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Charge Transport Simulator User`s Manual

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1. Capture process name ion 1 lt el capture Captured cariers charge 1 Initial trap charge qcs_2_1 1 e 7 Insert before function name Short name ion Capture cross section cs_2_1 Selected traps coincide with these traps of an adjacent layer Constant X 1e 010 cm 2 de pi Parameters of charge carier release from selected traps Se Naber af processes T Selected process iy Release process name ion 0 gt el release Released cariers ei charge 1 Initial trap charge qgs_2_1 0 e Release coefficient as 2 1 L ok J cn Ay Fig 7 An example of the parameter sheet Interface trap parameters when the selected interface traps coincide with interface traps that are defined in the adjacent layer 21 22 Position of the new interface traps mi D The traps are on the left edge of the layer corresponding to the least coordinate D The traps are on the right edge of the layer corresponding to the greatest coordinate The traps are on both edges of the layer Fig 8 The dialog window for specifying the position of new interface traps CarrierParms 83 Should the new traps coincide with previously defined traps belonging to the adjacent layer Fig 9 The message box that pops up when new interface traps are being created and the adjacent layer already has interface traps defined Selection of primary
2. 4 On Ji x t q NE g D gt 1 2 Ox OE _ p x t 1 3 Or MEN where the following notations are used t time x coordinate n concentration of the charge carriers of i th type qi electric charge of the charge carriers of i th type Qo the rate of bipolar generation including photogeneration of the charge carriers of i th type cx concentration of the filled states of k th type which act as a source of charge carriers of i th type Qik the rate coefficient of generation of the charge carriers of i th type from the states of k th type Px concentration of the empty states of k th type which can accept charge carriers of i th type ix the rate coefficient of capture of the charge carriers of i th type to the states of k th type ji the conduction current density of the charge carriers of i th type 14 mobility of the charge carriers of i th type D diffusion coefficient of the charge carriers of i th type E electric field strength p the total space charge density the high frequency dielectric permittivity Ey the electric constant For brevity the term charge carriers will be applied not just to free charge carriers but to traps of a particular charge state as well The program can simulate interface traps too in order to simplify Equations 1 1 and 1 2 the terms corresponding to interface traps have been omitted Charge carrier
3. Constant and a constant capture coefficient Known capture coefficient e The group of controls Parameters of charge carrier release from selected traps is used to specify names of release processes and parameters that define the release rates A release process can be added or removed using the button Add or remove a process When defining a new release process the released carriers must be selected in the drop down list Released carriers and the trap charge prior to carrier release must be specified in the text box Initial trap charge in units of the elementary charge e The drop down list Release coefficient is used to select the method that should be used to compute the release coefficient The release coefficient appears in the expression of the release rate G G an 5 4 where n is the concentration of traps whose charge state corresponds to the charge value entered in the text box Initial trap charge Three choices are available in the drop down list Release coefficient 1 A constant release coefficient 2 The Shockley Read Hall model According to this model the release coefficient is proportional to the capture coefficient o Vy corresponding to the opposite process charge carrier capture where ois the capture cross section and Vp is the average thermal velocity of the released charge carriers E A 0U4N 2 5 5 where N is the state density of free charge carri
4. the simulation time step At will be computed from other parameters e The parameter minR is the minimum value of the ratio of the time interval between adjacent argument values of computed f t functions i e between two adjacent points of their graphs and the simulation time step At When that ratio is increased the minimum number of the simulation time steps that can fit between two adjacent argument values of a computed f t function increases as well If neither maxChange nor dtMax is used then at first the maximum graph time step is computed then it is divided by minR The result is the maximum simulation time step At e The parameter maxR is the maximum value of the ratio of the time interval between adjacent argument values of computed f t functions i e the graph time step and the simulation time step At If at least one of the mentioned parameters maxChange and dtMax is used then at first Af is computed then it is multiplied by maxR The result is the maximum graph time step later on it can be decreased on the basis of other parameters which are described below An increase of maxR causes an increase of the maximum number of simulation time steps between adjacent argument values of computed f t functions As a result the spacing of the points in f t plots increases In order to decrease computer memory requirements that parameter should be increased The final v
5. 1 the short name of the traps don meaning donors 2 the initial trap charge prior to capture expressed in elementary charges 1 3 the short name of the captured charge carriers el meaning electrons Since the mentioned three components of the process name are read from other fields they are represented by special character sequences which indicate the place where a given component must be inserted Therefore when entering the mentioned name into a corresponding text box the following character sequence must be entered Ys Yo d lt s capture This is a format sequence of the C programming language s means any string i e any sequence of characters and d means an integer number with the sign or This example is one of default names Any name can be modified For example one or two last format specifiers can be removed In the mentioned example the format string could be modified to contain only the first two format specifiers s and d or only the first one s The name may also be entered without any format specifiers In such a case it would not depend on the content of any other input fields Note Since a single percent character in the format specification has a special meaning it indicates that a value of a certain variable must be inserted in its place in order to display a single percent character in the final text it mus
6. Constant which corresponds to a constant trap concentration its value must be entered in the nearby text box However this value is not the total concentration but just a constant term in the total concentration see below The parameter group Trap region parameters is used to specify non uniform trap distribution in the layer This is achieved by defining the so called bulk trap regions Each region corresponds to one coordinate dependent term in the total dependence of the trap concentration on the coordinate x N N f 5 1 i l where No is the mentioned constant term f x is a coordinate dependent term corresponding to the i th region of the current traps and I is the total number of regions A trap region consists of three parts the region of increasing concentration on the left the constant concentration region in the middle and the region of decreasing concentration on the right The dependence of concentration on the coordinate in the first and third mentioned parts is Gaussian Thus the general expression of the term f x is Model parameters Ea Initial camier distribution Simulation algorithm parameters Extemal parameters Photogeneration parameters Layer thickness and other parameters Free camier parameters Interface transmission Bulk trap parameters Interface trap parameters Number of I gt 1 Number of bulk trap types 3 Parameters of charge carier capture to se
7. and Q is an integer number indicating the charge state Name Meaning A O concentration 1 cm 3 Concentration of bulk traps A whose charge is equal to O elementary charges Functions corresponding to processes of charge carrier capture into bulk traps and release from them Those functions have the meaning of a process rate i e the number of elementary events per unit volume and unit time The default name of those functions is P rate 1 s cm 3 where P stands for the name of the given process The process name is specified by the user also see Section 2 Introduction to user interface The function names corresponding to the default process names are given in the table below in this table A and B stand for the short names of bulk traps and free charge carriers respectively and Q is an integer number indicating the initial charge state of those traps Name Meaning A Q lt B capture rate 1 s cm 3 The rate of capture of charge carriers B into bulk traps A whose charge is equal to Q elementary charges this is the trap charge prior to capturing the charge carrier A Q gt B release rate 1 s cm 3 The rate of release of charge carriers B from bulk traps A whose charge is equal to Q elementary charges this is the trap charge prior to releasing the charge carrier 3b Object time f
8. photogeneration Caniers of opposite charge fry Photogeneration quantum yield qy1 Soectedforezchcomponent v Spectral components stimulating this process b Cor Come Ca Fig 11 Examples of the parameter sheet Photogeneration parameters a photogeneration quantum yields of all spectral components are equal b quantum yield is specified for each spectral component separately 25 1 Blue 450 nm 2 Green 550 nm Fig 12 Examples of the dialog window for selecting active spectral components of light when a constant photogeneration quantum yield is used In the example on the left all three spectral components are turned on In the example on the right only the third spectral component is turned on active whereas the first two spectral components are turned off inactive i e their photogeneration quantum yields are zero Quantum yields of spectral components Change the selected quantum yield 1 Blue 450 nm 1 2 Green 550 nm 0 3 Red 680 nm 0 a Fig 13 An example of the dialog window for specifying quantum yields of individual spectral components a the initial view b the view after selecting a spectral component and opening the corresponding text box 26 8 Parameters of charge carrier transport through interfaces The parameter sheet Interface transmission is used to define conditions of charge carrier tr
9. 1 4 Ox Xk 1 7 Xk X Xk where j1 jx and jx are the values of current density at three adjacent nodes x1 x and x are the coordinates of those nodes and r is the ratio of the two adjacent inter node intervals Xy E r k k 1 Keam r In a special case when r 1 i e when both adjacent intervals are of equal width the expression 1 4 becomes Oj E Jan Fra p 1 5 Ox x x AT ai Similar expressions are used when calculating the derivative of the charge carrier concentration which is used in the expression of the conduction current density 1 2 The expression 1 4 or 1 5 is called the two sided derivative because its value at the k th node depends on function values at two adjacent nodes with numbers k 1 and k 1 At the edges of the system and at interfaces between different layers of the system one sided derivatives are used For example the derivative of the charge carrier concentration at the left edge of the layer which corresponds to the minimum value of x i e to the first node x x is calculated as follows on Ox The system of equations 1 1 1 3 is solved using the so called explicit algorithm I e increments of the charge carrier concentrations during one time step of simulation are calculated by multiplying the time derivatives of the concentrations by the simulation time step At For example if the concentration of a given type of charge carriers at the k th node at the th moment
10. Capture cross section a constant cross section Constant and a constant capture coefficient Known capture coefficient Number of layers 2 Number of interface trap types T Parameters of charge camier capture to selected traps Naber at pocesses T Ai mne taioe inos a r Gi Selected interface traps 1 y io as iii Interface trap name ion 1 lt el capture Larni Interface traps Captured caniers el 7 charge 1 0 the traps are at the right edge of the layer T TAR ao a Initial trap charge qes_1_1 1 e El Insert after function name Capture cross section cs_1_1 Surface density of the traps Ns_1_1 Known capture coefficient Z 1e 012 1 cm 2 parameters of charge canier release from selected traps Number of processes 1 Add or remove a process Selected process iy Release process name ion 0 gt el release Released camiers charge 1 Initial trap charge qgs_1_1 0 e Release coefficient gs_1_1 Shockley ReadHallmodel y Medel param lok Comcel too Fig 6 An example of the parameter sheet Interface trap parameters Initial carier distribution Simulation algorithm parameters Extemal parameters i I esi Bulk trap parameters Interface trap parameters Parameters of charge camier capture to selected traps Number of processes Selected process e
11. an 0 733 Exponent of impurity concentration for electrons Ba 2 0 Exponent of electric field for electrons LA 49 7 em V s Minimum hole mobility u 479 em V s Maximum hole mobility Mai 1 6 10 cm Reference impurity concentration for holes Usat p 1 06 107 cm s Hole saturation velocity Vp 2 2 Exponent of normalized temperature in the numerator for holes Ep 3 7 Exponent of normalized temperature in the denominator for holes Op 0 7 S Exponent of impurity concentration for holes b 1 0 Exponent of electric field for holes e The drop down list Diffusion coefficient is used to select the carrier diffusion coefficient computation method Two choices are available 1 a constant diffusion coefficient 2 Einstein relation which states that the diffusion coefficient D is directly proportional to mobility 4 ql where k is the Boltzmann constant T is the absolute temperature and q is the carrier charge e The drop down list State density is used to select the method that should be used when computing the state density of the selected free charge carriers Two choices are available 1 constant state density 2 free particle approximation According to this approximation the free carrier state density is 3 2 N ae 4 4 where k is the Boltzmann constant T is the absolute temperature m is the carrier effective mass and h is the Planck constant If the carrier spin is 1 2 then the actual state dens
12. charge carriers to the end of the list this is the default action insertion of the new carrier type before the current type of free charge carriers and deletion of the current type of free charge carriers e The drop down list Selected charge carriers is used to select charge carriers when there are two or more types of charge carriers in the current layer All parameters shown in this parameter sheet correspond to the selected type of free charge carriers e The next two text boxes are used to enter the full name and the abbreviated name of the carriers e The next two text boxes are used to enter the charge value of the current charge carriers in units of the elementary charge and the carrier effective mass in units of the electron rest mass in vacuum Note In the current version of the program v0 75 the carrier effective mass in only used in these cases a to compute the thermionic emission flux see Section 8 b to compute the average thermal velocity it is used to compute the rate of bipolar recombination or capture into traps when the corresponding cross section is known c to compute the state density when the free particle approximation is applied Model parameters a Initial camier distribution Simulation algorithm parameters _ Extemal parameters Photogeneration parameters Layer thickness and other parameters Free carier parameters Interface t
