Abstract:
A method for characterizing a semiconductor sample, said method comprising: shining light on one or more points in said semiconductor sample; measuring one or more voltage decay curves corresponding to said shining of light on said one or more points in said semiconductor sample; extracting one or more intermediate voltage decay curves corresponding to one or more measured voltage decay curves; obtaining one or more normalized decay curves corresponding to one or more intermediate voltage decay curves, each of the said one or more normalized decay curves corresponding to one or more discrete estimates of survival functions; and analyzing said obtained one or more normalized decay curves, said analyzing comprising obtaining one or more discrete estimates of the probability of recombination corresponding to the one or more normalized decay curves, and computing one or more summary statistics corresponding to each of said obtained one or more discrete estimates.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation-in-part of and claims priority to U.S. patent application Ser. No. 14/336,046 filed on Jul. 21, 2014; and claims the benefit of U.S. Provisional Patent Application No. 62/019,460, filed on Jul. 1, 2014, each of which is incorporated herein by reference in its entirety. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present disclosure relates to characterization of semiconductors. 
       BRIEF SUMMARY 
       [0003]    A method for characterizing a semiconductor sample, said method implemented using an analysis system comprising: a transient photoconductive decay measurement subsystem, a database, a data analysis subsystem, and a statistical analysis subsystem, said transient photoconductive decay measurement subsystem, database, data analysis subsystem and statistical analysis subsystem connected to each other by an interconnection, said method comprising the steps of: shining, using the transient photoconductive decay measurement subsystem, light on one or more points in said semiconductor sample; measuring, using the transient photoconductive decay measurement subsystem, one or more voltage decay curves corresponding to said shining of light on said one or more points in said semiconductor sample; extracting, using the data analysis subsystem, one or more intermediate voltage decay curves corresponding to one or more measured voltage decay curves; obtaining, using the data analysis subsystem, one or more normalized decay curves corresponding to one or more intermediate voltage decay curves, each of the said one or more normalized decay curves corresponding to one or more discrete estimates of survival functions; analyzing, using the statistical analysis subsystem, said obtained one or more normalized decay curves, said analyzing comprising obtaining one or more discrete estimates of the probability of recombination corresponding to the one or more normalized decay curves, and computing one or more summary statistics corresponding to each of said obtained one or more discrete estimates; and determining, by either the statistical analysis subsystem or the data analysis subsystem, the presence of nonuniformities within said semiconductor sample based on results of said analyzing. 
         [0004]    A method for characterizing a semiconductor sample, said method comprising: measuring a plurality of voltage decay curves corresponding to a plurality of points in said semiconductor sample; extracting a plurality of intermediate voltage decay curves corresponding to said plurality of measured voltage decay curves; converting said extracted plurality of intermediate voltage decay curves to a plurality of minority carrier population decay curves; performing one or more comparisons of survival behavior using said plurality of minority carrier population decay curves; and determining presence of nonuniformities within said semiconductor sample using said results from said performing of one or more comparisons of survival behavior. 
         [0005]    A method for comparing a plurality of semiconductor samples, said method comprising: measuring a plurality of voltage decay curves corresponding to a plurality of points from said plurality of semiconductor samples; extracting a plurality of intermediate voltage decay curves corresponding to said plurality of measured voltage decay curves; converting said extracted plurality of intermediate voltage decay curves to a plurality of minority carrier population decay curves; performing one or more comparisons of survival behavior using said plurality of minority carrier population decay curves; and determining presence of nonuniformities within said plurality of semiconductor samples using said results from said performing of one or more comparisons of survival behavior. 
         [0006]    The foregoing and additional aspects and embodiments of the present disclosure will be apparent to those of ordinary skill in the art in view of the detailed description of various embodiments and/or aspects, which is made with reference to the drawings, a brief description of which is provided next. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]    The foregoing and other advantages of the disclosure will become apparent upon reading the following detailed description and upon reference to the drawings. 
           [0008]      FIG. 1  shows the steps used in the prior art analysis approaches 
           [0009]      FIG. 2  shows an example analysis setup. 
           [0010]      FIG. 3  shows a sample voltage decay curve  300 . 
           [0011]      FIG. 4  shows a grid of points for mapping spatial nonuniformities across a semiconductor sample. 
           [0012]      FIG. 5A  shows one embodiment of an analysis performed on a voltage decay curve. 
           [0013]      FIG. 5B  shows an example of extracting readings to form an intermediate voltage decay curve V(t). 
           [0014]      FIG. 6  shows one embodiment of step  504 . 
           [0015]      FIG. 7  shows an embodiment of a process to perform comparison between two or more samples, or two or more locations within the same sample. 
           [0016]      FIG. 8  shows an example flowchart to calculate and compare summary statistics. 
           [0017]      FIG. 9  shows a process to estimate an upper bound t win  to calculate the proportion of recombination events taking place within a localized region of interest compared to the number of recombination events taking place within the entire semiconductor sample. 
           [0018]      FIG. 10  shows a process to estimate an upper bound t win  to calculate the proportion of minority carriers within the localized area as compared to the entire semiconductor sample. 
       
