Simulation methods, simulation programs, simulation systems, and information processing devices.

Abstract

This invention provides a simulation method, simulation program, simulation system, and information processing device capable of reproducing carrier transfer delays in actual devices. [Solution] The simulation method uses Poisson's equation, the electron current continuity equation, and the hole current continuity equation to calculate a new free carrier density that takes into account the capture of residual carriers at trap sites and the excitation of captured carriers. Furthermore, the simulation method repeatedly calculates the number of residual carriers and determines whether residual carriers are captured at trap sites and whether captured carriers are excited, based on the newly calculated free carrier density, until the number of residual carriers converges.

Description

[Technical Field]

[0001] This disclosure relates to a simulation method, a simulation program, a simulation system, and an information processing device. [Background technology]

[0002] In the design of semiconductor devices, device simulations can be used to determine the electrical characteristics of the semiconductor device through numerical analysis. For example, in the design of photodiodes provided in each pixel of an image sensor, transfer simulations can be used to calculate the transient state in which carriers accumulated in the photodiode are transferred to floating diffusion. Device simulations are disclosed, for example, in Patent Documents 1 and 2. [Prior art documents] [Patent Documents]

[0003] [Patent Document 1] U.S. Patent Publication No. 2014 / 0019101 [Patent Document 2] Japanese Patent Publication No. 2008-028328 [Overview of the Initiative] [Problems that the invention aims to solve]

[0004] Conventional transfer simulations have the problem of not being able to reproduce the carrier transfer delay in actual photodiodes. This problem is not limited to image sensors, but can also occur in devices such as flash memory, DRAM, MOSFETs, and Taiyo Yuden. It is desirable to provide a simulation method, simulation program, simulation system, and information processing device that can reproduce the carrier transfer delay in actual devices. [Means for solving the problem]

[0005] The simulation method relating to the first aspect of this disclosure includes the following five steps: (1) Calculate the number of remaining carriers in the semiconductor device by spatially integrating the free carrier density in the semiconductor device during a transient state. (2) For each of the remaining carriers, determine whether or not the remaining carrier will be captured at the trap site. (3) For each capture carrier captured at the trap site, determine whether or not the capture carrier is excited. (4) Using Poisson's equation, the electron current continuity equation, and the hole current continuity equation, calculate a new free carrier density that takes into account the capture of the remaining carriers at the trap site and the excitation of the captured carriers. (5) Based on the new free carrier density, the number of remaining carriers in the semiconductor device is calculated, and the trapping of the remaining carriers at the trap site and the excitation of the trapped carriers are determined, and this process is repeated until the number of remaining carriers in the semiconductor device converges.

[0006] The simulation program relating to the second aspect of this disclosure causes a computer to perform the five steps ((1) to (5)) described above.

[0007] The simulation system relating to the third aspect of this disclosure comprises an input device, a calculation device, and an output device. The input device is capable of acquiring boundary conditions. The calculation device is capable of executing the five steps ((1) to (5)) described above. The output device is capable of outputting the electrical characteristics of the semiconductor device obtained by the calculation device.

[0008] The information processing device relating to the fourth aspect of this disclosure comprises an input device and an output device. The input device is capable of receiving inputs of the impurity density and structure of a semiconductor device, the trapping cross-section of the trapping sites of the semiconductor device, the thermal velocity and trapping concentration, and an external bias applied to the semiconductor device as set values. The output device is capable of outputting the time evolution of the number of carriers, the carrier density and electrostatic potential, and the time evolution of the density distribution of trapped carriers and excited carriers as transient data of the semiconductor device, calculated using the set values ​​obtained via the input device. [Brief explanation of the drawing]

[0009] [Figure 1] Figure 1 is a diagram showing an example of the schematic configuration of a simulation system according to one embodiment of the present disclosure. [Figure 2] Figure 2(A) shows an example of the density distribution of substrate impurities in a photodiode. Figure 2(B) shows an example of the density distribution in Figure 2(A) discretized. [Figure 3] Figure 3 shows an example of the density distribution of each discretized donor impurity. [Figure 4] Figure 4 shows an example of the free carrier density distribution within a photodiode. [Figure 5] Figure 5 shows an example of a trap site in a photodiode where free carriers are captured. [Figure 6] Figure 6(A) shows an example of the density distribution of free carriers in a photodiode before capture and reflection. Figure 6(B) shows an example of the density distribution of free carriers in a photodiode after capture and reflection. [Figure 7] Figure 7(A) shows an example of the density distribution of free carriers in a photodiode before excitation reflection. Figure 7(B) shows an example of the density distribution of free carriers in a photodiode after excitation reflection. [Figure 8] Figure 8 shows an example of the transfer characteristics of a photodiode. [Figure 9] Figure 9 shows an example of the procedure for executing a transfer simulation in the simulation system shown in Figure 1. [Figure 10] Figure 10 is a diagram illustrating an example of the execution procedure for the transient calculation shown in Figure 9. [Figure 11] Figure 11 is a diagram illustrating an example of the procedure for executing the capture calculation shown in Figure 10. [Figure 12] Figure 12 is a diagram illustrating an example of the procedure for performing the excitation calculation shown in Figure 10. [Figure 13] Figure 13(A) shows an example of the density distribution of free carriers in a photodiode before capture and reflection. Figure 13(B) shows an example of the density distribution of free carriers in a photodiode after capture and reflection. [Figure 14] Figure 14(A) shows an example of the density distribution of free carriers in a photodiode before excitation reflection. Figure 14(B) shows an example of the density distribution of free carriers in a photodiode after excitation reflection.