13. computation method Two choices are available 1 a constant value of the generation rate 2 impact ionization The rate of impact ionization by charge carriers of i th type i e the number of carrier pairs generated in unit volume per unit time at point x is calculated using the expression E I EDI 4 where c and d are positive constants model parameters E is electric field strength j is the current density of carriers of i th type and q is their charge Impact ionization is the only asymmetric process of bipolar generation recombination 1 e a process when primary and secondary carriers are not equivalent to each other During an elementary event of an impact ionization process the primary carrier is the one that creates a pair of new carriers the primary carrier does not disappear in the process Thus the factor j in 4 14 is the current density of the primary charge carriers Consequently if charge carriers of both types can cause impact ionization then at least two impact ionization processes must be defined In general c and d may depend on electric field strength This dependence can be modeled by defining several impact ionization processes each one corresponding to a given interval of electric field strengths In order to define an impact ionization process completely four parameters must be entered the maximum and minimum electric field strength values they define the electric field range where that process
14. difference V Difference between potentials of the right edge and the left edge of the given layer Average conduction current density A cm 2 The average conduction current density taking into account all free charge carriers of the given layer 3 Object functions 3a Object coordinate functions Functions corresponding to free charge carriers in this table A stands for the carrier name which must be specified by the user Name Meaning A concentration 1 cm 3 Concentration of charge carriers A A current density A cm 2 Conduction current density of charge carriers A 1 e the sum of drift and diffusion current densities given by the formula 1 2 A drift cur dens A cm 2 Drift current density of charge carriers A 1 e the first term in the expression 1 2 A dif cur dens A cm 2 Diffusion current density of charge carriers A 1 e the second term in the expression 1 2 A mobility cm 2 V s Charge carrier A mobility A conc gradient 1 cm 4 Derivative of charge carrier A concentration with respect to coordinate x Functions corresponding to processes of bipolar recombination or generation of free charge carriers and charge carrier transmutations i e processes that change the type of a free charge carrier but do not change its charge Those f
15. done using the checkboxes Insert before function name and Insert after function name which are on the left side of some parameter sheets under the text box with the layer name see Fig 1 This option may be useful if two or more layers of the system contain objects 1 e carriers traps etc with identical names For example if there are two layers containing charge carriers of the type Electron then after selecting the GraphiXT menu command Select f x t const functions the function selection list will contain two functions with identical names Electron concentration 1 em 3 If the names of different layers are different then such identical function names can be eliminated by inserting the layer name into them Then it is easier to distinguish between functions corresponding to different layers The default layer name is the layer number The corresponding C format specifier is 02d That format specifier corresponds to an integer number consisting of at least two digits and if that number is less than 10 then the first digit is 0 the format specifications are also mentioned in Section 2 Introduction to user interface In the just mentioned example after checking the checkbox Insert before function name the names of the mentioned two functions would become 01 Electron concentration 1 cm 3 and 02 Electron concentration 1 cm 3 It is sufficient to check that checkbox i
16. in the parameter sheet External parameters below are the expressions of potentials at various points of the circuit as well as the expressions of the total current in terms of those potentials in the two electrode configuration Depending on the electrode whose potential is varied i e the right electrode or the left electrode two schematic circuit diagrams are possible They are shown in Fig 15a and Fig 15b The following notations are used in them Uo potential of the left electrode of the simulated system U potential of the right electrode of the simulated system U potential of one pole of the external voltage source which is connected via the resistor R to one of the electrodes the one whose potential is varied R external resistance C external capacitance i total current of the simulated system If the right electrode potential is varied as in Fig 15a then its time dependence is described by the following differential equation dU U U i de RC C If the left electrode potential is varied as in Fig 15b then its time dependence is described by the following differential equation 9 1a dU U U i dt RC C 9 1b The total current i is defined as follows i SU j 5 irae 9 2 where S is the electrode area j is the conduction current density 1 e the sum of expressions of the type 1 2 corresponding to all free charge carriers at a given point ja is the displac
17. interface traps exe Layer 0 y on the left Traps Interface trap 4 Fig 10 An example of the dialog window for selecting the previously defined interface traps of the adjacent layer that should coincide with the new interface traps of the current layer The group of controls Parameters of charge carrier release from selected traps is used to specify names of release processes and parameters that define the release rates A release process can be added or removed using the button Add or remove a process When defining a new release process the released carriers must be selected in the drop down list Released carriers and the trap charge prior to carrier release must be specified in the text box Initial trap charge in units of the elementary charge The drop down list Release coefficient is used to select the method that should be used to compute the release coefficient The release coefficient q appears in the expression of the release rate G G n QM 6 2 where n is the surface density of interface traps whose charge state corresponds to the charge value entered in the text box Initial trap charge G is defined as the number of free charge carriers that are released from the interface traps in the unit area per unit time Three choices are available in the drop down list Release coefficient 1 A constant release coefficient 2 The Shockley Read Hall mod
18. is active and the corresponding values of c and d i The maximum electric field strength corresponding to the last interval is co The corresponding entered value must be such that it is always greater than the actual electric field strength for example a number 10 could be entered The electric field intervals and the corresponding values of parameters c and d for impact ionization processes in silicon are given in Table 4 2 data from the software package MicroTec user s manual MicroTec Software Package for Two Dimensional Process and Device Simulation Version 4 0 for Windows User s Manual Siborg Systems Inc 1998 Table 4 2 Impact ionization model parameters for electrons and holes in silicon 4 14 Symbol Value Units Meaning Emin a 0 V cm Minimum electric field strength for electrons Eris 00 V cm Maximum electric field strength for electrons d 1 4 10 V cm Field exponent for electrons Cn 7 10 l cm Pre exponential factor for electrons Eini p 0 V cm Lower bound of the first electric field interval for holes Emaxlp 6 07 10 V cm Upper bound of the first electric field interval for holes dip 2 09 10 V cm Field exponent for holes in the first electric field interval Cip 1 3 10 1 cm Pre exponential factor for holes in the first electric field interval Emin p 6 07 10 V cm Lower bound of the second electric field interval f
19. layer and of the primary interface traps see Fig 7 The group Parameters of charge carrier capture to selected traps contains the controls that are used to specify names and cross sections of charge carrier capture to current interface traps A capture process can be added or removed using the button Add or remove a process When defining a new capture process the captured carriers must be selected in the drop down list Captured carriers and the trap charge prior to capturing a carrier must be specified in the text box Initial trap charge in units of the elementary charge The drop down list Capture cross section is used to select the method that should be used to compute the cross section of charge carrier capture into the interface traps The capture cross section o appears in the expression of the capture rate R Ron OUP gt 6 1 where Un is the average thermal velocity of the captured charge carriers it is calculated according to 4 12 n is their concentration and p is the surface density of interface traps whose charge state corresponds to the charge value entered in the text box Initial trap charge R is defined as the number of free charge carriers that are captured into the interface traps in the unit area per unit time The factor ov is sometimes called the capture coefficient In the current version of the model v0 75 two choices are available in the drop down list
20. left electrode potential is varied 31 where x is the coordinate of the right edge of the k th layer xp corresponds to the left edge of the system By inserting expressions 9 3a and 9 3b into equations 9 1a and 9 1b respectively and expressing dU dt and dU dt the following two differential equations are obtained if the right electrode potential is varied dU 1 U U dt C amp S Wep R and if the left electrode potential is varied po a Sha 9 6b dt C ES Wey R Since at a given moment of time t t all quantities on the right hand side of equations 9 6a and 9 6b are known the value of potential U or Up at a later moment of time tm t Af can be computed in the same way as charge carrier concentrations i e by applying the explicit algorithm see formula 1 7 1f the right electrode potential is varied Sha 9 6a Ult U 4 ar 9 7a and if the left electrode potential is varied Volt U y t 4 ar 9 76 Examples of the parameter sheet External parameters are shown in Fig 15a and Fig 15b Below are descriptions of all parameters of that parameter sheet e The parameter Temperature T is the absolute ambient temperature in kelvins e The three radio buttons that are in the group Electrode configuration are used to specify the number of electrodes one or two and if there is only one electrode then its position is specified either t
21. not in a suspended state fully utilizes one processor here the term processor is used synonymously with the term core if a multi core processor is used Consequently if for example the computer has four processors and only one thread is used for simulation then simulation would only utilize 25 of available computing power and if 4 threads were used then all available computing power would be utilized so that processing time could decrease 3 or 4 times However this would be only possible in the ideal case when different threads are practically independent i e communication between threads and their synchronization is relatively infrequent and when absolutely all operations are equally distributed among the threads In this respect simulation of charge carrier kinetics is not very amenable to multithreading because different threads must be synchronized several times during each time step of simulation Besides some steps are done by a single thread Threads are synchronized with each other as follows Each thread that has finished the current stage of simulation checks if the other threads have finished their tasks too If not then the thread enters the suspended state and waits until all other threads finish their tasks Threads are suspended and resumed using special signals of the operating system The mentioned periodic checks for task completion and thread response to the mentioned signals are not instantaneous h
22. release from interface traps Examples of this parameter sheet are shown in Fig 6 and Fig 7 Below are explanations of all controls of this parameter sheet 20 The button Add or remove interface traps is used to open a dialog window where one of three actions can be selected addition of a new type of interface traps to the end of the list this is the default action insertion of the new trap type before the current type of interface traps and deletion of the current type of interface traps After selecting addition or insertion of new traps the user has to specify the trap position Three choices are available left edge of the current layer right edge of the current layer or both edges of the layer see Fig 8 In the case of both edges the values of surface density of those traps on both edges of the layer will be equal to each other If the adjacent layer of the simulated system has already interface traps defined then it is possible to use the same traps when defining carrier capture and release processes of the current layer In such case the message box shown in Fig 9 pops up before the mentioned trap position dialog If the button Yes is clicked the dialog window similar to the one shown in Fig 10 is opened It is used to select one of the previously defined interface trap types which belongs to the adjacent layer of the system and which should coincide with the new traps that are being created This option i
23. the bipolar mobility model for holes in silicon The bipolar mobility model for electrons in silicon is defined by the following two formulas 11 max T min B 1 T O kio hsa lE lisa N ue au u N E ps NY 1 582 42 T N On V 3 300 Nret n where N is the total concentration of impurities donors and acceptors measured in units of cm T is the absolute temperature measured in kelvins and E is the electric field strength V cm The definition of the bipolar mobility model for holes in silicon is similar in the case of holes the subscript n should be replaced by p The values of the bipolar mobility model parameters are listed in Table 4 1 data from the software package MicroTec user s manual MicroTec Software Package for Two Dimensional Process and Device Simulation Version 4 0 for Windows User s Manual Siborg Systems Inc 1998 sat n Table 4 1 Bipolar mobility model parameters for electrons and holes in silicon Symbol Value Units Meaning a 55 2 em V s Minimum electron mobility ue 1430 em V s Maximum electron mobility Nisin 07 210 cm Reference impurity concentration for electrons Vsat n 1 07 10 cm s Electron saturation velocity Vin 2 3 5 Exponent of normalized temperature in the numerator for electrons En 3 8 Exponent of normalized temperature in the denominator for electrons
24. values of f t model functions 1 e abscissas of the corresponding curves plotted in time graphs in this case would be equal to Os 1 s 2 s 10 s The parameter maxTime is the maximum duration of the simulated physical process corresponding to a single value of the varied external factor external voltage or illumination intensity The parameter group Adjacent node interval calculation parameters contains parameters that define the rule for computing node coordinates in each layer of the simulated system If all inter node intervals were equal to each other then it would be sufficient to specify the total number of nodes in each layer the first and the last nodes are on the opposing surfaces of the layer The corresponding parameter is denoted nX see Fig 2 However this program can also use coordinate dependent inter node intervals which are either monotonously increasing or monotonously decreasing with distance to the layer surface For example the interval could be largest at one surface and smallest and the opposite surface or smallest at the layer center and largest at its surfaces or largest at the layer center and smallest at its surfaces The remaining parameters of that parameter group define the rule for computing inter node intervals when nodes are not equidistant The parameter dxR is the ratio of the maximum and minimum intervals in the case of equal intervals it should be equal to 1 In the exam