    
    
       [0019]    While the present disclosure is susceptible to various modifications and alternative forms, specific embodiments or implementations have been shown by way of example in the drawings and will be described in detail herein. It should be understood, however, that the disclosure is not intended to be limited to the particular forms disclosed. Rather, the disclosure is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of an invention as defined by the appended claims. 
       DETAILED DESCRIPTION 
       [0020]    While the description here focuses on analysis of results obtained using the transient photoconductive decay technique, it is equally applicable to the analysis of results obtained using other similar techniques, that is, where a population is generated, and a measurable output related to the survival of the generated population is measured. 
         [0021]    Similarly while many of the examples here are discussed in relation to mercury cadmium telluride (HgCdTe), the analyses are equally applicable to any semiconductor material. 
       Introduction 
       [0022]    In the transient photoconductive decay technique, in one embodiment a current is passed through a sample which is to be measured. As a result, a voltage is created across the sample. The voltage is proportional to the resistance of the sample, which in turn is proportional to the resistivity and hence inversely proportional to the conductivity of the sample. 
         [0023]    Then, light is shone upon the sample to generate electron hole pairs. Depending on whether the sample is p-type or n-type, either electrons or holes are the minority carriers. As a result, the conductivity of the sample increases, leading to a drop in the resistance of the sample. As a consequence of the resistance drop, the voltage across the sample will also drop. 
         [0024]    The generated minority carriers drift under the influence of the bias field created by the voltage and diffuse throughout the sample due to the concentration gradient. Over time, these minority carriers will also recombine within the sample. As a consequence of the recombination of the minority carriers, the conductivity will decay to its value before the light was shone upon the sample. As the conductivity decays so does the resistance, and consequently the voltage across the sample will also increase. 
         [0025]    The aim of the transient photoconductive decay technique is to analyse the decay of the induced transient increase in conductivity by measuring the change in voltage across the sample. The sample can be characterized using this technique. 
         [0026]    A variation of the transient photoconductive decay technique is spatial mapping. In this variation, light is shone on different locations of the sample, and the resultant transient photoconductive decay curves are measured. By doing so, spatial variations across a sample can be characterized. 
       Material Parameters 
       [0027]    In previous works, several parameters of interest have been characterized using the transient photoconductive decay technique. Some of these parameters are described in section 1.3 of R. Rajaduray, “Investigation of Spatial Characterisation Techniques in Semiconductors,” Honours Thesis 1998, University of Western Australia, herein incorporated by reference as if reproduced in its entirety. 
         [0028]    Two parameters of interest which have been characterized in previous works are the bulk minority carrier lifetime and the surface recombination velocity. 
         [0029]    The bulk minority carrier lifetime τ b  is the average time a minority carrier identified at a particular instant and location within the bulk of a semiconductor will exist until recombination. It is defined by: 
         [0000]    
       
         
           
             
               τ 
               b 
             
             = 
             
               
                 p 
                 - 
                 
                   p 
                   o 
                 
               
               R 
             
           
         
       
     
         [0030]    where
       p is the total minority carrier density   p o  is the equilibrium minority carrier density   R is the minority carrier recombination rate       
 
         [0034]    The bulk minority carrier lifetime τ b  is highly dependent upon the nature of the recombination mechanisms within the bulk of the semiconductor. 
         [0035]    The surface recombination velocity s is a measure of the recombination rate of minority carriers at the surface of a semiconductor. It is defined for excess holes in an n-type semiconductor with a surface at x=0 by: 
         [0000]    
       
         
           
             s 
             = 
             
               
                 
                   D 
                   a 
                 
                  
                 
                   
                     ∂ 
                     p 
                   
                   
                     ∂ 
                     x 
                   
                 
                  
                 
                   1 
                   p 
                 
               
                
               
                 | 
                 
                   x 
                   = 
                   0 
                 
               
             
           
         
       
     
         [0036]    where
       D a  is the ambipolar diffusion coefficient   p is the excess hole concentration       
 
         [0039]    For excess electronics in a p-type semiconductor with a surface at x=0: 
         [0000]    
       
         
           
             s 
             = 
             
               
                 
                   D 
                   a 
                 
                  
                 
                   
                     ∂ 
                     n 
                   
                   
                     ∂ 
                     x 
                   
                 
                  
                 
                   1 
                   n 
                 
               
                
               
                 | 
                 
                   x 
                   = 
                   0 
                 
               
             
           
         
       
     
         [0040]    where
       D a  is the ambipolar diffusion coefficient   n is the excess hole concentration       
 
         [0043]    Physically, the surface recombination velocity can be understood as follows: A current of holes or electrons of density p or n drift with an average velocity equal to the surface recombination velocity s into the surface and the holes or electrons are then removed. Thus, as the surface recombination velocity increases, the excess hole or electron concentration at the surface decreases. 
       Recombination Mechanisms 
       [0044]    A detailed explanation of examples of various bulk recombination mechanisms in, for example, HgCdTe is given in Sections 2.2.1 to 2.2.3 of R. Rajaduray, “Investigation of Spatial Characterisation Techniques in Semiconductors,” Honours Thesis 1998, University of Western Australia. 
         [0045]    For example, with reference to HgCdTe three important bulk recombination mechanisms are Auger, radiative and Shockley-Read-Hall (SRH) recombination. Auger and radiative recombination are strongly dependent upon the carrier concentrations and energy gap. SRH recombination is associated with the presence of defect states within the bandgap, known as traps. 
         [0046]    There may be other bulk recombination mechanisms present in other semiconductor materials. 
         [0047]    Similarly, an explanation of surface recombination mechanisms in HgCdTe is given in section 2.3 of R. Rajaduray, “Investigation of Spatial Characterisation Techniques in Semiconductors,” Honours Thesis 1998, University of Western Australia. 
         [0048]    Three surface recombination mechanisms in HgCdTe are:
       Thermal transitions through Shockley-Read-Hall centres in the depletion region: This process is similar to the SRH bulk recombination mechanism.   Thermal transitions via fast surface states.   Tunnel transitions through the Shockley-Read-Hall centres in the depletion region       
 