[0010] <1. Embodiment> [composition] A simulation system 1 according to one embodiment of this disclosure will be described. Simulation system 1 is a transfer simulation that calculates the transient state in which carriers accumulated in a photodiode are transferred to floating diffusion. Hereinafter, the "transient state in which carriers accumulated in a photodiode are transferred to floating diffusion" will be simply referred to as the "transient state".

[0011] Simulation System 1 corresponds to a specific example of the "simulation system" in one embodiment of the present disclosure. The photodiode corresponds to a specific example of the "semiconductor device" in one embodiment of the present disclosure. In the transfer simulation, several governing equations are solved with the electrostatic potential and carrier density as variables, and the impurity (donor, acceptor) density and boundary conditions of the photodiode being constant values. This calculates the transient state of carrier transfer in the photodiode.

[0012] Figure 1 shows an example of the schematic configuration of simulation system 1. Simulation system 1 includes, for example, an input device 10, a storage device 20, an arithmetic unit 30, and an output device 40, as shown in Figure 1. The input device 10 corresponds to one specific example of the "input device" in one embodiment of the present disclosure. The storage device 20 corresponds to one specific example of the "storage device" in one embodiment of the present disclosure. The arithmetic unit 30 corresponds to one specific example of the "arithmetic unit" in one embodiment of the present disclosure. The output device 40 corresponds to one specific example of the "output device" in one embodiment of the present disclosure. A standalone information processing device may include the input device 10 and the output device 40.

[0013] The input device 10 is an information processing device having, for example, a GUI (Graphical User Interface) or CUI (Character-based User Interface) for setting initial conditions and boundary conditions. The input device 10 is capable of receiving input of initial conditions and boundary conditions. The input device 10 may be capable of acquiring numerical data, for example, by direct input by the user. The input device 10 may be capable of acquiring a data file with the input file name from the input file path, for example, by inputting the file name and file path by the user. The input device 10 is capable of transmitting the acquired initial conditions and boundary conditions to the arithmetic unit 30.

[0014] The storage device 20 is a device having a non-volatile recording medium such as EEPROM (Electrically Erasable Programmable Read-Only Memory) or flash memory. The storage device 20 is a device having a non-volatile storage medium that can be read by the arithmetic unit 30 (computer). The storage device 20 stores, for example, an arithmetic program 21 that includes a series of procedures for performing a transfer simulation, and data generated by the execution of the arithmetic program 21. The data generated by the execution of the arithmetic program 21 includes, for example, capture carrier data 22 and excitation carrier data 23.

[0015] The output device 40 is an information processing device having a GUI for visualizing data on the electrical characteristics of a photodiode, which is the result of calculations performed by the arithmetic unit 30. The output device 40 is an information processing device having a GUI for visualizing the time evolution of the number of free carriers, carrier density and electrostatic potential, as well as the time evolution of the density distribution of captured carriers and excited carriers, as transient data of the photodiode. The output device 40 is capable of acquiring data on the electrical characteristics of a photodiode, which is the result of calculations performed by the arithmetic unit 30, from the arithmetic unit 30. The output device 40 is also capable of acquiring the above transient data from the arithmetic unit 30. The output device 40 is capable of outputting (for example, displaying) the data acquired from the arithmetic unit 30. The output device 40 may, for example, display the data acquired from the arithmetic unit 30 as a graph on the screen. The output device 40 may, for example, generate the data acquired from the arithmetic unit 30 as a data file and store it in a specified memory.

[0016] The arithmetic unit 30 is an information processing device (computer) having an arithmetic unit such as a CPU (Central Processing Unit). The arithmetic unit 30 can perform a transfer simulation by reading an arithmetic program 21 from the storage device 20 and executing the read arithmetic program 21. When executing the arithmetic program 21, the arithmetic unit 30 can obtain initial conditions and boundary conditions from the input device 10. The arithmetic unit 30 can store the data obtained by executing the arithmetic program 21 (data about the electrical characteristics of the photodiode) in the storage device 20.

[0017] The arithmetic unit 30 can acquire initial conditions and boundary conditions via the input device 10. Using the initial conditions and boundary conditions obtained via the input device 10, the arithmetic unit 30 can calculate the steady state and transient state of the photodiode, and can also calculate the electrical characteristics of the photodiode based on the calculated steady state and transient state. The initial conditions are the impurity density (donor density N) of the photodiode. D and acceptor density N A ) includes. Donor density N D This represents a continuous donor density distribution with position r within the photodiode as a variable. Acceptor density N A This represents a continuous acceptor density distribution with position r within the photodiode as a variable. Initial conditions further include the trapping cross-section of the trapping site, thermal velocity, and trap concentration. Boundary conditions include the structure of the photodiode (dimensions and electrode positions) and the external bias applied to the photodiode (gate voltage, etc.). The external bias (gate voltage, etc.) includes, for example, the gate voltage when the photodiode is in its initial state (e.g., 0V) and the gate voltage when the photodiode is in a transient state (e.g., 3V).