25. 2 Fig 16 An example of the parameter sheet Initial carrier distribution 11 Layer thickness and other parameters The parameter sheet Layer thickness and other parameters is used to specify thickness of each layer of the simulated multi layer system as well as all parameters that do not belong to any of the previously mentioned categories In the current version of the model v0 75 that parameter sheet contains only two parameters layer thickness and high frequency dielectric permittivity of the layer see Fig 17 Initial camer distribution Simulation algorithm parameters Extemal parameters Photogeneration parameters Layer thickness and other parameters Free camer parameters Interface transmission Bulk trap parameters Interface trap parameters Number of layers 1 ze pe Layer thickness w 1 um Add or delete a layer Selected layer Layer name 01 Highfrequency dielectric permittivity epshf 11 8 Insert before function name Insert after function name ok J cancel Apply Fig 17 An example of the parameter sheet Layer thickness and other parameters 35 12 Model functions Model functions are the quantities that are computed by the simulator plug in and stored in computer memory by the program GraphiXT Most of those quantities characterize the state of the simulated system but there are also severa
26. Name Meaning Interval between nodes um The distance between the node that is at a given point and the nearest node on the left um 2b Layer time functions Name Meaning Number of nodes in the layer The number of nodes in the layer the first and last nodes are on the opposite surfaces of the layer Right edge surface charge density e cm 2 The total surface charge density of the right edge of the given layer it includes the charge of the interface traps and the free surface charge which was defined in Section 8 If that edge is the interface between two simulated layers 1 e if it is not the right edge of the entire system then surface charges of both layers are taken into account Note This charge does not include the polarization surface charge caused by the difference of high frequency dielectric permittivities of the layers Right edge free surface charge density e cm 2 Free surface charge density of the right edge of the given layer If that edge is the interface between two simulated layers 1 e if it is not the right edge of the entire system then free surface charges of both layers are taken into account Average space charge density e cm 3 Average space charge density Average electric field V cm Average electric field strength Right edge potential V Potential of the right edge of the given layer Right and left edge potential
27. Simulator of charge transport in multi layer systems GraphiXT plug in v0 75 User s Manual by Andrius Po kus Vilnius University Faculty of Physics 2014 04 09 Copyright 2014 by Andrius Poskus E mail andrius poskus live com Web http www graphixt com Contents Ne Simulate Oni Method eva E ai i i a a i E a S a k a a R i 1 2 Introdi ctionto user interface ii a ia 3 3 Simulation algorithm parameters eeen a dd A A E EE ia E 5 4 Free charge Carrier Par ra 11 5 Bulk trap parameters ata 17 6 Interface trap parameters a 20 7 Photogeneration parameters a 24 8 Parameters of charge carrier transport through interfaces cccecccecsceeseeseeeseesecesecesecesecneeeeeeseeeeeneeses 27 9 External Parameters siz dd iia 30 10 Initial carrier distrib ti n esena acordando eiei dean ridad demana danesa dadas 34 11 Layer thickness and other parameters cccecccecscesssesseeeseceseceseceseeesecseeseeeeeeeeeseeeseeeseecsaecsaeenseeeseseaeens 35 122 Model funcions aanse sa cole torres ins 36 1 Simulation method The charge carrier kinetics is simulated by numerically solving the following system of partial differential equations On 1 Oj P tig 45 ic n Pr 1 1 at g ax i0 gt ik k i2 PaP
28. alue of the spacing between adjacent points in f t plots depends on another parameter too see below Model parameters Layer thickness and other parameters Free canier parameters Interface transmission Bulk trap parameters Interface trap parameters Initial canier distribution Simulation algorithm parameters Extemal parameters Photogeneration parameters Maximum relative change of concentrations during one time step of simulation maxChange 0 001 Maximum time step of simulation dtMax 0 s Minimum and maximum ratios of graph time step and sim time step minR 1000 maxR 3000 Maximum graph time step dtMax_graph 0 s Minimum ratio of the simulated process duration and the graph time step minR_graph 200 The maximum relative change of concentrations at each node must be defined relative to cariers or traps whose charge density at that node is greatest varies fastest The factor f for calculation of threshold concentration f n lt f nMax then nis assumed to be 0 1e 025 Smooth camier concentrations every 10 time steps f 0 then no smoothing Concentration smoothing method 9 three point five point Smooth time dependences of concentration rates V Stop simulation upon reaching stationary state The stationarity criterion minimum value of the adaptive exponential smoothing factor alphaMin 02 Y Calculate a sequence of stationary states coresponding to several constant values of extemal voltage and or i
29. ame Meaning A Q lt B capture rate 1 s cm 2 The rate of capture of charge carriers B into interface traps A whose charge is equal to O elementary charges this is the trap charge prior to capturing the charge carrier A Q gt B release rate 1 s cm 2 The rate of release of charge carriers B from interface traps A whose charge is equal to Q elementary charges this is the trap charge prior to releasing the charge carrier Note If the given interface traps are at both edges of the layer then each of the latter five functions is replaced by a pair of functions one function for each edge of the layer The name of the function corresponding to the left edge of the layer is obtained by adding the symbol lt to the name defined above The name of the function corresponding to the right edge of the layer is obtained by adding the symbol gt to the name defined above In addition for each object coordinate function there is a corresponding time function which is equal to the average value of that coordinate function over the layer where it is defined The name of that time function is obtained by inserting the abbreviation avg into the coordinate function name e g Electron avg concentration 1 cm 3 It is possible to insert the layer name into the names of all layer and object functions defined in that layer This is
30. and if the primary charge carriers are electrons then E is the difference between the conduction band edge and the trap energy level and E is the difference between the trap energy level and the valence band edge i e the sum E E is the width of the forbidden energy band also called the energy gap The energy level of recombination centers is usually assumed to be at the center of the energy gap Then E E E 2 where E is the energy gap width The SRH recombination rate is defined as follows Ras R G R G 4 13 where R and R are the rates of primary charge carrier capture and secondary charge carrier capture into the traps respectively and Gjand G are the corresponding release rates Thus both positive and negative values of Rsry are possible a positive value means that the dominant process is the charge carrier capture recombination and a negative value means that the dominant process is the charge carrier release generation During simulation of the time evolution of the system state the SRH recombination model violates conservation of charge if the difference of the primary carrier capture and release rates were equal to the difference of the secondary carrier capture and release rates then concentrations of traps of both charge states would be constant but the concentration of charge state A traps calculated according to 4 11 is not constant because it depends on the free carrier concentration
31. ansport through interfaces between the layers of the system Those conditions are defined separately for each type of charge carriers The charge carrier type is selected in the drop down list Selected charge carriers see Fig 14 Mathematically the mentioned conditions determine the boundary conditions of the solved system of differential equations Four options are available see Fig 14 e If the option The interface blocks the selected charge carriers completely is selected then conduction current of the selected charge carriers on that interface will always be zero e Ifthe option The interface acts as an ohmic contact for the selected charge carriers is selected then the conduction current density of the selected charge carriers on that interface will be computed using the same expression as in the bulk of the layer i e 1 2 but with the concentration gradient On Ox computed as a one sided derivative rather than a double sided derivative The influence of this option on the charge distribution depends on presence or absence of an electrode on the other side of the interface If that interface is a free surface of the system or if it separates two simulated layers of a multi layer system then the conduction current flowing through that interface will cause a variable surface charge In this case the charge carriers do not cross the interface but accumulate on it as surface charge in the names of model fu
32. ayers The just mentioned drop down list contains the names of charge carriers of the adjacent layer with charge equal or opposite to the charge of the selected charge carriers of the current layer Thus the charge carriers crossing a transparent interface may be replaced by charge carriers of the same charge e g electrons of one layer become electrons of another layer or by charge carriers of the opposite charge e g electrons on one side of the interface and holes on the other side In order to ensure continuity of conduction current density and of its derivative with respect to coordinate the values of the conduction current density at the interface nodes and at two nearest bulk nodes are replaced by linearly fitted values Thus four values of current are involved the values of current of the selected charge carriers on the interface node of the selected layer and on the nearest bulk node of the same layer and the values of current of the corresponding charge carriers on the interface node of the adjacent layer and on the nearest bulk node of the same layer Those four values are replaced by values computed according to the equation of a straight line two middle values are equal to each other because interface nodes of adjacent layers coincide with each other If the option The interface acts as a potential barrier for the selected charge carriers is selected then an additional term appears in the expression of the conductio
33. b The parameter group External illumination parameters contains parameters that define the external illumination intensity and its time dependence Two radio buttons that are in the group Exposed edge of the system are used to specify the position of the light source either on the left of the system or on the right of it The text box Initial photon flux density is used to specify the constant term in the expression of the external photon flux density photon flux density is the number of photons incident upon one square centimeter of the layer per one second The drop down list Time dependence of photon flux density is used to select one of several standard time dependences The selected time dependence is the deviation from the mentioned initial photon flux density Le the instantaneous value of the total photon flux density is equal to the sum of the initial value and the selected time dependence In the current version of the model v0 75 one of four standard functions can be selected Constant illumination intensity Periodic square pulse sequence Periodic sequence of short pulses and Linear variation The other text boxes are used to enter the parameters of the selected time dependence e g period pulse height etc If all parameters of the time dependence are zero then the corresponding time dependence is also assumed to be zero If at least one parameter is non zero then the parame
34. c mode see below then the mentioned parameters do not affect the memory requirements e The first two radio buttons of this parameter sheet are used to specify the reference charge carriers or traps whose concentration is used to define the maximum allowed relative change of the charge carrier concentration during one time step of simulation That relative change may be defined relative to charge carriers or traps of one charge state whose concentration at a given node either is greatest or varies fastest This is done in the following way First at each node of the layer the program determines charge carriers whose space charge density varies fastest in time those charge carriers could be electrons holes traps of a particular type and charge state etc If the button varies fastest is selected then the value of At for that node is calculated by dividing the space charge density of those charge carriers by the absolute value of its time derivative and multiplying the result by the mentioned parameter maxChange If the button is greatest is selected then the program determines charge carriers whose space charge density is largest at that node and divides that charge density by the absolute value of the same time derivative as in the previous case Thus this option only affects the numerator of the ratio but not its denominator the denominator is always equal to the absolute value of the time derivative of the fastest varyi
35. capture to interface traps and release from them causes an additional term in the expression of charge carrier flux density at the given interface The value of that term depends on the charge carrier concentration at that interface and on the surface density of interface traps of the given charge state The number of free charge carriers of a given type in a given space region can change due to their transport through the boundaries of that region or due to their bipolar generation or recombination or due to their capture or release or due to their transmutation into other free charge carriers The number of bulk or interface traps of a given charge state can only change due to capture or release of free charge carriers Consequently equations that describe the time dependence of the trap concentration of each charge state do not contain the first two terms that are present on the right hand side of Equation 1 1 and summing is done over free charge carriers that can be captured into those traps or released from them The system of equations 1 1 1 3 is solved by the finite difference method The entire range of x values corresponding to each layer of the system is discretized by replacing it with a set of fixed values which will be referred to as nodes The partial derivative of current density which appears in Equation 1 1 at the k th node of the layer is calculated as follows i p 1 j Chest j r H j Jka J 4 gt
36. carriers or traps in it If there can be several parameter sets of a given type e g free charge carriers of several types or traps of several types then a parameter set of that type can be added or removed using a corresponding button For example four such buttons are visible in Fig 1 Under each such button there is a drop down Model parameters x Initial camer distribution Simulation algorithm parameters Extemal parameters Photogeneration parameters Layer thickness and other parameters Free carrier parameters Interface transmission Bulk trap parameters Interface trap parameters Number of layers 1 Number of carrier types 2 State density Nc1 Constant w 164020 1 cm 3 E is Add or remove charge caniers Selected charge camiers 1 Bipolar recombination parameters Selected layer 1 7 Oreja iausi Short name Number of processes 1 Add or remove a process Layer name Electron el Selected process 1 X i Charge a1 1 e Mass aij 1 Jimo Becombnaon process name el h recombination Insert before function name Mobility mu Insert after function name Poole Frenkel mobility model Model param Caniers of opposite charge h X Diffusion coefficient D1 Recombination cross section r1 Constant X 0 au 25 Langevin recombination model v Canier transmutation parameters Bipolar generation parameters Number of transm 0 Add or remove transmu
37. concentration i e the parameter a or b in the expression 5 2 H4 is the left half distance i e the distance between the left boundary of the region a and the point x lt a where f x Nimax 2 see Fig 5 H is the right half distance i e the distance between the right boundary of the region b and the point x gt b where f x Nmax 2 see Fig 5 A new region of bulk traps is defined by clicking the button Add or remove a region then clicking OK in the dialog window that is opened and then entering the region parameters in the five text boxes that are below the drop down list Selected region see Fig 4 Those parameters are a the width of the trap region which is equal to b a it is denoted w in Fig 5 b the distance between the left edge of the region and the left edge of the layer sublayer to which those traps belong in the parameter sheet Bulk trap parameters this parameter is called Region offset and in Fig 5 it is denoted a c the maximum trap concentration in the region Nmax d the left half distance Ha e the right half distance H In order to define a trap region with purely Gaussian profile i e symmetric and without the intervening region of constant concentration the width w must be set to zero and parameters H and H must be set equal to each other In this case the parameter a would mean the distance from the left edge of the current layer to the poi