         [0052]    There may be other surface recombination mechanisms present in other semiconductor materials. 
         [0053]    Previous Analysis Approaches 
         [0054]    Many of the existing analysis approaches are based on parametric techniques.  FIG. 1  shows the steps involved in the prior art analysis approaches:
         101 : Creating a model based on one or more assumptions     102 : Setting up one or more differential equations with boundary conditions based on the assumptions     103 : Solving the differential equations to obtain a solution with one or more parameters which shows the expected behavior of the decay of the minority carrier population over time, and     104 : Fitting experimental results to the obtained solution using, for example, least squares fitting to extract the one or more solution parameters.       
 
         [0059]    Two examples of steps  101 - 103  are explained below. The first example uses the approach detailed in W. Van Roosbroeck, “Injected Current Carrier Transport in a Semi-Infinite Semiconductor and the Determination of Lifetimes and Surface Recombination Velocities.” Journal of Applied Physics 26.4 (1955): 380-391. The solution is given as: 
         [0000]        p ( U )= p (0)exp[ U ( S   2 −1)] erfc[S√{square root over (U)}]   (1)
 
         [0060]    where
       U is time t normalized with respect to τ b      τ b  is the bulk minority carrier lifetime   p(U) is the minority carrier population at normalized time U or at time t=U×τ b      p(0) is the minority carrier at normalized time U=0 or equivalently t=0   S is the normalized surface recombination velocity. S is further given by:       
 
         [0000]    
       
         
           
             S 
             = 
             
               
                 s 
                  
                 
                     
                 
                  
                 
                   τ 
                   b 
                 
               
               L 
             
           
         
       
     
         [0066]    where
       s is the surface recombination velocity   L is the minority carrier diffusion length, given by √(D a ×τ b )       
 
         [0069]    The second example of steps  101 - 103  provides a solution for a finite rectangular sample of dimensions  2 A,  2 B and  2 C and uses the approach detailed in J. S. Blakemore, Semiconductor Statistics, Oxford Pergamon 1962. The solution is given as a series of eigenfunctions for different modes (i,j,k) and is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ABC 
                     
                      
                     
                       
                         ∑ 
                         ijk 
                       
                        
                       
                         
                           K 
                           ijk 
                         
                         × 
                         
                           exp 
                            
                           
                             [ 
                             
                               - 
                               
                                 t 
                                  
                                 
                                   ( 
                                   
                                     
                                       v 
                                       b 
                                     
                                     + 
                                     
                                       v 
                                       ijk 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0070]    where
       p(t) is the minority carrier population at time t   p(0) is the minority carrier at time t=0   K ijk  is the constant for mode (i, j, k)   ν ikj  is the inverse of the time constant for mode (i, j, k)   ν b  is the inverse of the bulk minority carrier lifetime τ b          
 