[0018] The computing unit 30 calculates the donor density N D and acceptor density N AIt is possible to spatially discretize. In the spatial discretization, the arithmetic unit 30, for example, as shown in FIGS. 2(A) and 2(B), sets a larger number of discrete positions in regions where the donor density N D is higher, and a smaller number of discrete positions in regions where the donor density N D is lower. The arithmetic unit 30 can further set a larger number of discrete positions in regions where the acceptor density N A is higher, and a smaller number of discrete positions in regions where the acceptor density N A is lower.

[0019] For each discretized donor impurity, the arithmetic unit 30 can set a density α(i, r) that is a function of the position r from the center of the discretized donor and the position R i of the center of the i-th discrete impurity. The arithmetic unit 30 can further set a density α(i, r) that is a function of the position r from the center of the discretized acceptor and the position R i of the center of the i-th discrete impurity. The density α(i, r) is represented by the following formula (1). The density α is the charge density created by the i-th discrete impurity at the position r, and has, for example, the distribution shown in FIG. 3.

[0020]

Equation

[0021] By adding the density α(i, r) set for each donor impurity, the arithmetic unit 30 can calculate the density (donor density N Ddis (r)) of the spatially discretized donor impurities in the entire calculation region. By further adding the density α(i, r) set for each acceptor impurity, the arithmetic unit 30 can calculate the density (acceptor density N Adis (r)) of the spatially discretized acceptor impurities in the entire calculation region. The donor density N Ddis(r) represents a discrete donor density distribution with position r in the photodiode as a variable, for example, as shown in Figure 2(B). Donor density N Ddis (r) is expressed by the following equation (2): Acceptor density N Adis (r) represents a discrete acceptor density distribution with position r within the photodiode as a variable. Acceptor density N Adis (r) is expressed by the following equation (3).

[0022]

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[0023] The computing unit 30 is capable of performing steady-state calculations using multiple governing equations. Steady-state calculation refers to calculating the carrier density (electron concentration n and hole concentration p) and electrostatic potential of a photodiode when the photodiode is in a steady state. A steady state refers to a state in the photodiode where the inflow and outflow of carriers are kept constant, and the overall state does not change over time.

[0024] The governing equations used in steady-state calculations include Poisson's equation, the electron current continuity equation, and the hole current continuity equation. Poisson's equation is expressed, for example, by equation (4) below. The electron current continuity equation is expressed, for example, by equation (5) below. The electron current density J included in equation (5) n The hole current continuity equation is, for example, represented by the following equation (6). The hole current density J included in equation (7) p This can be expressed, for example, by the following equation (8).

[0025]

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[0026] ψ: Electrostatic potential of a photodiode e: Elementary charge ε: Dielectric constant of the photodiode N D Photodiode donor density (continuous value) N Ddis Photodiode donor density (discrete value) N A Photodiode acceptor density (continuous value) N Adis Photodiode acceptor density (discrete value) n: Electron concentration of the photodiode (number density of free electrons) p: Hole concentration (number density of holes) of the photodiode J n :electron current density G n : rate of electron production R n : electron recombination rate μ n : Electron mobility D n : Electron diffusion constant J p : Hole current density G p : Hole generation rate R p : Hole recombination rate μ p : Hole mobility D p : Hole diffusion constant

[0027] The computing device 30 can calculate the electrostatic potential ψ, electron concentration n, and hole concentration p by solving equations (4) to (8). At this time, the boundary conditions for Poisson's equation, the electron current continuity equation, and the hole current continuity equation are fixed to the gate voltage (e.g., 0V) when the photodiode is in its initial state. The electrostatic potential ψ, electron concentration n, and hole concentration p obtained by the steady-state calculation become the initial input parameters in the transient calculation described later.

[0028] The arithmetic unit 30 is capable of performing transient calculations using multiple governing equations. Transient calculation refers to calculating the changes in the carrier density (electron concentration n and hole concentration p) and electrostatic potential of a photodiode when it is in a transient state. The arithmetic unit 30 is capable of discretizing time in transient calculations. The multiple governing equations used in transient calculations include the temporally and spatially discretized Poisson equation, and the temporally discretized electron current continuity equation and hole current continuity equation.

[0029] The arithmetic unit 30 can use the initial input parameters (electrostatic potential ψ, electron concentration n, and hole concentration p) obtained in steady-state calculations as initial values ​​for transient calculations. In transient calculations, the arithmetic unit 30 can set the boundary conditions for the Poisson equation, the electron current continuity equation, and the hole current continuity equation to the gate voltage (e.g., 3V) when the photodiode is in a transient state. The arithmetic unit 30 can calculate the carrier density after dt seconds using the time-discretized electron current continuity equation and the hole current continuity equation. In this embodiment, the carriers are electrons. The arithmetic unit 30 can calculate the electrostatic potential after dt seconds using the time- and spatially discretized Poisson equation.

