Abstract:
A nano structure CdTe/CdS Photon Trapping Solar Cell (PTSC) comprising an ultra thin film stack construction of 2 layers, one each of Cadmium Sulfide and Cadmium Telluride, and wherein the bottom electrode is a reflecting sheet or plate that serves to reflect Photons unabsorbed by their pass through the material back through the layers of Cadmium Telluride and Cadmium Sulfide. The bottom metallic plate also serves as an electron conductor for allowing the photoelectric current to exit the device and perform useful work. The cell has a maximum efficiency of 40%.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Application No. 61/137,822 filed Aug. 4, 2008, the contents of which are incorporated herein by reference. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    Solar cell technology and design is rapidly converging to thin film CdTe techniques to reduce the high production cost of Silicon based solar cell technology. Thin film p-type CdTe with n-type CdS solar cells and systems have emerged as the one possible solution to replace Silicon solar cells. However, the conversion efficiency of thin film CdTe/CdS solar cells presently available are far short of Silicon Cells and have not reached a point where government subsidies are not required for effective commercialization to compete with conventional energy production. Typical proven efficiencies of only 9 percent are presently reported by large commercial companies. Today&#39;s CdTe/CdS solar cell systems designs cannot compete directly with the economic costs of oil and natural gas unless a major breakthrough in efficiency and cost is achieved by solar cell systems. 
         [0003]    Researchers from across the country and the world are intensely searching for thin film solutions such as CdS/CdTe that exploits the low cost potentials of thin films and still produce the relatively high efficiencies of silicon solar cells. Researchers are on the right track as to the thickness issue by concentrating on thin film technologies. However, at optical wavelengths measured in angstroms, micron technologies appear to be extremely thick when dealing with energy conversion and propagation and reported efficiencies are extremely low as compared to Silicon cells. For examples, Jordan in U.S. Pat. No. 5,261,968 discusses an improved photovoltaic panel and method of forming a photovoltaic high efficiency CdTe/CdTe cell with a relatively thick design solution, wherein the relatively thick CdS layer is initially formed on a substrate, but its thickness is reduced during regrowth of the active photovoltaic layers. The effective thickness of the CdS layer would be much less but details on the physics of optimizing efficiencies based on exact thickness solutions of the phovoltaic junction of the p and n type material is not discussed (from the National Renewable Energy Laboratory, 0-7803-1220-1/93 IEEE). Zweibel et. al. reports a 13.4% efficient CdTe solar cell in “The U.S. DOE/NREL Polycrystalline Thin Film Photovoltaics Project”, Zweibel, K.; Ullal, H. S.; Mitchell, R. L.; Noufi, R., Photovoltaic Specialists Conference, 1991., Conference Record of the Twenty Second IEEE Volume, Issue , 7-11 Oct. 1991 Page(s): 1057-1061 vol. 2). In other reported efficiencies by Zweibel, Conventional Cadmium Telluride solar cells have efficiencies of less than 16.5% (K. Zweibel et. al., “Perspectives And Opportunities In Polycrystalline Thin-Film Photovoltaic Technologies” (Zweibel, K.; Ullal, H. S.; von Roedern, B. G.; Noufi, R.; Coutts, T. J.; Al-Jassim, M. M., Photovoltaic Specialists Conference, 1993., Conference Record of the Twenty Third IEEE Volume, Issue, 10-14 May 1993 Page(s): 379-388). General trends have been recognized as stated by Akhlesh Gupta, et.al. in his paper on “Effect of Thickness Reduction in High Efficiency CdS/CdTe Cells”, Materials Research Society, observing that increased efficiencies are possible as thickness of the photovoltaic junction is reduced. 