         [0076]    Then, once a solution such as in equations (1) and (2) above have been provided, experimentally obtained decay curves can be fitted to these curves using, for example, least squares regression. The parameters used to obtain the best fit are extracted and recorded. For example, using the Van Roosbroeck model, the surface recombination velocity s and bulk minority carrier lifetime τ b  to obtain the best fit are extracted. 
         [0077]    There are other analysis approaches which are variations of these 2 approaches. Usually, these variations employ slightly different assumptions to create a model. However many of these approaches are flawed for several reasons. 
         [0078]    Many of the existing approaches use models which employ unrealistic assumptions and then set up differential equations and boundary conditions based on these unrealistic conditions. 
         [0079]    For example, firstly many of the models assume that recombination parameters such as the bulk minority carrier lifetime and the surface recombination velocity are spatially and temporally constant within the analyzed semiconductor sample. This has clearly been shown not to be the case. Studies such as those performed by V. C. Lopes et at “Characterization of (Hg, Cd)Te by the Photoconductive Decay Technique,” J. Vac Sci, vol. 8, no. 2, pp. 1167-1170, March/April 1990; and R. G. Pratt et at “Minority carrier lifetime in n-type Bridgman grown Hg 1-x Cd x Te,” J. Appl. Physics vol 54, no. 9 pp. 5152-5157, 1983; showed spatial nonuniformity of bulk lifetime across semiconductor samples. Furthermore, as shown in Chapter 6 of Ramesh Rajaduray, “Investigation of Spatial Characterisation Techniques in Semiconductors,” Honours Thesis 1998, University of Western Australia, the extracted parameters clearly exhibited temporal nonuniformity, that is, when segments of a voltage decay curve with differing temporal extents were fitted to equation (1), the values of the extracted parameters were non-uniform. 
         [0080]    Secondly, many of the models assume that the dominant recombination mechanism is independent of the minority carrier density. This is also unrealistic, when it has been shown in that in certain situations, minority carrier concentration dependent recombination mechanisms such as Auger recombination will dominate in materials such as HgCdTe. As an example, in pages 53 and 54 of Ramesh Rajaduray, “Investigation of Spatial Characterisation Techniques in Semiconductors,” Honours Thesis 1998, University of Western Australia, it was shown that at time t=0, a population of minority carriers equivalent to 19% of the total number of minority carriers is generated within a small area. It was shown in, for example, G. Nimtz, et al. “Transient carrier decay and transport properties in Hg 1-x Cd x Te.” Phys. Rev. BIO p 3302 (1974); and F. Bartoli et al. “Auger-limited carrier lifetimes in HgCdTe at high excess carrier concentrations.” Journal of Applied Physics vol. 45 no. 5 pp. 2150-2154 (1974); that under such conditions Auger recombination is likely to dominate over radiative and Shockley-Read-Hall mechanisms. 
         [0081]    Furthermore, as was pointed out by D. A. Redfern et al “On the transient photoconductive decay technique for lifetime extraction in HgCdTe” in Optoelectronic and Microelectronic Materials Devices, 1998. Proceedings. 1998 Conference on, pp. 275-278. IEEE, 1999, “none of the current models unambiguously [explained] experimental results and that detailed lifetime extraction by photoconductive decay is still not a quantitative technique.” 
         [0082]    In addition, the generation and recombination of minority carriers which occur within a semiconductor sample each time light is incident on the sample, are random processes. As a consequence, carrier concentrations at particular points within a semiconductor sample are also likely to vary randomly as well. This means that diffusion based movements, which are highly dependent on concentration gradients, are also likely to be random in nature. As a consequence, this further intensifies the random behavior of the carrier concentration at a particular point within a semiconductor sample. If carrier concentration dependent recombination mechanisms dominate, then the random behavior is even further intensified. However many of the differential equations set up in steps  101 - 104  of  FIG. 1  above are assumed to be deterministic in nature. 
         [0083]    As a consequence of the above, many of the previously proposed models employ unrealistic assumptions which lead to an incorrect understanding of the evolution of the population of generated minority carriers over time within a semiconductor sample. 
         [0084]    As a further consequence, analysis approaches which use such models to perform spatial mapping of recombination parameters, such as, for example, spatial bulk minority carrier lifetime mapping are inherently flawed. Not only is the understanding of the evolution of the population over time wrong, but the incorrect behavior is then used to detect parameter variations which is fundamentally opposite to the assumptions employed. 
         [0085]    Therefore there is a need for analysis approaches which are less reliant on using models with inherently unrealistic assumptions to perform parametric-based analysis, or worse still: Using models built on certain assumptions with the aim of detecting properties which are in direct opposition to these assumptions. 
       New Analysis Approaches 
       [0086]    This section demonstrates several analysis approaches which overcome the problems due to the parametric analysis approaches used previously. 