[0030] The time-discretized electron current continuity equation can be expressed, for example, by the following equation (9). The electron current density J included in equation (9) nThis can be expressed, for example, by the following equation (10). The time-discretized hole current continuity equation can be expressed, for example, by the following equation (11). The hole current density J included in equation (11) p This can be expressed, for example, by equation (12) below. The arithmetic unit 30 can calculate the electron concentration n and hole concentration p after dt seconds when the photodiode is in a transient state by solving equations (9) to (12).

[0031]

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[0032] The computing device 30 is capable of performing calculations regarding the capture and excitation of free carriers when the photodiode is in a transient state. The computing device 30 can set a calculation domain in the photodiode, for example, as shown in Figure 4. The computing device 30 can calculate the number of discrete free carriers by spatially integrating the free carrier density of the entire set calculation domain. Specifically, the computing device 30 can calculate the number of discrete electrons X by spatially integrating the electron concentration n of the entire set calculation domain. The number X corresponds to the number of free carriers remaining in the photodiode (residual carriers). In this way, the computing device 30 can calculate the number of residual carriers in the photodiode by spatially integrating the free carrier density in the photodiode during the transient state.

[0033] The arithmetic unit 30 can determine the capture state of residual carriers in the photodiode for each residual carrier at the trap sites. Specifically, the arithmetic unit 30 can determine whether each residual carrier in the photodiode is captured by the trap sites. Based on the capture probability determined by the capture cross-sectional area, thermal velocity, and trap concentration of the trap sites, the arithmetic unit 30 can set the capture range and non-capture range between 0 and 1. Here, let the number of residual carriers be X. The arithmetic unit 30 can issue a random number between 0 and 1 for each residual carrier, and when the issued random number is included in the capture range, it can be determined that the residual carrier is captured by the trap site. The arithmetic unit 30 can issue a random number between 0 and 1 for each residual carrier and calculate the number Y of random numbers included in the capture range among the X issued random numbers.

[0034] The arithmetic unit 30 can set the number of trap sites corresponding to the trap concentration in the calculation region of the photodiode. The arithmetic unit 30 can uniformly arrange the set number of trap sites in the calculation region of the photodiode. The arithmetic unit 30 can select Y (Y < X) trap sites with a uniform probability from among the arranged plurality of trap sites. The positions of the selected trap sites correspond to the capture positions of the residual carriers. When Y = 4, the arithmetic unit 30 can select, for example, four trap sites (r a , r b , r c , r d ) as shown in FIG. 5. At this time, four of the residual carriers in the photodiode are captured at four locations (r a , r b , r c , r d ).

[0035] The calculation unit 30 is capable of setting trap levels for each selected trap site. The calculation unit 30 is capable of selecting a level with uniform probability from among a plurality of trap levels uniformly set in the band gap of the photodiode and setting it as the trap level for the selected trap site. The calculation unit 30 is capable of setting trap levels for each selected trap site. When Y=4, the calculation unit 30, for example, sets trap levels at 4 locations (r a ,r b ,r c ,r d ) Trap level E of the trap site t a ,E t b ,E t c ,E t d It is possible to set these parameters. The computing device 30 can store the capture position and trap level of the free carriers obtained in this way as capture carrier data 22 in the storage device 20.

[0036] Based on the captured carrier data 22, the computing unit 30 calculates the density N equivalent to one free carrier from the electron concentration n for each location of a free carrier (captured carrier) captured at a trap site. t Subtracting this, the electron concentration n is equal to the density N corresponding to one capture carrier. t It is possible to add the density N. t This is represented, for example, by equation (13). The arithmetic unit 30 determines the position of the capture carrier as shown, for example, by equation (14), r j If (1 ≤ j ≤ Y), then position r j In the right-hand side of the Poisson equation in , from the free carrier density, density N t Subtracting this, the capture carrier density is obtained as density N t It is possible to add this to position r. j In the right-hand side of the Poisson equation in this case, for example, as shown in Figures 6(A) and 6(B), the density N from the free carrier density is tSubtraction of and density N as capture carrier density t The addition cancels out, and the total charge remains unchanged.

[0037]

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[0038] The computing device 30 is capable of determining whether or not each capture carrier is excited. The computing device 30 calculates the capture probability and the Boltzmann factor (exp(-(E c -E t Based on the excitation probability determined by () / kT), the excitation range and non-excitation range can be set between 0 and 1. Here, the number of capture carriers is Y. At this time, the arithmetic unit 30 can issue a random number between 0 and 1 for each capture carrier, and when the Y issued random numbers fall within the excitation range, it can determine that the capture carrier is excited. The arithmetic unit 30 can issue a random number between 0 and 1 for each capture carrier and calculate the number Z of random numbers that fall within the excitation range out of the Y issued random numbers. The arithmetic unit 30 can store the excitation position of the excited capture carrier (excited carrier) obtained in this way as excitation carrier data 23 in the memory device 20.