38. corresponding to one absorbed photon The drop down list box Photogeneration quantum yield is used to select the method of specifying quantum yields for different spectral components of light Two choices are available Constant and Specified for each component a If Constant is selected then quantum yields of all spectral components will be equal to the value entered in the nearby text box see Fig 11a Besides in this case it is possible to turn off some spectral components 1 e to set their quantum yield equal to 0 This is done by clicking the button Spectral components stimulating this process Then a dialog window similar to the ones shown in Fig 12 is opened A spectral component can be turned on or turned off by left clicking its name b After selecting Specified for each component the mentioned text box is replaced by the button Quantum yields see Fig 11b By clicking that button the dialog window with photogeneration quantum yields of all spectral components is opened see Fig 13a A value of a quantum yield can be changed by left clicking the corresponding line of this list and then clicking the button Change the selected quantum yield or by right clicking the line and then selecting the command Change quantum yield in the context menu or by double clicking the line In all cases the dialog window Quantum yield value is opened see Fig 13
39. e sequence If a parameter value is outside those bounds then a corresponding error message appears 33 10 Initial carrier distribution The parameter sheet Initial carrier distribution is used to define the initial coordinate dependence of concentrations of free charge carriers as well as the initial charge state of bulk and interface traps An example of this parameter sheet is shown in Fig 16 Below are explanations of all controls of this parameter sheet 34 The group of controls Initial concentration of free charge carriers is used to specify initial coordinate dependence of concentrations of all free charge carriers of the current layer After selecting a free carrier type in the drop down list Selected charge carriers a method of calculating the mentioned dependence must be selected in the drop down list Initial carrier concentration There are 2 choices 1 Constant concentration In this case the concentration value must be entered in the nearby text box 2 A concentration value proportional to concentration of one type of bulk traps belonging to the same layer the reference traps In this case another drop down list appears on the right of the mentioned text box That drop down list contains the names of all bulk traps belonging to the same layer as the selected free charge carriers The reference bulk traps must be selected in that drop down list and the proportionality factor must be entered i
40. el According to this model the release coefficient is proportional to the capture coefficient o Ua corresponding to the opposite process charge carrier capture where o is the capture cross section and Vp is the average thermal velocity of the released charge carriers E a ov N exp E 5 6 3 where N is the state density of free charge carriers and E is the energy depth of the traps The factor N in 6 3 is computed using the method specified in the parameter sheet Free carrier parameters drop down list State density Thus the Shockley Read Hall model may only be selected when the opposite process capture exists If there is no corresponding capture process then a warning appears after an attempt to start simulation 3 The SRH model with modified state density is also defined by 6 3 but with an arbitrary state density N whose value must be entered by the user In this case the value of N may be different from the state density defined in the parameter sheet Free carrier parameters 23 7 Photogeneration parameters Examples of the photogeneration parameter sheet are shown in Fig 11 Descriptions of all parameters of that sheet are given below 24 The parameter group Parameters of spectral components contains three parameters the name of the light spectral component the weight factor of its photon flux and the absorption coefficient in the selected layer the layer select
41. em are grouped into parameter sheets which are accessed by left clicking a corresponding tab see Fig 1 Each sheet corresponds to a particular category of model parameters In the current version of the model v0 75 there are 9 parameter sheets Their titles are seen in Fig 1 Parameter sheets contain controls of various types edit controls for entering parameter values and parameter set names drop down lists for selecting layers and parameter sets push buttons radio buttons and checkboxes Each physical material parameter e g charge carrier mobility trap concentration etc belongs to one layer of the simulated system All material parameters that are visible in each parameter sheet correspond to a particular layer the current layer The current layer can be selected using the drop down list Selected layer which is on the left side of most parameter sheets see Fig 1 Above that drop down list there is a button Add or delete a layer That button is used to open a dialog window where one of three actions can be selected addition of a new layer on the right i e after the layer with the greatest value of the x coordinate this is the default action insertion of the new layer before the current layer and deletion of the current layer A layer can only be deleted when the simulated system consists of two or more layers After inserting a new layer it is empty i e there are no free charge
42. ement current density is the electric constant is dielectric permittivity and dE dt is the time derivative of the electric field strength the values of and dE dt correspond to the same point of the system as the value of j The total current i does not depend on coordinate although its components the conduction current and the displacement current are in general coordinate dependent The total current 9 2 may only depend on time t In order to derive the equation for U or Up the total current i must be expressed in terms of the time derivative dU dt or dU dt respectively That expression is given below if the right electrode potential is varied Ey dU RES pa 9 3a l i Woe di and if the left electrode potential is varied 8 Do el 9 3b Wore dt where Weg is the effective width of the system which is defined as L wee 9 4 k l Ek where L is the number of the layers in the system excluding the electrodes k is a layer number wx is thickness of the k th layer amp is dielectric permittivity of the k th layer and jer is the effective conduction current density which is defined as je YL jd 9 5 Wert k 1 Ek y 30 b Initial carer distribution Simulation algorithm parameters Exdemal parameters i Temperature T 300 K Initial potential of the left electrode U0 0 Bectrode configuration Initial potential of the righ
43. ence when synchronization is especially frequent the multithreading mode can become slower than single threaded mode This is the case when the processing time of a single time step of simulation is especially short i e when the simulated system is especially simple i e has few nodes and few processes to simulate or when a very fast computer is used Note Another reason why the multi threaded mode can become slower than the single threaded mode is running other programs during simulation Each executed program consists of at least one thread If the total number of processor intensive threads exceeds the number of processors then the operating system can slow down 10 Each thread processes a part of the nodes of the simulated system For example if two threads are active then one thread processes the data corresponding to even numbered nodes and the other thread processes the data corresponding to odd numbered nodes If three threads are used then each of them processes every third node etc This ensures that all threads have approximately equal work loads even when simulating an asymmetric system In multithreading mode the program can use a user specified constant number of threads This is done by selecting the radio button Use constant number of simulation threads see Fig 2 and entering the required number of threads nThreads in the corresponding text box Alternatively the program can optimize the number of
44. ers and E is the energy depth of the traps For example if the selected release process corresponds to liberation of electrons from donor atoms in a semiconductor then N is the effective conduction band state density and E is the difference between the conduction band edge and the donor energy level The factor N in 5 5 is computed using the method specified in the parameter sheet Free carrier parameters drop down list State density Thus the Shockley Read Hall model may only be selected when the opposite process capture exists If there is no corresponding capture process then a warning appears after an attempt to start simulation If the selected traps act as recombination centers then there must be two capture processes and two release processes defined with two types of charge carriers participating in them also see the part of Section 4 where the Shockley Read Hall recombination is described 3 The SRH model with modified state density is also defined by 5 5 but with an arbitrary state density N whose value must be entered by the user In this case the value of N may be different from the state density defined in the parameter sheet Free carrier parameters 19 6 Interface trap parameters The parameter sheet Interface trap parameters is used to enter parameters of each type of interface traps as well as parameters that define processes of charge carrier capture into interface traps and
45. f all interface traps of the current layer Those controls are identical in function and appearance to controls of the group Initial filling of bulk traps which are described above the only difference is that the bulk trap concentration is replaced by the surface density of interface traps ok Cance Apply pz Model parameters Layer thickness and other parameters Free carier parameters Interface transmission Buk trap parameters interface trap parametes Initial carier distribution Simulation algorthm parameters Extemal parameters Photogeneration parameters Initial concentration of free charge camiers 1 Number of layers Selected charge caniers Add or delete a layer 1 Bectron zZ Initial camier concentration n01 Reference bulk traps Selected layer 1 non E product of this factor and concentration of specified traps X 0 9 1 Donor Z Layer name 01 Initial filing of bulk traps Initial filling of interface traps F Insert before function name Selected bulk traps Selected interface traps Insert after function name 1 Donor X Initial charge of all selected traps is equal to e nitial charge of all selected traps is equal to e 9 Initial trap charge is either 1 or 0 nitial trap charge is either or o Concentration of traps with specified charge nt01 Surface density of traps with specified charge Fraction of all traps y 09 1 cm
46. f the surface charge density of those traps and the width of the system Next the maximum absolute value of all those averages is determined taking into account not only free charge carriers and traps but also the free surface charges of the interfaces see Section 8 and the electrode surface charges Finally that maximum value is multiplied by the parameter f which is specified in the parameter sheet Simulation algorithm parameters and divided by the absolute value of the charge of a single charge carrier of that type The result is the threshold concentration for that type of carriers e The next parameter defines the smoothing frequency of free charge carrier concentrations as functions of the coordinate see Section 1 Simulation method The smaller this parameter the more frequent the smoothing When the simulation time step is sufficiently small smoothing is not needed In order to turn off smoothing this parameter must be assigned a zero value e The next two radio buttons allow selection of the concentration smoothing method three point smoothing or five point smoothing see Section Simulation method the three point smoothing is also briefly described in the GraphiXT user s manual Section Data smoothing e The checkbox Smooth time dependences of concentration rates is used to turn on or turn off exponential smoothing of time derivatives of charge carrier concentrations as functions of ti
47. ferred option is varies fastest However if the magnitude of the conduction current is determined by charge carriers whose concentration is greatest for example in the case of discharge of an electrophotographic layer then this option has no significant effect on simulation accuracy hence the more appropriate option is is greatest because then the simulation is shorter e Atsome points of the system the concentration of some types of charge carriers can become so small that it will have practically no effect on the simulated physical phenomena In such a case there is no need to use the concentration of those charge carriers during simulation In order to be able to detect such situations the so called threshold concentration is defined It is the concentration value that must be exceeded for the given charge carriers to be taken into account at any given node Otherwise the concentration of those charge carriers is assumed to be zero and is not used for simulation for example it is not taken into account when calculating the mentioned relative change of charge carrier concentrations The threshold concentration is defined using the parameter f see Fig 2 as follows First the program computes the average space charge densities of free charge carriers of each type and of traps of each type and each charge state the average space charge density of interface traps of a given type and charge state is defined as the ratio o
48. harge carriers and that concentrations are smoothed not at every time step of simulation but at every time step of the graph specified by the parameter dtMax_graph e The next checkbox is used to turn on or turn off the current voltage characteristic mode I V mode In this mode the program computes a sequence of stationary states corresponding to a specified sequence of fixed values of external voltage or external illumination intensity Each of the mentioned fixed values is computed from corresponding time dependences which are defined in parameter sheets External parameters and Photogeneration parameters Thus in this mode the term simulation time has dual meaning one time is used to compute values of external factors 1 e external voltage and illumination intensity and the other time is the true duration of the simulated physical process Those two times are independent of each other The first mentioned time is displayed in the graphs 1 e it is used as the argument of the plotted model functions hence it may be called graph time whereas the second mentioned time process time is shown during simulation in the status bar of the GraphiXT main window and in the GraphiXT dialog window Time limits and amount of data A point is added to the stored model data only when the process time exceeds a given maximum value see description of the parameter maxTime below or when a s
49. he left edge of the system or the right edge e The text boxes System left edge coordinate x0 and System right edge coordinate x1 specify the position of the layer on the X axis Since the width of the layer is specified in a different parameter sheet Layer thickness and other parameters the mentioned two coordinates are not independent their difference must be equal to the width of the system Consequently the mentioned two text boxes can not be enabled simultaneously The radio buttons that are beside them are used to specify which of those text boxes must be enabled After changing a number in that text box and left clicking any other control of this parameter sheet the number in the other text box will be automatically updated so as to ensure that the difference of the two coordinates is equal to the width of the system e All remaining controls of this parameter sheet are used to specify the initial potentials of the electrodes and their time dependence Those controls are enabled only in the two electrode configuration The two text boxes at the top are used to enter initial potentials of the left electrode and the right electrode Uo and Uj respectively e The next two radio buttons are used to specify the electrode whose potential is varied during simulation either the left electrode or the right electrode This option determines the schematic circuit diagram of the simulated system which is shown at the top r