         [0087]    An example analysis setup is shown in  FIG. 2 . Transient photoconductive decay measurement subsystem  201  is used to obtain voltage decay curves for a semiconductor sample. In one embodiment, transient photoconductive decay measurement subsystem  201  is similar to that detailed in section 5.3 of Ramesh Rajaduray, “Investigation of Spatial Characterisation Techniques in Semiconductors,” Honours Thesis 1998, University of Western Australia. In another embodiment, transient photoconductive decay measurement subsystem  101  is similar to that used in Pratt et at “Minority carrier lifetime in n-type Bridgman grown Hg 1-x Cd x Te” Journal of applied physics 54, no. 9 (1983): 5152-5157, and herein incorporated by reference in its entirety. A further example is given in T. Tomlin, “Spatial Mapping of Minority carrier lifetime in Mercury Cadmium Telluride,” Honours Thesis 1995, University of Western Australia. In one embodiment, as shown in  FIG. 3 , an obtained voltage decay curve  300  denoted as V me (t) comprises a plurality of measurements comprising measurements  305 ,  306  and  307  of the voltage across the sample at corresponding times  301 ,  303 , and  304 . The time instant corresponding to each measurement within the plurality is separated from the time instant corresponding to the preceding measurement by a time interval Δt ( 302 ), such as shown in  FIG. 3 . 
         [0088]    Statistical analysis subsystem  204  performs statistical analyses which will be described later. In one embodiment, the statistical analysis subsystem  204  is implemented in hardware. In one embodiment, the statistical analysis subsystem  204  is implemented in software. In yet another embodiment, statistical analysis subsystem  204  is implemented in a combination of hardware and software. Different programming languages and systems can be used to implement statistical analysis subsystem  104 , including, for example, SPSS, S, R, STATA, Matlab™, SAS, Microsoft™ Excel™, SQL and C++. 
         [0089]    Data analysis subsystem  203  performs various functions, including preparing data for statistical analysis subsystem  204 , collating the results of analysis performed by statistical analysis subsystem  204 , performing further analysis of the results from statistical analysis subsystem  204  and presenting the results of these analyses. In one embodiment, the data analysis subsystem  203  is implemented in hardware. In one embodiment, the data analysis subsystem  203  is implemented in software. In yet another embodiment, data analysis subsystem  203  is implemented in a combination of hardware and software. Different programming languages and systems can be used to implement statistical analysis subsystem  204 , including, for example, SPSS, S, R, STATA, Matlab™, SAS, Microsoft™ Excel™, SQL and C++. 
         [0090]    Database  205  is used to store voltage decay curve data obtained from transient photoconductive decay subsystem  201 , and data for intermediate processing performed by statistical analysis subsystem  204 , data analysis subsystem  203 . Different programming languages and systems can be used to implement statistical analysis subsystem  204 , including, for example, SQL and Microsoft™ Access™. 
         [0091]    Interconnection  202  is used to connect the different subsystems together. These could include, for example, local area networks (LAN), campus area network (CAN), wide area networks (WAN). Interconnection  202  could encompass one or more subnetworks. Interconnection  202  could be implemented using various media including wireless, wired, optical network, and could encompass various technologies including Ethernet and IP-based networks. 
         [0092]    In one embodiment, the system illustrated in  FIG. 2  is used to compare one or more semiconductor samples. Then, for each of the one or more semiconductor samples, one or more voltage decay curves such as voltage decay curve  300  in  FIG. 3 , is measured using, for example, transient photoconductive decay measurement subsystem  201 . 
         [0093]    In another embodiment, the system illustrated in  FIG. 2  is used to map spatial non-uniformities across a single semiconductor sample. In this embodiment, using transient photoconductive decay measurement subsystem  201 , light is shone at different points across a semiconductor sample, and for each point a voltage decay curve is obtained. For example, in one embodiment, light is shone at each point, for example points  401 ,  402  and  403  within a grid of points  404  such as shown in  FIG. 4  for sample  400  is created. Voltage decay curves such as voltage decay curve  300  of  FIG. 3  as shown above are then obtained for each point. 
         [0094]    As explained previously, the generation, movement and recombination of minority carriers which occurs within a semiconductor sample each time light is incident on a spot on the semiconductor sample using the transient photoconductive decay measurement subsystem  201  are random sub-processes which are part of a single overall random or stochastic process. Consequently, each obtained voltage decay curve represents the evolution of the population of minority carriers with time for one realization of this overall random process. 
         [0095]    In one embodiment, in order to remove the impact of noise, light is shone on the same spot on the semiconductor sample a plurality of times. Then, each time light is shone on the spot, a corresponding voltage decay curve is obtained. 
         [0096]    In one embodiment, the following analysis is applied to each obtained voltage decay curve as shown in  FIG. 5A . 
         [0097]    In step  501 , using for example, data analysis subsystem  203 , the segment of the measured voltage decay curve with times greater than the time corresponding to the peak of the voltage decay curve is extracted to form an intermediate voltage decay curve V(t). An example is shown in  FIG. 5B . The time  5 A- 01  corresponds to the peak ( 5 A- 02 ) of the obtained voltage decay curve  5 A- 00 . Then, the segment  5 A- 03  of the voltage decay curve  5 A- 00  for all times greater than time  5 A- 01  is extracted, to form intermediate voltage decay curve  5 A- 04 . Each time instant on intermediate voltage decay curve  5 A- 04  is separated from the next time measurement by Δt ( 5 A- 08 ), such as, for example, time instants  5 A- 05 ,  5 A- 06  and  5 A- 07 . The intermediate voltage decay curve  5 A-04 can be represented as V(nΔt), n=0, 1, 2 . . . N where nΔt are the time instants. 
         [0098]    In the embodiment where light is shone on the same spot a plurality of times and a corresponding measured voltage decay curve is obtained for each time, a plurality of corresponding intermediate voltage decay curves V k (nΔt) are obtained, k=1, 2, 3 . . . K. The corresponding intermediate voltage decay curves are then averaged out to provide a smoothed intermediate voltage decay curve V s (nΔt), that is: 
         [0000]    
       
