼ T  flim Þ structure within acommon geometrical construct. The reason for the existence of anoptimum fin height is as follows: the PHJ structure is excellent forcharge carrier transport, but has a very poor efficiency for excitoncollection. On the other hand, the ordered BHJ collects almost allthe excitons, but the large surface area increases interfacerecombination and longer carrier extraction length increases‘series’ resistance, therefore, the structure is non-optimal for thecharge transport. The fin-like morphology balances the twodesirable properties of OPV: excellent carrier transport in thePHJ cell and the exciton collection property of ordered cell. InFig. 4b, we plot the efficiency of FIN–OPV as a function of finheight. The figure clearly shows that the maximum efficiency isclose to  Z max  6.5%, corresponding to  H  fin ð opt Þ 75 nm. Theoptimum fin dimension and the corresponding efficiency dependon material parameters such as mobility and recombination T film H D H A W A W D Design of Optimum MorphologyH fin = 0H fin = T film H fin PHJOHJ05010034567    E   f   f   i  c   i  e  n  c  y   (   %   ) H fin (nm) η opt  = 6.5 % η PHJ  = 3.5 % H fin (opt) η OHJ  = 6.1 %0501000.60.70.805010051015    J    S   C    (  m   A  c  m   -   2    ) H fin (nm)0501000.50.60.7H fin (nm)    V    O   C    (   V   ) H fin  (nm)    F   F < η BHJ  >= 5.5 % Fig. 4.  Design rules for fabricating the optimum morphology (FIN–OPV). (a) The structure of the optimum morphology. For  H  fin ¼ 0 the structure corresponds to PHJ celland for  H  fin ¼ T  film , it is an OHJ. (b) Efficiency variation of FIN-OPV is plotted for the various fin heights. The fin width is kept fixed to the value of   W  fin ¼ L ex . We findoptimum fin height to be  H  fin ð opt Þ T  flim  x L ex , where  x  is a number in the range (1 o x o 2). The variation of other solar cell performance metrics, i.e.,  J  SC ,  V  OC , and  FF   areplotted in (c), (d), (e), respectively. B. Ray, M.A. Alam / Solar Energy Materials & Solar Cells 99 (2012) 204–212 208
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