         [0004]    As a result, there remains a great need to produce solar cells at low cost with high efficiency that require small amounts of critical raw materials. 
       BRIEF SUMMARY OF THE INVENTION 
       [0005]    The present invention provides a solution to the efficiency barrier of CdS/CdTe cells by radically changing the design process for obtaining solutions for material thicknesses in the photovoltaic region and in the capturing of energy resulting in much thinner CsS/CdTe cell designs. 
         [0006]    The design approach for efficient solar energy designs was invented using first Principles of Quantum Mechanics coupled with Particle-Wave Duality. This unique process led to extremely thin CdTe/CdS Cell thicknesses whereby freed electrons traverse through the cell media with only a minimum chance of recombining with the CdTe molecules thus resulting in preservation of energy. In addition, the design is optimized for trapping reflected electrons to add the energy on multiple passes through the region thus further enhancing the energy conversion. 
         [0007]    Although Cadmium is readily available, the implication of large CdS/CdTe solar cell systems when implemented worldwide would increase raw material demands unless only a small amount of raw Cadmium Sulfide and Cadmium Telluride material is used in the designs. Thin film solar cell designs are an answer but they would strain the demands of the raw CdS/CdTe material. A typical 3 micron thick CdTe solar cell would use an estimated 3 grams of CdTe per square meter, whereas the Nano Structure Photon Trapping Solar Cell (PTSC) would use 0.25 grams CdTE per square meter. The use of Nano Structure PTSC systems would put less impact to the world demands of Cadmium Telluride when PTSC is implemented in large power producing plants of miles in diameter, producing Giga Watts of prime power for the industrial grid. To the inventors&#39; best knowledge, the efficiency of the PTSC System is estimated to be as much as four times higher than that of the known reported CdS/CdTe solar cell designs based on a recent patent search and available literature. Typical reported proven efficiencies of only 9 percent are presently reported by large commercial companies. 
         [0008]    The implications of this Invention to the US and the world energy policies is extremely broad and PTSC would open a new era of energy production. The supply of energy from the sun is almost limitless and if PTSC can harness this energy at affordable costs, the world can generate almost limitless energy in electricity. The energies of the oceans can be harnessed. Ocean waters can be converted to hydrogen gases for supplying heat and fuel cell engines, water can be obtained for deserts and farming at low costs. Miles and miles of open desert areas can be turned into energy farms supplying giga watts of energy to the power grid. 
         [0009]    The Nano Structure CdTe PTSC solar cell unique multi layer construction allows the formation of a photon trapping geometry which results in high energy conversions. A preferred embodiment of the present invention comprises a 2 stack design between 96.75 nm and 387 nm thick with the following parameters: the upper layer comprises n-type photoelectric material, preferably CdS, and the bottom layer comprises p-type photoelectric material, preferably CdTe. Further, electrodes are attached to the top and bottom of the 2 layer stack, wherein the bottom electrode is a metallic reflective conductive material, for example aluminum. The PTSC may further comprise an antireflective  1  over the top electrode that also functions to protect the structure underneath 
     