         
           
             
               
                 V 
                 S 
               
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                   n 
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                   Δ 
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             = 
             
               
                 
                   Σ 
                   
                     k 
                     = 
                     1 
                   
                   K 
                 
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                     k 
                   
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         [0099]    This is performed for n=0, 1, 2 . . . N. 
         [0100]    In optional step  502 , using for example, data analysis subsystem  203 , the intermediate voltage decay curve V(nΔt) or smoothed intermediate voltage decay curve V s (nΔt) obtained in step  501  is converted to a minority carrier population decay curve p(t). Various approaches to perform this conversion are known to those of skill in the art and will not be explained further within this specification. 
         [0101]    In step  503 , in one embodiment, using for example, data analysis subsystem  203  the intermediate voltage decay curve V(nΔt) or smoothed intermediate voltage decay curve V s (nΔt) obtained in step  501  is normalized to the voltage at time t=0 to obtain a normalized decay curve V no (nΔt). In an alternate embodiment, if optional step  502  is performed, the p(t) obtained in step  502  is normalized to the minority carrier population at time t=0 to obtain a normalized decay curve p no (t). 
         [0102]    The normalized decay curve represents a discrete estimate S e (nΔt), n=0, 1, 2 . . . N of the continuous time survival function S(t)=P[τ&gt;t]. S(t) is the probability that minority carriers will survive, that is not recombine, until beyond time t. If, in step  501 , a smoothed intermediate voltage decay curve is provided as an output, then the obtained normalized smoothed decay curve is a better discrete estimate S e (t) of S(t). 
         [0103]    In step  504 , S e (t) is analysed using, for example, statistical analysis subsystem  204 . One embodiment of step  504  is shown in  FIG. 6 . The cumulative distribution function CDF(t)=P[τ≦t] is obtained by P[τ≦t]=1−S(t). In an optional embodiment, in step  601 , a discrete estimate CDF e (nΔt) of the cumulative distribution function CDF(t) is obtained by computing 1−S e (nΔt), n=0, 1, 2 . . . . 
         [0104]    In step  602 , in one embodiment, an estimate of the probability of recombination between time [nΔt] and [(n+1)Δt], n=0, 1, 2 . . . is obtained. The probability is given by the probability mass function PMF e [(n+1)Δt], which is obtained by taking successive differences of the estimate of the survival function S e (t). That is, the estimate PMF e [(n+1)Δt], n=0, 1, 2 . . . is given by S e [nΔt]−S e [(n+1)Δt]. Alternatively, if in step  603 , CDF e (nΔt) is calculated, then PMF e [(n+1)Δt] is given by CDF e [(n+1)Δt]−CDF e [nΔt]. 
         [0105]    In step  603 , one or more summary statistics are computed. In one embodiment, an estimate of the mean of τ denoted as E[τ] or μ τ  is calculated. In one embodiment, S e (t) is used directly to calculate E[τ] performing the summation of S e (t) from n=1 onwards. In another embodiment PMF e [(n+1)Δt] is used. 
         [0106]    In another embodiment, the variance of τ denoted as Var(τ) or alternatively σ τ   2  is computed. 
         [0107]    Other summary statistical computations can be performed including moment generation, Laplace transform and characteristic function generation. Moments can also be calculated. Other expectations can also be calculated using the generalized formulae such as, for example E(1/τ 3 ) and E(1/τ 2 ). 
         [0108]    In another embodiment, in step  604 , one or more survival statistical computations are applied. In one embodiment, the discrete time hazard probability λ e [(n+1)Δt], which is the probability of recombination for a minority carrier between times [nΔt] and [(n+1)Δt] given that the minority carrier has not recombined before time [nΔt] is calculated. Mathematically this is given by: 
         [0000]    
       
         
           
             
               
                 λ 
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                       1 
                     
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                 ] 
               
             
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                   ) 
                 
               
             
           
         
       
     
         [0109]    Alternatively it can be calculated as: 
         [0000]    
       
         
           
             
               
                 λ 
                 e 
               
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                 [ 
                 
                   
                     ( 
                     
                       n 
                       + 
                       1 
                     
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                    
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                 ] 
               
             
             = 
             
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                     e 
                   
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                     ] 
                   
                 
                 