[0039] Based on the excitation carrier data 23, the computing unit 30 calculates the density N equivalent to one free carrier for each position of the excitation carrier, according to the free carrier concentration. t Adding these together, the density N equivalent to one captured carrier is obtained from the free carrier concentration. t It is possible to subtract the following. Specifically, based on the excitation carrier data 23, the computing device 30 subtracts the density N, which corresponds to one free carrier, from the electron concentration n for each position of the excitation carrier. t By adding these together, the density N corresponding to one capture carrier is obtained from the electron concentration n. tIt is possible to subtract from this. The arithmetic unit 30, for example, as shown in equation (15), determines the position of the excited carrier r m If (1 ≤ m ≤ Z), then position r m In the right-hand side of the Poisson equation in , the free carrier density is given by density N. t By adding this, it is possible to make the captured carrier density zero. At this time, position r m In the right-hand side of the Poisson equation in this case, for example, as shown in Figures 7(A) and 7(B), the density N to the free carrier density t The sum of the two, and the density N from the capture carrier density. t The subtraction cancels out, and the total charge remains unchanged.

[0040]

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[0041] The computing device 30 can obtain a spatially and temporally discretized Poisson equation (equation (16) below) by introducing the free carrier density and captured carrier density obtained as a result of considering the capture and excitation of free carriers into the right-hand side of the Poisson equation. The computing device 30 can obtain equation (16) based on the captured carrier data 22 and the excited carrier data 23.

[0042] The computing device 30 can use the obtained equation (16) to calculate a new electrostatic potential that takes into account the capture of residual carriers at trap sites and the excitation of captured carriers. By solving the obtained equation (16), the computing device 30 can calculate a new electrostatic potential ψ that takes into account the capture position of residual carriers and the excitation position of captured carriers. By solving the obtained equation (16), the computing device 30 can calculate the electrostatic potential ψ(t+dt) after dt seconds. In equation (16), N c This refers to the number of carriers captured. r This refers to the number of excited carriers.

[0043]

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[0044] The computing device 30 can use the free carrier density after dt seconds, obtained by considering carrier capture and excitation, as the electron concentration n(t). Furthermore, the computing device 30 can use the hole concentration p(t+dt) after dt seconds, obtained by solving the hole current continuity equation (equation (11)), as the hole concentration p(t). These are expressed by the following equation (17).

number

[0045] The computing device 30 can substitute the electron concentration n(t) obtained by equation (17) into n(t) in equations (9) and (10). The computing device 30 can substitute the hole concentration p(t) obtained by equation (17) into p(t) in equations (11) and (12). The computing device 30 can use the electrostatic potential φ(t+dt) obtained after dt seconds by solving Poisson's equation (equation (16)) that reflects carrier capture and excitation as the electrostatic potential φ(t) and substitute it into equations (10) and (12).

[0046] The computing device 30 can calculate a new free carrier density that takes into account the capture of residual carriers at trap sites and the excitation of captured carriers by solving the updated equations (9) to (12) as described above. The computing device 30 can calculate a new carrier density after dt seconds by solving the updated equations (9) to (12) as described above. The computing device 30 can then calculate a new electrostatic potential by performing calculations on carrier capture and excitation when the photodiode is in a transient state using equation (16). The computing device 30 can calculate a new electrostatic potential φ(t+dt) after dt seconds by performing calculations on carrier capture and excitation when the photodiode is in a transient state using equation (16).

[0047] The computing unit 30 can repeatedly calculate the number of remaining carriers in the photodiode, determine the capture of the remaining carriers at the trap site, and determine the excitation of the captured carriers, based on the new free carrier density, until the number of remaining carriers in the photodiode converges to a predetermined threshold. The computing unit 30 can repeatedly calculate the carrier density and electrostatic potential after dt seconds until the number of remaining free carriers in the photodiode converges to a predetermined threshold. When the number of remaining free carriers in the photodiode converges to a predetermined threshold, the computing unit 30 can generate data about the electrical characteristics of the photodiode based on the data obtained when the photodiode was in a transient state. For example, the computing unit 30 can generate the transfer characteristics of the photodiode based on the data obtained when the photodiode was in a transient state.

[0048] The transfer characteristics of a photodiode are expressed, for example, by the time change in the number of residual carriers in the photodiode, as shown in Figure 8. Figure 8 shows the simulation results of the comparative example, the simulation results of the embodiment, and the measured results. The simulation results of the comparative example were obtained by solving for the carrier density and electrostatic potential after dt seconds using Poisson's equation, the electron current continuity equation, and the hole current continuity equation. The simulation results of the embodiment were obtained using the simulation system 1 described above. The measured results were obtained based on the measured potential of the floating diffusion when the charge accumulated in the photodiode was transferred via the floating diffusion by turning on the transfer transistor in a photodiode, floating diffusion, and transfer transistor formed on a semiconductor wafer. From Figure 8, it can be seen that the simulation results of the embodiment are in general agreement with the measured results.