    
     
       BRIEF DESCRIPTION OF THE VIEW OF THE DRAWING 
         [0010]      FIG. 1  is a cross-sectional view of a preferred embodiment of the photon trapping Cadmium Telluride solar cell invention. 
           [0011]      FIG. 2  is a diagram illustrating the operation of computer program, optical_stack.f. 
           [0012]      FIG. 3  is a diagram illustrating the operation of computer program, optical_stack_opt.f. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0013]    The operation of this solar cell is similar to that of a standard optical antireflection coating used on windows and eyeglasses. An antireflection coating relies on the wave nature of light for its operation. 
         [0014]    Referring to the invention represented by  FIG. 1  in more detail,  FIG. 1  as shown comprises a top transparent layer  10 . Layer  10  comprises an antireflection layer that transmits the incident visible light energy to the layers underneath. Layer  20  is an electrode affixed to  10 . Affixed underneath the conducting layer  20  is a layer comprising a n-type photoelectric material, shown as  12 , preferably Cadmium Sulfide. Affixed underneath  12  is a layer comprising a p-type photoelectric material compatible with layer  12 , preferably Cadmium Telluride, shown as  14 . Affixed underneath  14  is a metallic reflective conducting electrode shown as  22 .  22  should have a mirror like surface at the boundary with  14 , in order to reflect photons that traverse the above layer without being absorbed, back through the photoelectric layers. 
         [0015]    The top layer  10  serves as an antireflection layer designed to operate over the entire spectrum of visible light, comprising relative electric permittivity and thickness chosen to maximize the transmission of photons to layers  12  and  14 . An antireflection layer has the material thickness designed so that light reflected at the boundary between two material layers with different indices of refraction has the opposite phase from the incoming light when it reaches the material boundary above and so is cancelled. This antireflection layer is necessary to transmit the light that strikes the top of  10  to the photoelectric layers  12  and  14 .  10  can comprise transparent or near-transparent materials including but not limited to borosilicate glass, acrylic glass, transparent plastics, soda-lime glass, and ordinary transparent glasses.  10  further functions to protect the layers below and so is at least about 70 nm thick. However, with a thicker antireflective coating beyond 70 nm, there is a reduction in the PTSC&#39;s efficiency. For example, a coating of ⅛ inch thickness will reduce the overall efficiency by approximately 6% depending on the initial efficiency. In a preferred embodiment,  10  comprises a top layer of ordinary window glass from 70 nanometers thick to 1 inch thick. 
         [0016]    Layer  20  comprises an electrically-conductive transparent layer also referred to as a top electrode, affixed between  10  and  12  that serves to allow electrons to return to the photovoltaic layers  12  and  14  below. A nonlimiting example of  20  is an open metallic grid that allows visible light through to the layers below. Preferably  20  comprises a transparent electrode with one or more sub-layers at least about 1000 nanometers in total thickness. More preferably such transparent electrode comprises at least one material including but not limited to Zinc Oxide and Indium Tin Oxide. Sub-layers of  20  may further comprise a high resistance transparent layer (HRT) to improve the cell efficiency, which is positioned as the bottom sub-layer of  20 . In a preferred embodiment  20  comprises an upper sub-layer of Zinc Oxide doped with Aluminum with a thickness of at least 1000 nanometers, and a lower HRT sub-layer of undoped Zinc Oxide with a thickness of between 50 and 150 nanometers. 
         [0017]    Layer  12  comprises an n-type photo-electric material, including but not limited to n-doped Silicon and Cadmium Sulfide. Preferably,  12  comprises Cadmium Sulfide. More preferably,  12  comprises Cadmium Sulfide with a thickness of between 26.75 nanometers and 107 nanometers. Layer  12  must be smoothly affixed to layer  14 . 
         [0018]    Layer  14  can comprise any p-type photoelectric material compatible with the dielectric constant of layer  12 , including but not limited to Cadmium Telluride and p-doped Silicon. Preferably, layer  14  is Cadmium Telluride. More preferably,  14  is Cadmium Telluride with a thickness of between 70 and 280 nanometers. In a preferred embodiment, the combined thickness of layers  12  and  14  is between 96.75 and 387 nanometers. Since Cadmium Sulfide and Cadmium Telluride have the same relative dielectric constant, the thicknesses of layers  12  and  14  can individually vary as long as the combined thickness is from 96.75 and 387 nanometers. Variations from this combined thickness range will decrease the efficiency. 
         [0019]    The overall efficiency of the photovoltaic structure is increased when the combined layers  12  and  14  are kept as thin as possible as long as the bottom CdTe layer is sufficiently effective in absorbing photons. This reduces the time for freed electrons to travel to the bottom conductor before being reabsorbed and losing the absorbed photon energy. 