                   
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         [0110]    Quantiles of S e (nΔt) can be calculated as well. For example, the lowest decile of S e (nΔt), that is, the time after which 90% of the minority carrier population at t=0 have not recombined can be calculated by determining when S e (nΔt) drops below 0.90. Similarly the highest quartile of S e (nΔt), that is, the time after which 25% of the minority carrier population at t=0 have not recombined can be calculated by determining when S e (nΔt) drops below 0.25. 
         [0111]    Another survival statistical computation which can be performed is calculating the mean residual time E(τ−nΔt|τ≧nΔt). This gives the expected time until recombination for a minority carrier, given that the minority carrier survived up to time nΔt. This can be calculated using well known mathematical formulas and will not be discussed in detail within this specification. 
         [0112]    It may be necessary to compare minority carrier decay behavior for two or more samples, or at two or more locations within the same sample, to determine if there is nonuniformity between samples or whether there is spatial nonuniformity within the same sample. Then, known mathematical techniques to compare survival behaviour for different populations of generated minority carriers can be employed. This involves using a process similar to that outlined in  FIG. 5A , except that step  503  is not performed. An embodiment is shown in  FIG. 7 . Steps  701  and  702  are identical to steps  501  and  502 , except that step  502  is not optional. These two steps are performed for every sample, or for every point or location within the same sample. These steps are implemented using for example, data analysis subsystem  203  and statistical analysis subsystem  204  as previously detailed. 
         [0113]    In step  704 , the minority carrier population decay curves obtained in step  702  are analyzed to perform comparisons between samples. In one embodiment, in step  704 , methods of semiparametric testing are used to detect spatial nonuniformities in the semiconductor sample or differences between semiconductor samples. In one embodiment, as described in, for example, p. 251-266 of N. Balakrishnan, and C. R. Rao “Handbook of statistics: advances in survival analysis. Vol. 23” Access Online via Elsevier, 2004. the Cox proportional hazard analysis model is used. This assumes that the discrete time hazard probabilities λ e [(n+1)Δt] for the samples are proportional to each other. In a further embodiment, results can be tested for validity of the proportional hazard assumption. Examples of tests for validity are described in D. Schoenfeld, “Partial residuals for the proportional hazards regression model.” Biometrika vol. 69 no 0.1 pp. 239-241 (1982); and T. M. Thernau et at “Modeling Survival Data: Extending the Cox Model” New York: Springer-Verlag 2000. 
         [0114]    In another embodiment, in step  704 , various nonparametric comparison techniques can be used to analyse the p(t) obtained in step  702 . In one embodiment, the Mantel-Cox or logrank test is used, as described in Section 7.3 and 7.7 of Klein et at “Survival Analysis: Techniques for Censored and Truncated Data” Springer, 1997. In another embodiment, the Gehan-Breslow test is used as described in the references E. A. Gehan, “A generalized Wilcoxon test for comparing arbitrarily singly-censored samples.” Biometrika vol 52, no. 1-2 pp. 203-223 (1965); and N. Breslow “A generalized Kruskal-Wallis test for comparing K samples subject to unequal patterns of censorship.” Biometrika vol. 57 no. 3 pp. 579-594 (1970). In another embodiment, the Tarone-Ware test is used as described in the reference R. E. Tarone et at “On distribution-free tests for equality of survival distributions.” Biometrika vol 64 no. 1 pp. 156-160 (1977). In yet another embodiment, the tests proposed in T. R. Fleming et at “Counting processes and survival analysis” Vol. 169. Wiley.com, 2011 are used. In another embodiment, one or more such comparisons are performed, depending on, for example, whether the survival curves to be compared cross with each other or the requirements of the analysis. In one embodiment, these tests are performed by statistical analysis subsystem  203 . In another embodiment, these tests are performed by a combination of statistical analysis subsystem  203  and data analysis subsystem  204 . 
         [0115]    In yet another embodiment, in step  704 , a combination of the previously described Cox proportional hazards analysis approach and the nonparametric approaches described above are used. Firstly, a visual check is performed to see if the discrete time hazard probabilities for the samples cross. If not, then the validity of the assumption of proportional hazards is tested. If the assumption of proportional hazards is valid, then the logrank test is used. If the discrete time hazard probabilities cross, a different test is used, such as the test outlined in A. Renyi “On the Theory of Order Statistics” Acta Mathematica Hungarica vol. 4 pp. 191-231, 1953. 
         [0116]    In a further embodiment, in step  704 , one or more combinations of analyses are performed. For example, once the non-parametric tests have been performed and differences between the populations have been observed, then the summary statistics for each sample or each point can be calculated and compared. An example flowchart is shown in  FIG. 8 . Steps  801 - 803  are similar to steps  601 - 603  respectively, and performed for each sample or each point within a sample using, for example, data analysis subsystem  203  and statistical analysis subsystem  204  as previously detailed. 
         [0117]    The advantage of the new analysis approaches over the previous parametric analysis approaches is that there are no assumptions of the form of the minority carrier population decay curve p(t). This therefore overcomes the problems due to relying on the use of models with unrealistic assumptions. By using minority carrier population decay curves p(t) or converting to a normalized decay curve and applying the understanding that this can be converted to a discrete time estimate of the CDF, methods of probabilistic analysis can be applied as described above without having to perform fitting to models which are inherently unrealistic. 
         [0118]    In an additional embodiment, one or more “windows” of interest are determined. Each of these windows comprises a lower bound and an upper bound, and the decay curve between these bounds is extracted. Then one or more statistical computations are performed using these one or more windows. For example, in one embodiment the mean of the values within the window given by E[τ|n 1 Δt≦τ≦n 2 Δt]; n 1 =0, 1, 2 . . . is computed using known formulas. Similarly other computations such as calculation of variance, Laplace transform, characteristic function, moments can also be performed. In another embodiment, the survival statistical computations and the comparison of survival behavior techniques outlined above and in  FIG. 7  are performed using these one or more windows. 
         [0119]    Various methods can be used to determine the window size. In one embodiment, in order to analyse the decay of the minority carrier population within a localized region of interest surrounding the point where minority carriers are generated by the incidence of light, a window with lower bound t=0 and upper bound t=t win  is set. The upper bound can be set in a variety of ways. 
         [0120]    In one embodiment, t win  is estimated by calculating the proportion of recombination events taking place within the localized region of interest compared to the number of recombination events taking place within the entire semiconductor sample, using one of the previously derived models such as in equations (1) and (2). An example is presented in  FIG. 9 .
       Step  901 : Determining the region of interest R   Step  902 : Using the model, determining the partial time derivative of the minority carrier concentration       
 
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             Step  903 : Spatially integrating 
           
         
       
     
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         [0000]    within the region of interest R using, for example, a triple integral 
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             Step  904 : Determining the proportion κ that the ∫∫∫ R  ∂t/∂[p(x, y, z, t)] comprises of the overall 
           
         
       
     
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             Step  905 : Determining a threshold proportion κ T    
             Step  906 : Denoting the time when κ drops below κ τ  as t win . 
           
         
       
     
         [0127]    In another embodiment, t win  is estimated by using one of the previously derived models to estimate the proportion of minority carriers within the localized area as compared to the entire semiconductor sample. An example is presented in  FIG. 10 :
       Step  1001 : Determining the region of interest R   Step  1002 : Spatially integrating p(x, y, z, t) within region of interest R using the triple integral ∫∫∫ R  p (x, y, z, t)   Step  1003 : Determining the proportion κ that the ∫∫∫ R  p (x, y, z, t) comprises of the overall p(t), that is       
 
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             Step  1004 : Determining a threshold κ T  for the proportion 
             Step  1005 : Denoting the time when κ drops below κ T  as t win . 
           