50. hort pulses 7 F Insert before function name Absorption coefficient abs2 2000 lem Time of the first pulse 11 0 s Insert after function name Photogeneration process parameters Number of photons in one pulse Fmax le 013 1 cm 2 Selected charge caniers 1 Becton vr Time interval between pulses TI 1 s Number of processes 1 Add or remove a process Selected process ly el h photogeneration Caniers of opposite charge Photogeneration quantum yield qy1 Constant r 1 a L ok J canei toy Initial canier distribution Simulation algorithm parameters Extemal parameters Photogeneration parameters Parameters of spectral components Extemal illumination parameters Number of layers 1 gji Number of spectral components 3 Exposed edge of the system aa Sitio ht eaten Selected layer fi Selected spectral component 2 gt Initial photon flux density 10 0 Vs em 2 eras Name of the spectral component Weight factor wf2 Time dependence of photon flux density 01 Green 550 nm 1 Periodic seguence of short pulses 7 F Insert before function name Absorption coefficient abs2 2000 1 cm Time of the first pulse 11 0 s Insert after function name Photogeneration process parameters Number of photons in one pulse Fmax le 013 1 cm 2 Selected charge camiers 1 Becton y Electron Time interval between pulses TI 1 os Number of processes 1 Add or remove a process Selected process Photogeneration process name el h
51. ic field strength at the interface in the simulated layer semiconductor or dielectric e is dielectric permittivity of that layer e is the elementary charge q is the carrier charge and amp 1s the electric constant The remaining part of the conduction current density i e the drift and diffusion current is set equal to the average of the expression 1 2 at the surface nodes of the two adjacent layers the coordinate values of those two nodes are equal to each other because they belong to the same interface Thus in this case the conduction current is continuous but its derivative with respect to the coordinate may have a discontinuity at the interface because the conduction current value at the surface node is computed without taking into account the value of the current at the nearest bulk node of the same layer 29 9 External parameters The external parameters are the ambient temperature electrode configuration two electrodes or one electrode and in the case of two electrodes time dependence of external voltage and parameters of the external circuit components resistance and capacitance The terms external voltage and external circuit are only applicable in the two electrode configuration If there is only one electrode then its potential is always zero and the total current i e the sum of conduction and displacement currents is also zero Prior to descriptions of the parameters whose values can be entered
52. ight corner of this parameter sheet compare Fig 15a and Fig 15b e The two radio buttons and the text box that are in the group Initial potential of external voltage source U2 are used to specify the initial potential of the external source pole which is connected via the resistor R to the previously specified electrode If a non zero value of R has been entered see below then that potential may be not equal to the potential of that electrode The difference of those two potentials is equal to the voltage drop across the resistor R In this case the radio button U2 must be clicked and the initial value of U must be entered in the nearby text box The radio button U2 is only enabled when R gt 0 The other pole of the external voltage source is directly connected to the other electrode so that the potential of that pole is constant and equal to the potential of that electrode which has been entered before If the initial voltage drop across the resistor R is zero 1 e either the initial current of the external voltage source is zero or R 0 then the other radio button of this group must be clicked when R 0 this is the only option available The text beside that radio button is either U2 U1 or U2 U0 depending on which electrode s potential is varied 32 The text box Electrode area S is used to enter the electrode area of the simulated layer That area is used when calculating the to
53. ination process has been defined using electrons as the primary charge carriers as in Fig 1 and Fig 3 then after selecting holes in the drop down list Selected charge carriers that recombination process must not be defined again otherwise there would be two identical recombination processes and the total recombination rate would increase by a factor of 2 This note also applies when defining bipolar generation and photogeneration processes excluding impact ionization see below The drop down list Recombination cross section is used to select the recombination cross section computation method The recombination cross section oO appears in the expression of the recombination rate R R O Vip gt 4 5 where n and n are concentrations of the recombining charge carriers and Ve is their average relative velocity i e the average absolute value of their velocity with respect to each other In the case of the Maxwell distribution of speeds en eee 4 6 TIN _ Mm T A gt m m where m and m are effective masses of both charge carriers Sometimes the recombination rate is expressed using the recombination coefficient 2 R fB nn 4 8 where m is the reduced mass 4 7 By comparing 4 5 and 4 8 we see that B OV ye 4 9 Four choices are available in the drop down list Recombination cross section 1 a constant value of the recombination cross section 2 Langevin recombination model Acc
54. ion drop down list is on the left side of this parameter sheet Spectral components can be added or removed by clicking the button Add or remove a spectral component the maximum allowed number of spectral components is 100 Prior to starting the simulation the program CarrierFunc dll normalizes all weight factors to unity i e it divides each weight factor by the sum of weight factors of all spectral components The physical meaning of a normalized weight factor is the fraction of photons corresponding to the selected spectral component the total photon flux density is defined in the parameter group External illumination parameters which is described below Consequently only the ratios of the weight factors are important but not their absolute values For example if there are three spectral components with weight factors 1 2 and 3 then the same simulation results would be obtained using the weight factors 50 100 and 150 The parameter group Photogeneration process parameters contains controls that make it possible to add remove or modify photogeneration processes A photogeneration process can be added or removed by clicking the button Add or remove a process The photogenerated charge carriers are specified in the drop down list boxes Selected charge carriers and Carriers of opposite charge Photogeneration quantum yield is defined as the average number of charge carrier pairs usually electron hole pairs
55. ity is greater than 4 4 by a factor of 2 this is the so called spin factor because there are two different spin states of the charge carrier corresponding to each state of its spatial motion However when the free particle approximation is selected the spin factor is not used i e it is assumed to be 1 because the state density is only needed to compute the rate of carrier release from traps according to the Shockley Read Hall model this option is in the parameter sheets Bulk trap parameters and Interface trap parameters and the carrier spin state does not change during its release from a trap e The parameter group Bipolar recombination parameters contains the controls that are used to specify names and cross sections of bipolar recombination processes Such a process can be added or removed 12 using the button Add or remove a process When defining a new recombination process the secondary carriers whose charge is opposite to the charge of the currently selected primary carriers specified in the previously mentioned drop down list Selected charge carriers must be selected in the drop down list Carriers of opposite charge Note When defining a bipolar recombination process it does not matter which charge carriers are treated as primary ones and which are treated as secondary ones In any case each such process must be defined only once For example if the recomb
56. l functions that characterize the computation procedure itself The functions can be divided into two categories according to their arguments f t functions whose only argument is time and f x t functions which have two arguments coordinate x and time t Further on fd functions will be called time functions and f x t functions will be called coordinate functions In addition all model functions can be categorized according to a different criterion 1 global functions which are computed always regardless of the details of the simulated system the coordinate x values corresponding to stored values of the global coordinate functions coincide with node coordinates of all layers of the system 2 layer functions which are computed for each layer even for empty layers the coordinate x values corresponding to stored values of the layer coordinate functions coincide with node coordinates of the corresponding layer 3 object functions which are associated with specific parameter sets e g with carriers of a particular type since each object belongs to a particular layer the coordinate x values corresponding to stored values of the object coordinate functions coincide with node coordinates of the same layer The default names of all functions are given below Those names include the measurement units which are written according to the same rules as model parameter uni
57. lected traps Number of processes 2 H delete slayer islam Add ordeletealayer r dias al Selected bulk traps 3 X Capture process name Bulk trap name Short name rec c 1 lt el capture Layer name Recombination center recc Captured caniers el charge 1 01 x Trap concentration Nt3 Initial trap charge qct1 1 e 7 Insert before function name Constant y 1e 009 1 cm 3 j 1 Ml lose arininn Capture cross section ct 1 Trap region parameters Constant X 5e 010 cm 2 Number of regions 0 Parameters of charge canier release from selected traps Number of processes 2 Add or remove a process Selected region Selected process aid Release process name Region offset um rec c 0 gt el release Maximum concentration 1 cm 3 Released carers el Y charge 1 Initial charge qat1 0 e Left half distance um trap charge ast Release coefficient at 1 um Shockley Read Hal model Modelparam Fig 4 An example of the parameter sheet Bulk trap parameters 17 18 T 2 2 Na 2 Me when x lt a n F x N rax whena lt x lt b 5 2 T 2 N max EXP AL N 2 CDH when x gt b L Main where Nmax is the concentration in the central part of the trap region Further on the term edge of the region will mean the coordinate value corresponding to one of the boundaries of the central part of the trap region with constant
58. llumination intensity Interval between time values used to calculate values of extemal voltage or exposure intensity according to corresponding time dependences interval 0 1 s The maximum duration of transition of the system to a stationary state maxTime 1e 006 s Adjacent node interval calculation parameters Multithreading parameters Selected layer 1 Z Initial number of nodes in the layer nX 201 D ctimzs the number of ainai Ratio of maximum and minimum intervals dxR 50 Use constant number of simulation threads Interval between two leftmost nodes is maximum 9 minimum Number of simulation threads n Threads Interval between two rightmost nodes is maximum minimum ek Les Fig 2 An example of the parameter sheet Simulation algorithm parameters e The parameter dtMax graph is the maximum spacing between adjacent points of computed f t functions i e the graph time step If the value of the graph time step computed only from the previously mentioned parameters exceeds dtMax graph than that value is decreased to dtMax_graph Consequently if the parameter maxChange or dtMax is used 1 e non zero and if it is desirable to use constant spacing between computed points of f t functions then the required value of the spacing must be assigned to dtMax graph and the parameter maxR must be assigned a very large value such that the value of the graph time ste
59. me When this checkbox is checked then time derivatives of concentrations of free charge carriers and traps of all types and all charge states at each node are replaced by their smoothed values which are calculated according to the formula f f 1 8 f where f is the smoothed value at the moment of time ti f 1s the smoothed value at the moment of time tm t At and f is the original non smoothed value at the moment of time f The value of the smoothing factor Ais always between 0 and 1 When Bis close to 1 then f f When Ais close to 0 then the smoothed value is close to the average of all previous values 1 e its time dependence is weak The smoothing factor Ais the same for all nodes and all types of charge carriers This parameter is re calculated at each time step of simulation so as to ensure that smoothed values accurately represent systematic 1 e non random variation of concentration time derivatives i e systematic increase or decrease and that their random component is as small as possible A similar approach is used when computing smoothed values of concentrations themselves rather than their time derivatives which are used to determine if the system has reached the stationary state see below When this option is selected computation time usually decreases especially in the stationary state simulation mode which is described below e The next checkbox is used to turn on or turn off the sta
60. n Constant is selected the text box Potential barrier height is disabled because in this case the potential barrier height is not used for calculating the injection current density If Free particles Maxwell distribution is selected then the charge carrier injection flux density will be calculated under the assumption that charge carrier speeds are distributed according to the Maxwell distribution and that the following two conditions are sufficient for a charge carrier to cross the potential barrier 1 kinetic energy of the charge carrier must exceed the potential barrier height 2 the x component of the charge carrier velocity vector must be directed outside of the layer 1 e towards the selected interface In this case the potential barrier height must be entered in the text box Potential barrier height Then the current density component corresponding to transport of the selected charge carriers through the interface will be calculated as follows j qn D meo 8 1 V2nkTm kT where q is the carrier charge n is their concentration at the interface m is the carrier effective mass d is the potential barrier height k is the Boltzmann constant T is the absolute temperature If the selected layer is an electrode then two more choices are available in the mentioned drop down list Elementary Richardson equation and Richardson equation with an additional factor If Elementary Richardson equa
61. n current 1 2 That term reflects the difference of two fluxes one is due to charge carrier transfer over the potential barrier in the positive x direction and the other one is due to charge carrier transfer over the potential barrier in the negative x direction The method of computing those two fluxes must be specified in the two drop down lists that are at the bottom of this parameter sheet see Fig 14 The first two choices in those drop down lists are Constant and Free particles Maxwell distribution If the selected layer is an electrode and Constant is selected then the nearby text box must contain the absolute value of the injection current density from that electrode into the layer If the selected layer is not an electrode i e if it is one of simulated layers and Constant is selected then the nearby text box must contain the ratio of injection flux density and charge carrier concentration at the corresponding interface node of the selected layer the measurement unit of that ratio is cm s In the latter case having selected Constant the injection flux density of the selected charge carriers from the selected layer into the adjacent layer or into the electrode will be calculated by multiplying that constant by the concentration of those charge carriers at the interface node of the selected layer the injection current density is equal to the product of the injection flux density and the carrier charge Whe