[0049] [Execution Steps] Next, the procedure for executing the transfer simulation in Simulation System 1 will be described. Figure 9 shows an example of the procedure for executing the transfer simulation in Simulation System 1. Figure 10 shows an example of the procedure for executing the transient calculation in Figure 9. Figure 11 shows an example of the procedure for executing the capture calculation in Figure 10. Figure 12 shows an example of the procedure for executing the excitation calculation in Figure 10.

[0050] The arithmetic unit 30 obtains initial conditions and boundary conditions via the input device 10 (step S101). The initial conditions are the impurity density of the photodiode (donor density N). D and acceptor density N A The boundary conditions include the trapping cross-section of the trap site, the thermal velocity, and the trap concentration. The boundary conditions include the structure of the photodiode (dimensions and electrode position) and the external bias applied to the photodiode (gate voltage, etc.).

[0051] Next, the arithmetic unit 30 discretizes the substrate impurities (step S102). Specifically, the arithmetic unit 30 spatially discretizes the impurity density (donor density N D and acceptor density N A ) of the photodiode. The arithmetic unit 30 sets a larger number of discrete positions in regions with a higher donor density N D , and sets a smaller number of discrete positions in regions with a lower donor density N D . The arithmetic unit 30 further sets a larger number of discrete positions in regions with a higher acceptor density N A , and sets a smaller number of discrete positions in regions with a lower acceptor density N A . The arithmetic unit 30 further calculates the spatially discretized donor density N Ddis and acceptor density N Adis using equations (2) and (3).

[0052] Next, the arithmetic unit 30 performs a steady-state calculation using a plurality of governing equations (step S103). Specifically, the arithmetic unit 30 calculates the electrostatic potential ψ, electron concentration n, and hole concentration p of the photodiode by solving the Poisson equation (equation (4)), electron current continuity equations (equations (5) and (6)), and hole current continuity equations (equations (7) and (8)).

[0053] Next, the arithmetic unit 30 performs a transient calculation using a plurality of governing equations discretized in time (step S104). Specifically, first, the arithmetic unit 30 obtains the initial input parameters (electrostatic potential ψ, electron concentration n, and hole concentration p) obtained from the steady-state calculation as the initial values for the transient calculation (step S201). Next, the arithmetic unit 30 obtains the gate voltage (e.g., 3V) when the photodiode is in a transient state as the boundary condition for the transient calculation. The arithmetic unit 30 changes the boundary condition to the gate voltage (e.g., 3V) when the photodiode is in a transient state (step S202).

[0054] Next, the arithmetic unit 30 calculates the carrier density (electron concentration n and hole concentration p) after dt seconds when the photodiode is in a transient state by solving the time-discretized current continuity equations (equations (9) to (12)) (step S203). Next, the arithmetic unit 30 calculates the number of free carriers remaining in the photodiode during the transient state (residual carriers) (step S204). The arithmetic unit 30 calculates the number of residual carriers in the photodiode by spatially integrating the free carrier density in the photodiode during the transient state.

[0055] Next, the arithmetic unit 30 performs a capture calculation for free carriers (residual carriers) (step S205). Specifically, first, the arithmetic unit 30 determines whether or not each residual carrier in the photodiode will be captured at the trap site. The arithmetic unit 30 determines whether or not each residual carrier will be captured at the trap site using, for example, a random number (step S301). The arithmetic unit 30 issues a random number between 0 and 1 for each residual carrier, and determines that the residual carrier will be captured at the trap site if the issued random number falls within the capture range. If, as a result, the residual carrier is captured at the trap site (step S302; Y), the arithmetic unit 30 determines the capture position and capture level of the captured residual carrier (captured carrier) (step S303). The arithmetic unit 30 updates the right-hand side of the Poisson equation. Specifically, the arithmetic unit 30 determines the position of the captured carrier as shown in equation (14) r j Therefore, position r j In the right-hand side of the Poisson equation in , from the free carrier density, density N t Subtracting this, the capture carrier density is obtained as density N t Add (step S304).

[0056] Next, after the capture calculation of all free carriers (remaining carriers) is completed (step S206; Y), the arithmetic unit 30 performs excitation calculation of the captured carriers (step S207). Specifically, first, the arithmetic unit 30 determines whether each captured carrier in the photodiode is excited. The arithmetic unit 30 determines whether a captured carrier is excited, for example, using a random number (step S401). The arithmetic unit 30 issues a random number between 0 and 1 for each captured carrier, and determines that the captured carrier is excited when the Y issued random numbers are within the excitation range. As a result, when the captured carrier is excited (step S402; Y), the arithmetic unit 30 determines the excitation position of the excited captured carrier (step S403). The arithmetic unit 30 makes a determination using a random number for each captured carrier in the photodiode, and when the captured carrier is excited, determines the excitation position of the excited carrier. The arithmetic unit 30 updates the right side of the Poisson equation. Specifically, as shown in Equation (15), the arithmetic unit 30 sets the position of the excited carrier as r m and then, in the right side of the Poisson equation at the position r m , adds the free carrier density with the density N t and subtracts the density N t from the captured carrier density (step S404).