         
       
     
         [0133]    An example of the approach in  FIG. 10  is provided in Chapter 7 of Ramesh Rajaduray, “Investigation of Spatial Characterisation Techniques in Semiconductors,” Honours Thesis 1998, University of Western Australia, for the model described in W. Van Roosbroeck, “Injected Current Carrier Transport in a Semi-Infinite Semiconductor and the Determination of Lifetimes and Surface Recombination Velocities.” Journal of Applied Physics 26.4 (1955): 380-391 as explained earlier. 
         [0134]    In another embodiment, t win  is determined using, for example, numerical simulations such as Monte Carlo simulations. 
         [0135]    In another embodiment, t win  is determined using, for example, historical results from previous experiments or other types of characterization techniques. 
         [0136]    In an embodiment, the setting of t win  is performed using data analysis subsystem  204 . In another embodiment, the setting of t win  is performed using a combination of data analysis subsystem  204  and statistical analysis subsystem  203 . 
         [0137]    In another embodiment, once t win  is known, then referring to  FIG. 3  the number of measurements (M) needed to perform a valid analysis is determined. Referring to  FIG. 3 , for example if a minimum of M min  samples are needed, then Δt ( 302 ) is set such that 
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         [0138]    In a further embodiment, clustering is performed. For example, in the case where tests are performed to spatially characterize a semiconductor sample, different data points are grouped into spatial clusters based on different clustering metrics. In one embodiment, the clustering metric is the probability that two samples are drawn from the same population, based on their survival curves. For example, referring to  FIG. 4 , if the probability that the survival curves belonging to points  401  and  402  are drawn from the same population is above a threshold, then it is likely that points  401  and  402  have very similar parameters, that is, there is no spatial variation between these points. Then  401  and  402  belong to the same cluster. However if the probability that the survival curves belonging to points  401  and  403  are drawn from the same population is below a threshold, then it is likely that there is spatial variation between points  401  and  403 . Then points  401  and  403  do not belong in the same cluster. 
         [0139]    Continuing the above example, assume that  402  and  403  also belong in the same cluster. Then two clusters for the points A, B and C can be created:
       Cluster 1: ( 401 ,  402 )   Cluster 2: ( 402 ,  403 )       
 
         [0142]    This is an example of overlapping clusters, that is, where a point belongs to a plurality of clusters. In the example above, point  402  belongs to clusters 1 and 2. It is also possible to stipulate that clusters are non-overlapping, that is, where a point belongs to only one cluster. Then a given point will be assigned to the cluster which is the closest match. 
         [0143]    In one embodiment, the clustering is performed using pairwise comparison, as demonstrated above. In another embodiment, the clustering is performed on the basis of summary statistics. In another embodiment, clusters are pre-defined using the results of other tests. 
         [0144]    In a further embodiment, the clustering demonstrated above is extended to a plurality of semiconductor samples. 
         [0145]    The methods explained above are not just limited to the transient photoconductive decay technique. The methods can be extended to other fields where a population is introduced and outputs related to the decay curves of the introduced population are readily available for measurement, so as to enable conversion into survival functions. 
         [0146]    Although the algorithms described above including those with reference to the foregoing flow charts have been described separately, it should be understood that any two or more of the algorithms disclosed herein can be combined in any combination. Any of the methods, algorithms, implementations, or procedures described herein can include machine-readable instructions for execution by: (a) a processor, (b) a controller, and/or (c) any other suitable processing device. Any algorithm, software, or method disclosed herein can be embodied in software stored on a non-transitory tangible medium such as, for example, a flash memory, a CD-ROM, a floppy disk, a hard drive, a digital versatile disk (DVD), or other memory devices, but persons of ordinary skill in the art will readily appreciate that the entire algorithm and/or parts thereof could alternatively be executed by a device other than a controller and/or embodied in firmware or dedicated hardware in a well known manner (e.g., it may be implemented by an application specific integrated circuit (ASIC), a programmable logic device (PLD), a field programmable logic device (FPLD), discrete logic, etc.). Also, some or all of the machine-readable instructions represented in any flowchart depicted herein can be implemented manually as opposed to automatically by a controller, processor, or similar computing device or machine. Further, although specific algorithms are described with reference to flowcharts depicted herein, persons of ordinary skill in the art will readily appreciate that many other methods of implementing the example machine readable instructions may alternatively be used. For example, the order of execution of the blocks may be changed, and/or some of the blocks described may be changed, eliminated, or combined. 
         [0147]    It should be noted that the algorithms illustrated and discussed herein as having various modules which perform particular functions and interact with one another. It should be understood that these modules are merely segregated based on their function for the sake of description and represent computer hardware and/or executable software code which is stored on a computer-readable medium for execution on appropriate computing hardware. The various functions of the different modules and units can be combined or segregated as hardware and/or software stored on a non-transitory computer-readable medium as above as modules in any manner, and can be used separately or in combination. 
         [0148]    While particular implementations and applications of the present disclosure have been illustrated and described, it is to be understood that the present disclosure is not limited to the precise construction and compositions disclosed herein and that various modifications, changes, and variations can be apparent from the foregoing descriptions without departing from the spirit and scope of an invention as defined in the appended claims.