62. n one parameter sheet only Note After checking or un checking the checkbox Insert before function name or Insert after function name the names of all functions of the corresponding layer will become equal to default names I e all changes of function names that were made in the program GraphXT exe will be lost only the changes made in the parameter editor will be retained After changing the name of a parameter set the names of object functions corresponding to that parameter set also become equal to default names After changing the number or meaning of model functions e g by inserting or removing a layer or a parameter set the names of all layer functions become equal to default names 39
63. n the mentioned text box see Fig 16 The group of controls Initial filling of bulk traps is used to specify the initial charge state of all bulk traps of the current layer After selecting a bulk trap type in the drop down list Selected bulk traps the initial charge state must be specified using the controls that are below The two radio buttons make it possible to select either a single charge state for all traps of the selected type or two possible charge states In the latter case the method of calculating the fraction of traps of each charge state must be selected in the drop down list Concentration of traps with specified charge The selected method is only applied to one charge state which is indicated by radio buttons that are under the two text fields where those charge values must be entered Two choices are available a constant concentration or a constant fraction of the total concentration which may be coordinate dependent as explained in Section 5 That constant concentration or the value of the mentioned fraction must be entered in the text box that is to the right of the mentioned drop down list If this charge state is called the first one then concentration of the second charge state will be calculated by subtracting concentration of the first charge state from the total trap concentration The group of controls Initial filling of interface traps is used to specify the initial charge state o
64. nctions that charge is called the free surface charge If the given interface separates a simulated layer and an electrode then the current that flows through that interface does not create any surface charge In addition after selecting the option The interface acts as an ohmic contact for the selected charge carriers it becomes possible to specify a constant value of the charge carrier concentration on that interface This option may decrease simulation time e g when simulating stationary current flowing through an ohmic contact between a metal and semiconductor semiconductor charge carrier concentrations at such a contact are always equal to their values at thermodynamic equilibrium During simulation charge carrier concentration at the corresponding surface node is kept constant by turning off all charge carrier generation recombination processes that influence concentration of the selected charge carriers In addition the values of the current density at the surface node and at the nearest bulk node of that layer which are computed according to 1 2 are replaced by their arithmetic average this ensures that the gradient of the current density at the surface node is zero e If the option The interface is transparent to the selected charge carriers is selected then charge carrier concentrations on both sides of the interface are computed on the basis of the continuity condition for the conduction current density a
65. nd its gradient This option can only be selected after Model parameters EEx Initial camer distribution Simulation algorithm parameters Extemal parameters Photogeneration parameters Layer thickness and other parameters Free camer parameters Interface transmission Bulk trap parameters Interface trap parameters Selected layer Selected charge caniers h z Selected layer Blectrode A The interface blocks the selected charge caniers completely The interface acts as an ohmic contact for the selected charge camiers E Conc of selected caniers at the interface 1 cm 3 1 cm 3 The interface is transparent to the selected charge camiers The interface acts as a potential barier for the selected charge caniers The interface acts as a potential barier for the selected charge camiers Potential barier height ev Potential barier height eV Schottky effect Ratio of injection flux density and concentration Injection current density cm s A cm 2 Fig 14 An example of the parameter sheet Interface transmission 27 28 specifying adjacent layer s charge carriers that correspond to the selected charge carriers of the current layer The drop down list for selecting charge carriers of the adjacent layer is not visible in Fig 14 because it appears only when the simulated system consists of more than one layer and when the selected interface is not the edge of the system 1 e when it separates two simulated l
66. ng space charge density component multiplied by maxChange In this manner the value of At is computed for each node of the system and then the minimum value of At is used as the simulation time step at interface nodes the surface charge and its variation rate are also taken into account The optimal value of that parameter depends on the simulated system For example if the simplest Schottky diode is being simulated then there are free charge carriers of one type electrons and traps of one type and one charge state positive donor ions The Schottky diode current is governed by conditions of electron transfer across the interface between the electrode and the semiconductor At that interface the electron concentration is much less than the donor concentration However the donor ion concentration stays constant assuming that all donor atoms are ionized and do not capture electrons only the electron concentration varies with time Thus in this case the denominator of the mentioned ratio would be equal to the absolute value of the time derivative of electron space charge density If the relative change of electron concentration were defined with respect to charge carriers whose space charge density is greatest 1 e donor ions the simulation time step would be too large and consequently large simulation errors would be possible for example negative values of electron concentration could be obtained Therefore in this example the pre
67. nt of the maximum concentration The group Parameters of charge carrier capture to selected traps contains the controls that are used to specify names and cross sections of charge carrier capture to current bulk traps A capture process can be added or removed using the button Add or remove a process When defining a new capture process the captured carriers must be selected in the drop down list Captured carriers and the trap charge prior to capturing a carrier must be specified in the text box Initial trap charge in units of the elementary charge max 0 a b x Fig 5 Dependence of bulk trap concentration on coordinate in a single region and the region parameters e The drop down list Capture cross section is used to select the method that should be used to compute the cross section of charge carrier capture into the bulk traps The capture cross section o appears in the expression of the capture rate R R OU NP 5 3 where Vn is the average thermal velocity of the captured charge carriers it is calculated according to 4 12 n is their concentration and p is the concentration of traps whose charge state corresponds to the charge value entered in the text box Initial trap charge The factor ov is sometimes called the capture coefficient In the current version of the model v0 75 two choices are available in the drop down list Capture cross section a constant cross section
68. o discard the latest modifications of parameter values 4 3 Simulation algorithm parameters An example of the parameter sheet Simulation algorithm parameters is shown in Fig 2 All parameters of this sheet are explained below e The parameter maxChange is the maximum allowed relative change of concentrations of free charge carriers and traps of each type and of each charge state during a single time step of simulation For example if its value is 0 001 as in Fig 2 then the simulation time step Af will be chosen such that the concentration of each carrier type at each node of the system does not change more than by 0 1 relative to charge carriers or traps whose charge density at that node is greatest or whose charge density at that node varies fastest see below for a more detailed explanation In order to increase accuracy of simulation this parameter must be decreased however then the processing time would increase The final value of At depends on other parameters too see below Note If maxChange is zero then it will not be used during simulation 1 e Af will be computed from other parameters e The parameter dtMax is the maximum value of the simulation time step At If the value of At computed only from the mentioned parameter maxChange exceeds dtMax then Ar is decreased to the value of dtMax Note If the parameter dtMax is zero then it will not be used during simulation i e
69. of time ti was n x ti then after one time step 1 e at the moment of time f tp1 f Af that concentration will be At 1 7 Xp st Thus it is assumed that the charge carrier concentrations at all nodes during the time interval from t to t At depend on time linearly and the rate of that dependence is determined by the state of the system at the moment of time 7 because the right hand sides of Equations 1 1 1 3 are computed using the charge carrier concentrations at the moment of time t The simulation time step At is computed based on the rate of change of the charge carrier concentrations the faster the change the less the time step Af The precision of finite difference estimates of the derivatives with respect to the coordinate can be increased by decreasing the inter node interval Ax but this would also cause an increase of the number of nodes and of the computer memory usage Ax may depend on the coordinate For example the node density may be largest at the edges of the layer and smallest at its center this option is in the parameter sheet Simulation algorithm parameters However in the current version of the model v0 75 the node positions do not depend on time i e they do not change during the entire simulation Since the explicit algorithm becomes unstable at large time steps At better results are sometimes obtained by smoothing the concentrations of free charge carriers The dependence of the f
70. or holes Emax2 p 00 V cm Upper bound of the second electric field interval for holes do 1 4 106 V cm Field exponent for holes in the second electric field interval Crp 4410 l cm Pre exponential factor for holes in the second electric field interval The parameter group Carrier transmutation parameters contains the controls that are used to specify names and rate coefficients of free carrier transmutation processes Each transmutation is a process whereby a free charge carrier is replaced by a different free charge carrier whose charge is equal to the charge of the original carrier Thus carrier transmutations are only possible when the given layer contains several types of charge carriers with equal charge A carrier transmutation process can be added or removed using the button Add or remove transmutation When defining a new transmutation process the secondary free charge carriers whose charge is equal to the charge of the 15 16 currently selected primary carriers specified in the previously mentioned drop down list Selected charge carriers must be selected in the drop down list Secondary carriers The drop down list Rate coefficient is used to select the transmutation rate calculation method In the current version of the model v0 75 only one choice is available Constant which corresponds to a constant rate coefficient its value must be entered in the nearby text box The
71. ording to this model the recombination coefficient depends on mobilities 4 and 1 of the recombining charge carriers p M 4 10 EqE where q is the absolute value of the charge is the electric constant sis the dielectric permittivity 3 A known recombination coefficient In this case the user must specify not the recombination cross section but the recombination coefficient A and the program calculates the recombination rate according to 4 8 4 Shockley Read Hall recombination This is recombination of free charge carriers via traps which are called recombination centers Each SRH recombination process is a sum of four processes a capture of primary charge carriers into bulk traps of charge state B forming charge state A b release of primary charge carriers from the traps of charge state A forming charge state B c capture of secondary charge carriers into the traps of charge state A forming charge state B d release of secondary charge carriers from the traps of charge state B forming charge state A Since those four processes are defined in the parameter sheet Bulk trap parameters described in Section 5 only the first of the mentioned processes has to be specified in the SRH recombination dialog window The program will find the other three processes automatically If the defined capture and release processes do not include all mentioned processe
72. p computed from maxR is always greater than the required value Note If the parameter dtMax_graph is zero then it will not be used during simulation 1 e the graph time step will be computed from other parameters However at least one of parameters maxChange dtMax and dtMax graph must be used i e non zero e The parameter minR_graph is the minimum ratio of the duration of the simulated process and the graph time step In other words it is approximately equal to the minimum total number of points in f t function graphs If the parameter dtMax_graph is not used 1 e zero or if it is greater than the ratio T minR graph then the latter ratio has the same role as dtMax_ graph i e it is approximately equal to the maximum value of the graph time step In this case minR_graph is approximately equal to the final number of points in f t function graphs Important Parameters maxR and dtMax graph must be chosen carefully trying to avoid too small values because otherwise there is a risk of insufficient computer memory during simulation For the same reason one should avoid too large values of the parameter minR graph During simulation it is possible to check how fast the amount of used computer memory is growing this is done using the GraphiXT menu command Simulation options Time limits and amount of data If the simulation is done in the current voltage characteristi
73. ple of Fig 2 the maximum inter node interval is 50 times larger than the minimum interval The next four radio buttons define the points of the layer where the interval is largest and where it is smallest For example in the case of Fig 2 the maximum interval is between the node that is on the right surface of the layer and the adjacent node on the left and the minimum interval is between the node that is on the left surface of the layer and the adjacent node on the right If it were specified that the interval is smallest at both surfaces of the layer then the maximum interval would be between the central nodes there it would be 50 times larger than near the surface of the layer Important The parameter nX as well as the previously mentioned parameters maxR dtMax graph and minR graph strongly affects the amount of used computer memory during simulation One should avoid too large values of that parameter because otherwise there is a risk of computer memory shortage during simulation Parameters of the parameter group Multithreading parameters control the multithreading mode The essence of multithreading is splitting the simulation into several threads which run in parallel The Windows operating system distributes the computer resources equally among all those threads If the computer has more than one processor and if the number of threads does not exceed the number of processors then each thread that is