         [0020]      22  is an electrically-conductive reflective layer also referred to as a bottom electrode, affixed to layer  14 , and gives the photovoltaic stack structural stability, serves as a conductor for the electrons freed by photons, and acts as a reflective surface that bounces photons that pass through the CdTe layer without being absorbed back through the CdTe layer. The CdS and CdTe layers create the photoelectric effect, a natural electric field that moves freed electrons to the bottom conductor. These electrons exit the bottom electrode through a wire, do work, and then are returned to the electrode on top of the CdS. The electrode must have a surface in contact with  14  that reflects light energy. The electrode comprises one layer or multiple sub-layers of any metal or combination of metals with a combined thickness of at least 100 nanometers, including but not limited to gold, copper, aluminum, and silver. In a preferred embodiment, the electrode comprises aluminum. 
         [0021]    In further detail, photovoltaic cells of this invention can be used to convert light radiation into electrical power. Light comprised of photons is incident on the top antireflective coating  10 , passing through to layer  14 . Some of the incident photons strike atoms of the Cadmium Telluride layer of  14 , freeing electrons. The freed electrons travel to the bottom electrode  22  where they flow to an attached wire to do work. The electrons then return to the top electrode  20 , enter layer  12 , then pass through to layer  14 , completing the circuit. Photons that do not free an electron on the initial pass are reflected by the reflective electrode back through layer  14  where they may strike a molecule of Cadmium Telluride. Photons that do not strike a molecule on the second pass through layer  14  travel through to the top of layer  12 . The thickness of  10 ,  12 , and  14  are independently selected to exploit the wave nature of light, as expressed in the quantum mechanical principle of particle-wave duality, and captured light photons described above destructively or constructively interfere with the light incident on the top of  10 , and bounce back through layers to the photoelectric layers  12  and  14 . The CdS and CdTe material have similar optical indices of refraction, eliminating reflections from the interface of the two materials. The photons repeatedly bounce through layers  12  and  14 , increasing their chances of sting a molecule and freeing an electron. 
         [0022]    The thicknesses of  10 ,  12 , and  14  are crucial to the operation of the device. The thicknesses depend upon the relative electric permittivity of layers  10 ,  12 , and  14 , and on the effective loss tangent, or imaginary part of the dielectric constant of the material of layer  14 . These thicknesses are found by an optimization computer program that minimizes the light reflected from the top surface of  10 . The optimization is performed for incident light wavelengths ranging from 400 nanometers to 796 nanometers. Photons with wavelengths longer than 796 nanometers are not energetic enough to free an electron from a Cadmium Telluride atom and are not absorbed because their energy is below that of the Cadmium Telluride band gap of approximately 1.56 electron volts. As non-limiting examples, for a relative Cadmium Telluride electric permittivity of 6.76 (relative index of refraction equal to 2.6), the top layer  10  is 70 nanometers thick, Layer  12  is 53.5 nanometers thick, layer  14  is 140 nanometers thick. 
         [0023]    The relative thinness of layer  14  increases the probability that electrons freed by photons can travel to the electrode  22  without being reabsorbed. This increases the overall efficiency of the solar cell shown in the drawing compared to conventional solar cells. Conventional solar cells are usually much thicker than the device shown here. Freed electrons have a much greater distance to travel and this increases the time needed before reaching the bottom conductor. This in turn increases the chances for the freed electrons to be absorbed, thus losing their energy. 
         [0024]    Here electric permittivity (for vacuum) is the quantity 8.8532×10 −12  farad/meter. Relative electric permittivity is the value of the electric permittivity relative to that of vacuum (relative permittivity=1.0). Antireflection coating refers to a  1  or multilayer transparent coating whose relative electric permittivities and thicknesses are chosen to minimize the reflection of any photons in the range of visible light (generally known in the art as 400 to 700 nanometers wavelength). 
         [0025]    The thicknesses of these layers are found by two computer programs written in Fortran 77, optical_stack.f and optical_stack_opt.f (see block diagram below). Optical_stack.f analytically solves Maxwell&#39;s equations 
         [0000]      {right arrow over (∇)}× {right arrow over (E)}=i ω μ {right arrow over (H)}−{right arrow over (M)}{right arrow over (∇)}·{right arrow over (D)}=σ   
         [0000]      {right arrow over (∇)}× {right arrow over (H)}=−i ω ε {right arrow over (E)}+{right arrow over (J)}{right arrow over (∇)}·{right arrow over (B)}=σ   m ( E   −iωt  time dependence) 
         [0000]    for an electromagnetic plane wave incident on the top layer of the stack at a given angle from normal incidence and at a given frequency. This is done by enforcing the correct boundary conditions at the boundaries of surfaces of the material stack. For example, for a material boundary at x=0, see  FIG. 2 : 
         [0000]        E   1   tan   =E   2   tan   =ZJ    (1) 
         [0000]        H   1   tan   =H   2   tan   +E   2   tan   /Z    (2) 
         [0000]    where
       E 1   tan ≡tangential component of incident+reflected E−field at x=0   E 2   tan ≡tangential component of transmitted E−field at x=0   H 1   tan ≡tangential component of incident+reflected H−field at x=0   H 2   tan ≡tangential component of transmitted H−field at x=0   η 0 ≡√{square root over (μ 0 /ε 0 )}≅377=free space impedance       
 