[0057] Next, after the excitation calculations for all captured carriers are completed (step S208; Y), the computing unit 30 calculates the electrostatic potential ψ(t+dt) after dt seconds by solving Poisson's equation (equation (16)) (step S209). If the computing unit 30 continues transient calculations (step S210; Y), it sets the electrostatic potential ψ(t+dt) after dt seconds as the electrostatic potential ψ(t) and further updates the free carrier density using equation (17). The computing unit 30 applies the updated free carrier density and electrostatic potential to equations (9) to (12) and solves these equations to calculate the carrier density (electron concentration n and hole concentration p) after dt seconds, taking into account the capture of remaining carriers at trap sites and the excitation of captured carriers (step S203). The computing unit 30 calculates the electrostatic potential after dt seconds by performing calculations about carrier capture and excitation (steps S204 to S209) using equation (16). The computing unit 30 repeatedly calculates the carrier density and electrostatic potential after dt seconds until the number of free carriers remaining in the photodiode converges to a predetermined threshold or less.

[0058] When the number of free carriers remaining in the photodiode converges to a predetermined threshold or less (i.e., when the transient calculation is terminated) (step S210;Y, step S105;Y), the arithmetic unit 30 generates and outputs data about the electrical characteristics of the photodiode based on the data obtained when the photodiode was in a transient state (step S206). In this way, the transfer simulation in the simulation system 1 is performed.

[0059] [effect] Next, the effects of the simulation system 1 according to this embodiment will be described.

[0060] In this embodiment, a new free carrier density is calculated that takes into account the capture of residual carriers at trap sites and the excitation of captured carriers. Then, based on the new free carrier density, the calculation of the number of residual carriers in the photodiode and the determination of the capture of residual carriers at trap sites and the excitation of captured carriers are repeated until the number of residual carriers in the photodiode converges. This makes it possible to reproduce the actual carrier transfer delay in a photodiode.

[0061] In this embodiment, a new electrostatic potential is calculated using equation (16), taking into account the capture of residual carriers at trap sites and the excitation of captured carriers. Then, by solving the electron current continuity equation and the hole current continuity equation, which are applied to the newly calculated electrostatic potential and the free carrier density before discretization, a new free carrier density is calculated. This makes it possible to reproduce the carrier transfer delay in an actual photodiode.

[0062] In this embodiment, by solving equation (16), a new electrostatic potential ψ that takes into account the trapping positions of residual carriers and the excitation positions of the trapped carriers is calculated. This makes it possible to represent trapping phenomena that take into account the trapping positions of residual carriers and excitation phenomena that take into account the excitation positions of the trapped carriers.

[0063] In this embodiment, a random number is issued for each remaining carrier, and if the issued random number falls within the capture range, it is determined that the remaining carrier is captured at the trap site. Furthermore, a random number is issued for each captured carrier, and if the issued random number falls within the excitation range, it is determined that the captured carrier is excited. This makes it possible to represent the capture phenomenon considering the capture location of the remaining carrier, and the excitation phenomenon considering the excitation location of the captured carrier.

[0064] <2. Variant> Next, a modified example of the above embodiment will be described.

[0065] [Differentiation A] In the above embodiment, the arithmetic unit 30 determines the impurity density of the photodiode (donor density N) D and acceptor density N A Alternatively, the distribution may be applied to the Poisson equation as a spatially continuous distribution without spatial discretization. In this case, the Poisson equation can be expressed, for example, as shown in equation (18) below.

[0066]

number

[0067] In this modified version, the arithmetic unit 30 can omit setting the locations of the set number of trap sites when a number of trap sites corresponding to the trap concentration is set during transient calculation. The arithmetic unit 30 can set a trap level selected with uniform probability from among a plurality of trap levels uniformly set within the band gap of the photodiode as the trap level of the captured carrier. The arithmetic unit 30 can set a trap level for each captured carrier. The arithmetic unit 30 can store the trap levels for each captured carrier obtained in this way as captured carrier data 22 in the storage device 20. At this time, the captured carrier data 22 does not include data on the capture location of the captured carrier.

[0068] In this modified version, the computing device 30 is able to calculate a new electrostatic potential by reflecting the density changes due to the capture of residual carriers and the excitation of captured carriers in the Poisson equation across the entire computational domain within the photodiode. Specifically, the computing device 30 calculates the density N corresponding to one free carrier from the electron concentration n. t When the density N is uniformly distributed across the entire computational domain of the photodiode, tavg Subtracting this, the electron concentration n is equal to the density N corresponding to one capture carrier. t When the density N is uniformly distributed across the entire computational domain of the photodiode, tavgIt is possible to add the density N on the right-hand side of the Poisson equation, for example, as shown in equation (19), from the free carrier density. tavg Subtracting this, the capture carrier density is obtained as density N tavg It is possible to add this. On the right-hand side of the Poisson equation, for example, as shown in Figures 13(A) and 13(B), the density N from the free carrier density tavg Subtraction of and density N as capture carrier density tavg The addition cancels out, and the total charge amount does not change. The calculation device 30 calculates the density N of the electron concentration n by the number of captured carriers included in the captured carrier data 22. tavg It is possible to perform addition and subtraction.