74. racterization of the stationary state rather than of the system relaxation to the stationary state then it is recommended to turn SRH recombination on because then the simulator reaches the stationary state faster On the other hand if the aim of the simulation is characterization of the time evolution of the system then more accurate results are obtained when the SRH recombination is turned off 2 In order to turn SRH recombination off it is sufficient to enter zero recombination cross section This is done by selecting Constant in the drop down list Recombination cross section and entering zero in the nearby text box it is also possible to remove the recombination process altogether but then all previous simulation results will be lost and all plotted model curves will be removed from the graphs e The parameter group Bipolar generation parameters contains the controls that are used to specify names and rates of bipolar generation processes Such a process can be added or removed using the button Add or remove a process When defining a new generation process the secondary carriers whose charge is opposite to the charge of the currently selected primary carriers specified in the previously mentioned drop down list Selected charge carriers must be selected in the drop down list Carriers of opposite charge e The drop down list Generation rate is used to select the generation rate
75. ransmission Bulk trap parameters Interface trap parameters Number of layers 1 Number of carrier types 2 State density Nc1 Constant y 282 019 1 0m 3 ___Addorremove charge caries Selected charge camiers 1 Bipolar recombination parameters Siei ad Naber ene Charge carier name Short name pro gt or remove a process Layer name Electron el Selected process 1 Zi 0 Charge a1 1 e Mass m1 032 mo Recombination process name SRH recombination Insert before function name Mobility mut E Insert after function name Constant 1200 cm2 W s Carers of opposte charge a Diffusion coefficient D1 Recombination cross section r1 l Einstein relation Shockley Read Hall recombination y _ Model param Camer transmutation parameters Bipolar generation parameters Number of transm 0 Addorremovetransmutation Number of processes 0 Selected transmutation Selected process Transmutation name Generation process name Secondary caniers Caniers of opposite charge Rate coefficient Generation rate Constant Ys Constant 1 cm 3 s Fig 3 An example of the parameter sheet Free carrier parameters e The drop down list Mobility is used to select the carrier mobility computation method Four choices are available 1 constant mobility 2 exponential dependence on the square root of electric field strength 1 aexp by E 4 1 3 the bipolar mobility model for electrons in silicon 4
76. rate coefficient y appears in the expression of the transmutation rate V V yn 4 15 where n is the concentration of the primary charge carriers The transmutation rate V is defined as the number of elementary transmutation events in the unit volume per unit time 5 Bulk trap parameters The parameter sheet Bulk trap parameters is used to enter parameters of each type of bulk traps as well as parameters that define processes of charge carrier capture into bulk traps and release from bulk traps An example of this parameter sheet is shown in Fig 4 Below are explanations of all controls of this parameter sheet The button Add or remove bulk traps is used to open a dialog window where one of three actions can be selected addition of a new type of bulk traps to the end of the list this is the default action insertion of the new trap type before the current type of bulk traps and deletion of the current type of bulk traps The drop down list Selected bulk traps is used to select bulk traps when there are two or more types of bulk traps in the current layer All parameters shown in this parameter sheet correspond to the selected type of bulk traps The next two text boxes are used to enter the full name and the abbreviated name of the traps The drop down list Trap concentration is used to select the trap concentration computation method In the current version of the model v0 75 only one choice is available
77. ree carrier concentration on the coordinate is smoothed when a seesaw type sequence of concentration values is detected with inflection points spaced periodically One of two smoothing methods can be chosen If the three point smoothing method is used then the mentioned sequence is only smoothed when its inflection points are at adjacent nodes If the five point smoothing method is used then the mentioned sequence is also smoothed when its inflection points are at every other node Only the concentration values corresponding to the mentioned sequence are smoothed and only when the number of inflection points is 2 or greater The smoothing methods ensure that the smoothing does not affect the average charge carrier concentration inside the smoothing range The user can specify the smoothing frequency this is done in the parameter sheet Simulation algorithm parameters The rate coefficients a and fj see Equation 1 1 mobilities and diffusion coefficients can be either constant or computed according to a user specified model For example charge carrier mobility can TM 1 6 X x X on ns UX 1 be calculated according to the formula y a exp b E such dependence is a result of the so called Poole Frenkel effect The number of charge carrier types trap types and processes that can be simulated using the programs CarrierFunc dll and CarrierParms exe is practically unlimited In the current ve
78. rface charges are smoothed by the method of adaptive exponential smoothing The definition of exponential smoothing is f af 1 f where f is the smoothed value at the moment of time T f is the smoothed value at the moment of time T and f is the original non smoothed value at the moment of time T The smoothing factor a is always between 0 and 1 A decrease of makes the resulting smoothed time dependence of f more smooth In this way random fluctuations of f are eliminated However along with random fluctuations the systematic changes of f are decreased too Adaptive exponential smoothing attempts to adjust the value of the smoothing factor a at each smoothing step so that the time dependence of f represents the systematic changes of fas accurately as possible I e if the systematic dependence of f on time becomes more pronounced then G is increased and if the time dependence of f becomes more random then a is decreased adaptive exponential smoothing is described in more detail in the GraphiXT user s manual Section Data smoothing In this manner values of q are computed for each node and for each charge type Then the average G is computed That average is used as the stationarity criterion Simulation is stopped when the average a decreases below the given threshold value alphaMin Notes 1 The exponential smoothing of time dependences of concentrations is only used to check if the state of the s
79. rsion v0 75 the only limitation is the requirement that the total number of parameters corresponding to one layer can not exceed 5000 and the total number of layers can not exceed 10 2 Introduction to user interface The GraphiXT plug in for simulation of charge transport in multi layer systems consists of two files the function file CarrierFunc dll and the model parameter editor CarrierParms exe If the function file and the parameter editor are loaded then the parameter editor can be invoked using the GraphiXT menu command Simulation options Model parameters and simulation can be started resumed or stopped using the menu command Start simulation or Stop simulation In order to load the function file the GraphiXT menu command Simulation options Model function file must be selected this action also causes the reference to the plug in user s manual to be added to the help system The parameter editor is loaded using the menu command Simulation options Parameter editor file Prior to starting simulation with the currently loaded plug in the user has to set the initial and final times of simulation the GraphiXT menu command Simulation options Time limits and amount of data The GraphiXT user s manual is in a separate file Further on only the GraphiXT plug in for simulation of charge transport will be described In the parameter editor window all parameters of the simulated syst
80. s or if there are several candidate processes defined then a corresponding error message will appear If an SRH recombination process has been defined then the change of concentration of traps of a given charge state during one simulation time step At is calculated not in the usual way 1 e not by multiplying the rate of concentration change by Af but on the basis of the requirement that the difference of the primary 13 14 carrier capture and release rates is equal to the difference of the secondary carrier capture and release rates This condition implies that concentration of state A traps is equal to E Vin 101 U y 20N eXP Er ale a EA ele ae cf Ey Vin 101 L N oo Zi FU 20 N E where N is the trap concentration N na ng n and m are concentrations of primary and secondary free charge carriers respectively Vn and Up are average thermal velocities of primary and secondary charge carriers which in the case of the Maxwell distribution are given by pois S 4 12 TIM ny N 4 11 o and o are capture cross sections of primary and secondary charge carriers respectively N and N are state densities of primary and secondary charge carriers respectively E and E are energy depths of the traps with respect to release of primary and secondary charge carriers respectively For example if the simulated system is a semiconductor with two types of charge carriers electrons and holes
81. s is centimeter cm the unit of energy is electronvolt eV the unit of mass is the electron rest mass in vacuum m m0 the unit of electric charge is the elementary charge e e other units correspond to the SI system of units The character A in unit notations means raising to a power The parameter values must be entered according to the following rules the number of significant digits must be such that all digits are visible and the decimal exponent must be specified using the format X xxxxxety for example the number 1 234 10 should be written 1 234e9 Note When displaying the previously entered or default parameter values with the decimal exponent the program uses the standard format of the C programming language where the exponent always consists of three digits and is preceded by the sign or for example the mentioned number would be displayed as 1 234e 009 but there is no need to insert zeros or the sign at the beginning of the exponent when entering a new value Some edit controls are used to specify the name of a particular parameter set 1 e of a particular type of charge carriers or traps or of a particular process Some default names contain names of other objects or charge values For example the name of a process of electron capture into traps could be the following don 1 lt el capture This name consists of the following components
82. s n and m which are not constant However if the concentration of recombination centers is by several orders of magnitude less than concentrations of other traps and free charge carriers then the space charge of recombination centers has only a weak influence on electric field strength Consequently in such a case the errors caused by the mentioned non conservation of charge are small too The dialog window Shockley Read Hall recombination parameters provides an option to specify whether the space charge of recombination centers should be taken into account If that charge is taken into account then its variation with time is not compensated by a corresponding variation of the other types of charges present in the system i e the total electric charge of the system is not conserved If the space charge of recombination centers is not taken into account then the overall effect is such as if that charge was zero Hence this eliminates the mentioned non conservation of charge Consequently this is the default option although in reality recombination centers have a certain charge Notes 1 If concentration of recombination centers is comparatively small then turning SRH recombination on or off has practically no effect on the simulation results this option only affects the method of calculation of trap population by charge carriers Besides the formula 4 11 is exact in the stationary state Therefore if the aim of the simulation is cha
83. s necessary when the interface traps act as an intermediate state for charge carrier transfer from one layer to another In this case the interface traps do not belong to a particular layer but belong to both adjacent layers to be more precise they characterize the interface between the two layers Since this program requires each parameter set to belong to one layer the concept of coinciding interface traps is used i e interface traps belonging to one layer coincide with interface traps belonging to the adjacent layer The drop down list Selected interface traps is used to select interface traps when there are two or more types of interface traps in the current layer All parameters shown in this parameter sheet correspond to the selected type of interface traps The next two text boxes are used to enter the full name and the abbreviated name of the traps The drop down list Surface density of the traps is used to select the trap surface density computation method In the current version of the model v0 75 only one choice is available Constant which corresponds to a constant trap surface density its value must be entered in the nearby text box If the selected interface traps coincide with interface traps defined in the adjacent layer the primary traps then the drop down list Surface density of the traps is replaced with two static text fields containing the names of the adjacent
84. t be entered twice The rate coefficients o and f see Equation 1 1 mobilities and diffusion coefficients can be either constant or computed according to a user specified model For example in the case of Fig 1 electron mobility is calculated according to the formula y a exp b E such dependence is a result of the so called Poole Frenkel effect If a coefficient is computed according to a particular model then parameters of that model can be modified by clicking the button Model param which is to the right of the model name see Fig 1 If the selected model does not have user adjustable parameters then the mentioned button is absent For example if the model Einstein relation is selected in the list of diffusion coefficient models then the diffusion coefficient is calculated from the values of mobility and temperature which are specified elsewhere In order to apply parameter changes the button OK or Apply must be clicked Some parameters e g some simulation algorithm parameters external voltage etc can be modified without stopping the simulation process In the case of a major change such as adding or deleting a parameter set or changing a parameter that is used to define the initial state of the system the simulation process will be terminated and all stored model data will be deleted In this case prior to applying the changes a waming message will appear with an option t
85. t electrode U1 1 O Teo decimos Electrode whose potential is varied let O right One electrode at the left edge of the system Initial potential of extemal voltage source U2 One electrode at the right edge of the system 6U2 U1 u2 9 System left edge coordinate lt 0 0 um Electrode area S 1 System right edge coordinate 61 02 jum Extemial resistance R 0 Om Extemal capacitance C 0 F Time dependence of the extemal voltage source potential Linear variation rii k A Start time of linear change U1 0 s Rate of potential change dU dt 1 Vis End time of linear change U2 1 s ek L em a Temperature T 300 K Initial potential of the left electrode U0 Hectrode configuration Initial potential of the right electrode U1 9 Two electrodes El y ilvai left One electrode at the left edge of the system Initial potential of extemal voltage source UZ One electrode at the right edge of the system U2 U0 u2 9 System left edge coordinate 0 0 um Electrode area S System right edge coordinate 61 02 um Edemalresistance R 0 Om Extemal capacitance C 0 F Time dependence of the extemal voltage source potential L iati Start time of linear change U1 Rate of potential change dU dt End time of linear change U2 ox come J ao Fig 15 Examples of the parameter sheet External parameters a when the right electrode potential is varied b when the