         [0031]    The program used for this work is designed to make this computation at optical frequencies. The output of this computer program is the reflection coefficient of the reflected wave, expressed in decibals (dB). A decibal is 10.0 times the log to the base  10  of the reflected power relative to the power of the incident wave. 
         [0032]    The second computer program as shown in  FIG. 3  is an optimization program that implements the downhill simplex method (see, e. g. Numerical Recipes, by W. H. Press et. al. (Cambridge, Cambridge University Press, 1986), page 289). This program writes an input file to be used by the first program, calls the first program to compute the reflection coefficient, and then reads the output file of the optical_stack program. It computes a figure of merit for the set of parameter values, the repeats the process. Both of these Fortran programs are compiled by the open source GNU Fortran compiler. The quantities to be optimized are the thickness of each layer and the effective relative loss of the CdTe layer. The figure of merit of this computation is the largest reflection coefficient of the incident light waves at frequencies down to wavelengths of 796 nanometers. 796 nanometers corresponds to the longest wavelength of light that has enough energy to free an electron from a CdTe molecule, or 1.56 electron volts. Longer wavelengths of light do not have enough energy to be absorbed by the CdTe layer. 
         [0033]    The three thicknesses of the three material layers as well as the effective relative electrical loss in the CdTe layer define a point on a four dimensional surface. The reflection coefficient of the incident light can be viewed as the height above this four dimensional surface. The downhill simplex, or amoeba method, starts with 5 initial sets of values. The incident values and the reflection coefficient, and 4 more sets of values consisting of the incident values with each of the four initial values increased by a percentage, for example, 20%. The optimizer then chooses a new set of values to move away from high points on the 4 dimensional surface. 
         [0034]    Eventually, the optimizer can no longer reduce the largest reflection coefficient. The entire process then repeats with the initial parameter values consisting of the values that previously gave the lowest reflection coefficient. This happens in case the best set of values represent a local null on the four dimensional surfaces. Restarting the process sometimes allows the search to climb out of a local null and find a true global null. This is done because there is no optimization procedure yet discovered that can assure a global null. 
         [0035]    This process optimizes a two layer antireflection coating along with the layers of CdS and CdTe. These programs are written to assume that all thicknesses are in thickness units of Angstroms. Since most optical thicknesses are currently expressed in nanometers (1 nanometer=10 Angstroms). The optical stack optimization code has been modified to compute the efficiency. It uses the measured solar spectrum. The maximum power in watts was computed for each 20 nanometer bin of the measured solar spectrum and power absorbed was computed for the incident light in each bin. The optimizer computes the power generated assuming that the photon absorption gives rise to an effective loss in the material stack. All photons at or above the 1.56 eV CdTe band gap are assumed to be absorbed and produce 1.56 eV of energy. Photon energy above 1.56 eV is assumed to be lost as heat. Photons with energy below 1.56 eV are not absorbed. The total solar energy incident on 1 square meter is 1000 watts using this solar power data set. A 165 nm thick CdTe/CdS layer was assumed. All freed electrons are assumed to travel to the bottom conductor without being re-absorbed. The optimization parameters are the thickness of the glass antireflection coating on top of the stack and the effective loss in the CdS/CdTe material. 
         [0036]    The efficiency of the Nano Structure CdTe/CdS PTSC is limited by the band gap of the CdTe (approximately 1.56 electron volts). Photons entering the top of the cell that have energy below the band gap cannot free an electron. They escape through the top of the solar cell or are absorbed as heat. Photons with energy at or above the band gap energy can strike a molecule on one of its bounces through the CdTe and free an electron. The energy of any absorbed photon above that of the CdTe band gap is lost as heat. The thinness of the CdTe layer makes it likely that many photons will traverse the entire layer without being absorbed. Photons reaching the bottom conductor are reflected back through the CdTe. The thickness of the CdTe layer is optimized in this Invention so that photons reaching the top surface are 180 degrees out of phase with incident photons, preventing the photons from escaping and forcing them to make another pass through the CdTe layer, further increasing the chance of being absorbed. In this way the CdS and CdTe layers themselves act similar to an antireflection coating making this Invention unique. 
         [0037]    The maximum efficiency of this type of solar cell is determined by the number of photons with energy above 1.56 electron volts that strike a CdTe atom and free an electron that travels to the bottom conductor. The PTSC solar cell design using the measured solar spectrum indicates that the maximum efficiency of this PTSC solar cell is 40%. This assumes that all photons with energy above the CdTe band gap are absorbed and that all freed electrons travel to the bottom conductor without being reabsorbed. When compared to the known existing standard CdTe solar cells of today, their efficiency is limited (Approximately 9%) because most electrons get reabsorbed by the thin film thickness (thousands of nanometers) CdTe before they can reach the bottom conductor. In this Invention, the Nano Structure PTSC solar cell, electrons need to travel only a maximum of 165 nanometers, greatly increasing the probability of them reaching the bottom conductor before being reabsorbed. Considering material impurities and crystalline imperfections in mass production of the PTSC, efficiencies of greater than 32 percent are achievable. 
       EXAMPLE 
       [0038]    Shown below is some of the output of a typical optimization run using programs optical_stack.f and optical_stack_opt.f. This run optimizes a two layer antireflection coating along with the layers of CdS and CdTe. These programs are written to assume that all thicknesses are in thickness units of Angstroms. Since most optical thicknesses are currently expressed in nanometers (1 nanometer=10 Angstroms), the final design thicknesses are converted from Angstroms to nanometers. 
         [0000]    
       
         
               
             
           
               
                   
               
             
             
               
                 #AMOEBA SUMMARY FILE GENERATED Sat Mar 1 11:10:44 2008 
               
               
                 #NUMBER OF ANGLES = 10 
               
               
                 #Layer 1 Thick. Layer 2 Layer 3 Layer 4 loss Layer 4 Figure of merit 
               
               
                 # 760.000 450.000 300.000 0.500 3000.000 −3.936 
               
               
                 # 988.000 450.000 300.000 0.500 3000.000 −4.426 
               
               
                 # 760.000 585.000 300.000 0.500 3000.000 −3.621 
               
               
                 # 760.000 450.000 390.000 0.500 3000.000 −3.598 
               
               
                 # 760.000 450.000 300.000 0.650 3000.000 −5.055 
               
               
                 # 760.000 450.000 300.000 0.500 3900.000 −3.557 
               
               
                 #END OF INITIALIZATION # 1 
               
               
                 # 851.200 504.000 336.000 0.560 2100.000 −2.292 
               
               
                 # 782.800 463.500 309.000 0.515 3450.000 −3.708 
               
               
                 #... ~ 200 more iterations 
               
               
                 # 
               
               
                 #TOTAL TIME FOR JOB (SEC) = 62 
               
               
                 #BEST PARAMETER VALUES AND FOM: 
               
               
                   764.746 434.571 96.430 2.138 4570.444 −19.775 
               
               
                   
               
             
          
         
       
     
         [0039]    The optimizer computes the power generated assuming that the photon absorption gives rise to an effective loss in the material stack. All photons at or above the 1.56 eV CdTe band gap are assumed to be absorbed and produce 1.56 eV of energy. Photon energy above 1.56 eV is assumed to be lost as heat. Photons with energy below 1.56 eV are not absorbed. The total solar energy incident on 1 square meter is 1000 watts using this solar power data set. A 165 nm thick CdTe/CdS layer was assumed. All freed electrons are assumed to travel to the bottom conductor without being re-absorbed. The optimization parameters are the thickness of the glass antireflection coating on top of the stack and the effective loss in the CdS/CdTe material. The optimizer came up with 32.5% efficiency for the example shown below. The last few iterations of the optimizer are: 
         [0000]    
       
         
               
             
               
               
               
               
             
               
               
               
               
               
             
               
               
             
               
               
               
               
               
             
               
               
             
               
               
               
               
               
             
               
               
             
           
               
                   
               
               
                 Efficiency Calculation 
               
             
          
           
               
                   
                 Anti reflection 
                 CdS/CdTe 
                 Efficiency 
               
               
                   
                   
               
             
          
           
               
                   
                 # 
                 769.219 
                 3.644 
                 0.675 
               
             
          
           
               
                   
                 EFFICIENCY FOR ABOVE PARAMETERS = 32.474% 
               
             
          
           
               
                   
                 # 
                 999.984 
                 3.644 
                 0.706 
               
             
          
           
               
                   
                 EFFICIENCY FOR ABOVE PARAMETERS = 29.428% 
               
             
          
           
               
                   
                 # 
                 769.219 
                 4.737 
                 0.679 
               
             
          
           
               
                   
                 EFFICIENCY FOR ABOVE PARAMETERS = 32.079% 
               
               
                   
                 # END OF P(MP,NP) INITIALIZATION # 3 
               
               
                   
                 #