86. tal current i S j Where ji is the total current density The text box External resistance R is used to enter the resistance of the mentioned resistor which separates the external voltage source and the previously specified electrode of the system The text box External capacitance C is used to enter capacitance of the external circuit that capacitance is connected in parallel with the simulated system The drop down list Time dependence of the external voltage source potential is used to select one of four standard time dependences Constant voltage Periodic square pulse sequence Linear variation and Sine function The instantaneous value of potential Uz of the external source pole which is connected to the previously specified electrode of the system via the resistor R is equal to the sum of the previously specified initial value and the selected time dependence The remaining text boxes are used to enter parameters of the selected time dependence e g period pulse height etc If all parameters of the time dependence are zero then the corresponding time dependence is also assumed to be zero If at least one parameter is non zero then the parameter editor checks if the parameter values satisfy the restrictions that may exist in the context of the selected time dependence for example if the periodic square pulse sequence is selected then pulse duration must be less than the period of th
87. tation Number of processes 0 Add or remove a process Selected transmutation Selected process Transmutation name Generation process name Secondary camiers Camiers of opposite charge Rate coefficient Generation rate Constant Ys Constant 1 cm 3 s Fig 1 An example of the parameter editor window the parameter sheet Free carrier parameters is active list for selecting a parameter set of the corresponding type The parameter values corresponding to the selected parameter set are under that drop down list Thus only one set of parameters of each type is shown at a time Some simulation parameters do not belong to any layer of the simulated multi layer system Those are the simulation algorithm parameters e g the parameters that control the time step of the simulation and external parameters e g temperature and external illumination parameters Note The program assigns those parameters to the layer No 0 This layer is not included in the total number of layers that is shown above the button Add or delete a layer see Fig 1 The real layers which compose the simulated system are numbered starting from 1 only such layers can be added or removed In the parameter editor window units of all parameters are shown to the right of their values Values of layer thickness and coordinates of various points of the system are given in microns um um the unit of length that is used in derived unit
88. tationary state is reached Then the graph time is incremented by a given amount see description of the parameter interval below and simulation of another stationary state starts using the previous stationary state as the initial state of the system Notes 1 In order to stop the simulation upon reaching the stationary state the previously mentioned stationary state simulation mode must be turned on In this mode the mentioned parameters minR maxR and dtMax_graph are only used to define the smoothed sequence of charge densities and not the points plotted in the graphs hence they do not affect the computer memory requirements 2 Intermediate computation results are also displayed in the graphs e The parameter interval defines the interval between time values used to compute values of external factors according to corresponding time functions For example let us assume that 1 zero initial voltage is specified in the parameter sheet External parameters 2 the chosen time dependence of the voltage is linear growth with rate 1 V s starting at the moment of time 0 3 the parameter interval is 1 s 4 the initial and final times of simulation specified in the GraphiXT dialog window Time limits and amount of data are 0 and 10 respectively Then the program will compute a sequence of 11 stationary states corresponding to external voltage values 0 V 1 V 2 V 10 V The corresponding argument
89. ter editor checks if the parameter values satisfy the restrictions that may exist in the context of the selected time dependence for example if the periodic square pulse sequence is selected then pulse duration must be less than the period of the sequence If a parameter value is outside those bounds then a corresponding error message appears Note If the line Periodic sequence of short pulses is selected then the pulse duration will be assumed to be zero I e in this case each pulse creates a certain concentration of charge carriers instantaneously That concentration depends on the number of photons per square centimeter in one pulse Consequently in this case the number of photons per square centimeter must be specified instead of the photon flux density Initial camer distribution Simulation algorithm parameters Extemal parameters Photogeneration parameters Pi of L Extemal illuminati 27 E pr arameters of spectral components o illumination E Number of spectral components 3 Exposed edge of the system Oti Oni ema Selected layer te Selected spectral component 2 gt Initial photon flux density 10 0 1 s cm 2 a Name of the spectral component Weight factor wf2 Time dependence of photon flux density m i Green 550 nm 1 Periodic sequence of s
90. threads this is done by selecting the radio button Optimize the number of simulation threads In the latter case the program determines the number of threads that corresponds to the maximum processing speed 1 e to the minimum processing time of a single simulation time step This is done as follows The number of active threads is periodically every 5000 simulation time steps increased or decreased by 1 then the processing time corresponding to 100 time steps is measured and the average processing time of a single simulation step is calculated and compared with the previous value Based on the result of this comparison it is determined if the simulation should continue using the new number of threads or if the previous number of threads should be restored When optimization of the number of simulation threads is turned on the number of threads can not exceed the number of processors in the computer 4 Free charge carrier parameters The parameter sheet Free carrier parameters is used to enter parameters of each type of free charge carriers as well as parameters that define processes of bipolar recombination bipolar generation and charge carrier transmutations An example of this parameter sheet is shown in Fig 3 Below are explanations of all controls of this parameter sheet e The button Add or remove charge carriers is used to open a dialog window where one of three actions can be selected addition of a new type of free
91. tion is selected then the current density due to thermionic emission from the electrode will be calculated according to the Richardson equation j AT on E 8 2 where A is the Richardson constant A 1 20173 10 Am K However when simulating thermionic emission from a metal into a semiconductor or a dielectric it is more appropriate to use the effective Richardson constant A m mo A where m is the charge carrier effective mass in the semiconductor in the transverse direction to the interface in the case of an isotropic semiconductor m is equal to the density of states effective mass m Then Richardson equation with an additional factor must be selected and the value of the ratio m mo must be entered as the additional factor When the selected layer is an electrode and the potential barrier height is non zero then the Schottky effect can be taken into account The term Schottky effect refers to the decrease of the potential barrier height in the presence of external electric field that accelerates the emitted charge carriers away from the electrode In order to take into account the Schottky effect the checkbox Schottky effect must be checked Then the expressions of the thermionic injection current density 8 1 and 8 2 will be modified by replacing with amp AB where AQ is the mentioned decrease of the potential barrier height AQ e MEAR 8 3 ANE E where E is the electr
92. tionary state simulation mode In this mode simulation is stopped when either of these two conditions is satisfied 1 the largest duration of the simulated process is reached that duration is entered using the GraphiXT menu command Simulation options Time limits and amount of data 2 the stationary state is reached it is such a state when all bulk and interface charges of the system stop depending on time e In the stationary state simulation mode it is possible to modify the stationarity criterion alphaMin The stationary state is defined as the state when all charge densities of all types of bulk and interface charges at all nodes of the system do not depend on time The stationarity can be exact or approximate Under conditions of exact stationarity the increments of all types of charge densities at all nodes during a single time step of simulation are less than the round off error the relative round off error is approximately 10 so that the newly computed values are exactly equal to the old ones Under conditions of approximate stationarity the charge densities fluctuate with time but those variations are random i e the average of several last values is approximately constant The approximate stationarity is detected as follows Time dependences of concentrations of free charge carriers bulk and interface traps of all types and charge states at all nodes as well as free interface charges and 7 electrode su
93. ts see Section 2 Note The quote symbols that appear in some function names given in tables below are only for clarity they do not appear in actual function names 1 Global functions la Global coordinate functions Name Meaning Space charge density e cm 3 Electric space charge density Electric field V cm Electric field strength a negative value means that the electric field vector is directed in the negative direction of the X axis Potential V Electrostatic potential Conduction current density A cm 2 The sum of drift and diffusion current densities of all free charge carriers 1b Global time functions Name Meaning Electric charge e cm 2 The total electric charge of the system the sum of the integral of space charge density over the width of the system and all surface charge densities excluding electrode charges Total current density A cm 2 The sum of conduction and displacement current densities Total current A The product of the total current density and the electrode area External voltage source current A The resistor R current ip see Fig 15a and Fig 15b External capacitor current A The capacitor C current ic see Fig 15a and Fig 15b External photon flux density 1 cm 2 s Number of photons incident on unit area per unit time Simulation time step s The value of the last sim
94. ulation time step At Processing time of one time step of the simulation s The processing time of the last simulation time step Number of simulation threads The number of active simulation threads Total processing time s The total processing time Total number of simulation time steps The total number of simulation time steps Left electrode surface charge density e cm 2 The left electrode surface charge density Right electrode surface charge density e cm 2 The right electrode surface charge density External voltage source potential V The potential U see Fig 15a and Fig 15b Voltage drop in the external resistor V If the right electrode potential is varied then the potential difference U U gt see Fig 15a and if the left electrode potential is varied then U2 Up see Fig 15b Potential of the left edge of the system V Potential of the point corresponding to the minimum value of the coordinate x in Fig 15a and Fig 15b it is denoted Up Left edge surface charge density e cm 2 The total surface charge density of the left edge of the system 1t includes the charge of the interface traps and the free surface charge which was defined in Section 8 Left edge free surface charge density e cm 2 Free surface charge density of the left edge of the system 36 2 Layer functions 2a Layer coordinate functions
95. unctions Functions corresponding to interface traps in the table below A stands for the interface trap name which is specified by the user Name Meaning A conc 1 cm 2 Surface density of interface traps A A charge dens e cm 2 Surface charge density of interface traps A Functions corresponding to charge states of interface traps in the table below A stands for the trap name which is specified by the user and Q is an integer number indicating the charge state Name Meaning A Q conc 1 cm 2 Surface density of interface traps A whose charge is equal to Q elementary charges Functions corresponding to processes of charge carrier capture into interface traps and release from them Those functions have the meaning of a process rate i e the number of elementary events per unit area and unit time The default name of those functions is P rate 1 s cm 2 where P stands for the name of the given process The process name is specified by the user also see Section 2 Introduction to user interface The function names corresponding to the default process names are given in the table below in this table A and B stand for the short names of interface traps and free charge carriers respectively and Q is an integer number indicating the initial charge state of those traps 38 N
96. unctions have the meaning of a process rate 1 e the number of elementary events per unit volume and unit time The default name of those functions is P rate 1 s em 3 where P stands for the name of the given process The process name is specified by the user also see Section 2 Introduction to user interface The function names corresponding to the default process names are given in the table below in this table A and B stand for the short names of the primary and secondary charge carriers which are specified by the user 37 Name Meaning A B recombination rate 1 s cm 3 Bipolar recombination rate A B generation rate 1 s cm 3 Bipolar generation rate A gt B rate 1 s cm 3 The rate of charge carrier A transmutation into the charge carriers B A B photogeneration rate 1 s cm 3 Photogeneration rate Functions corresponding to bulk traps in the table below A stands for the trap name which is specified by the user Name Meaning A concentration 1 cm 3 Concentration of bulk traps A A charge density e cm 3 Space charge density of bulk traps A taking into account all charge states of those traps Functions corresponding to charge states of bulk traps in the table below A stands for the trap name which is specified by the user
97. ystem is stationary All stored values of model functions are computed using original non smoothed concentrations In this respect this type of smoothing differs from the previously mentioned smoothing of concentration dependences on the coordinate which modifies concentrations of free charge carriers and the exponential smoothing of concentration time derivatives which also modifies those time derivatives 2 Sometimes when the state of the simulated system is close to stationary charge carrier concentrations begin oscillating in time If those oscillations are relatively smooth i e if their period is at least several times longer than the time interval between the smoothed values then the smoothing factor a will begin oscillating as well and its decrease will slow down significantly In order to avoid such a situation the smoothed sequence of values only includes the points where the concentration time derivative changes sign 1 e the concentration maxima and minima Then the smoothing factor continues decreasing even in the presence of the mentioned oscillations hence it eventually becomes less than the user specified threshold value alphaMin and simulation of the stationary state is stopped 3 Another difference between exponential smoothing of concentrations and the previously mentioned exponential smoothing of concentration time derivatives is that values of a are different for different nodes and different types of c

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