[0069]

number

[0070] The arithmetic unit 30 is capable of storing the number of excited carriers Z calculated using random numbers as excited carrier data 23 in the memory device 20. For example, as shown in equation (20), the arithmetic unit 30 adds a density N to the free carrier density on the right-hand side of the Poisson equation. tavg By adding this, it is possible to make the captured carrier density zero. On the right-hand side of the Poisson equation, for example, as shown in Figures 14(A) and 14(B), the density N to the free carrier density tavg The sum of the two, and the density N from the capture carrier density. tavg The subtraction cancels out, and the total charge amount does not change. The computing device 30 calculates the density N of the electron concentration n by the number of excited carriers included in the excited carrier data 23. tavg It is possible to perform addition and subtraction.

[0071]

number

[0072] In this modified example, in step S209, the Poisson equation is used to calculate a new electrostatic potential by reflecting the density changes due to the capture of residual carriers and the excitation of captured carriers across the entire computational domain within the photodiode. Specifically, the impurity density (donor density N) of the photodiode is used. D and acceptor density N A The distribution is not spatially discrete, but rather a spatially continuous distribution that can be fitted to the Poisson equation. Furthermore, the trapping and excitation positions of free carriers when the photodiode is in a transient state are not set, and the addition and subtraction of densities corresponding to the trapping and excitation of free carriers is performed uniformly across the entire computational domain of the photodiode on the right-hand side of the Poisson equation. As a result, the computational load is reduced compared to the above embodiment because discretization is not required. Also, compared to the above embodiment, the computational stability is higher because there is no discreteness.

[0073] [Variation B] In the above embodiment, the carriers may be holes. In this case, when the arithmetic unit 30 performs calculations on the capture and excitation of free carriers when the photodiode is in a transient state, it is possible to calculate the discrete number of holes X by spatially integrating the hole concentration p of the entire set calculation domain.

[0074] In this modified example, the computing device 30 calculates the density N equivalent to one free carrier from the hole concentration p for each location of a captured free carrier (captured carrier) based on the captured carrier data 22. t Subtracting this, the hole concentration p is equal to the density N, which corresponds to one capture carrier. t It is possible to add r. The calculation device 30, for example, as shown in equation (21), determines the position of the capture carrier r j If (1 ≤ j ≤ Y), then position r j In the right-hand side of the Poisson equation in , from the free carrier density, density N t Subtracting this, the capture carrier density is obtained as density N t It is possible to add them.

[0075]

number

[0076] In this modified example, the computing device 30 calculates the density N corresponding to one free carrier for each position of the excited carrier, based on the excited carrier data 23. t Adding these together, the density N equivalent to one captured carrier is obtained from the free carrier concentration. t It is possible to subtract the following. Specifically, based on the excitation carrier data 23, the computing device 30 subtracts the density N, which corresponds to one free carrier, from the hole concentration p for each position of the excitation carrier. t By adding these together, the density N corresponding to one capture carrier is obtained from the hole concentration p. t It is possible to subtract from this. The arithmetic unit 30, for example, as shown in equation (22), determines the position of the excited carrier r m If (1 ≤ m ≤ Z), then position r m In the right-hand side of the Poisson equation in , the free carrier density is given by density N. t By adding this, it is possible to reduce the capture carrier density to zero.

[0077]

number

[0078] The computing device 30 can obtain a spatially and temporally discretized Poisson equation (equation (23) below) by introducing the free carrier density and captured carrier density obtained as a result of considering the capture and excitation of free carriers into the right-hand side of the Poisson equation. The computing device 30 can obtain equation (23) based on the captured carrier data 22 and the excited carrier data 23. By solving the obtained equation (23), the computing device 30 can calculate the electrostatic potential ψ(t+dt) after dt seconds.

[0079]

number

[0080] The computing device 30 can use the electron concentration n(t) obtained after dt seconds by solving the electron current continuity equation (equation (9)) as the electron concentration n(t). Furthermore, the computing device 30 can use the free carrier density obtained after dt seconds, which is the result of considering carrier capture and excitation, as the hole concentration p(t). These are expressed by the following equation (24).

number

[0081] The computing device 30 can substitute the electron concentration n(t) obtained by equation (24) into n(t) in equations (9) and (10). The computing device 30 can substitute the hole concentration p(t) obtained by equation (24) into p(t) in equations (11) and (12). The computing device 30 can substitute the electrostatic potential φ(t+dt) obtained after dt seconds by solving Poisson's equation (equation (22)) that reflects carrier capture and excitation, as the electrostatic potential φ(t), into equations (10) and (12).

[0082] The arithmetic unit 30 can calculate the carrier density after dt seconds using the updated equations (9) to (12) as described above. The arithmetic unit 30 can then calculate the electrostatic potential after dt seconds by performing calculations on carrier capture and excitation when the photodiode is in a transient state using equation (22). The arithmetic unit 30 can repeatedly calculate the carrier density and electrostatic potential after dt seconds until the number of free carriers remaining in the photodiode converges to a predetermined threshold or less. When the number of free carriers remaining in the photodiode converges to a predetermined threshold or less, the arithmetic unit 30 can generate data on the electrical characteristics of the photodiode based on the data obtained when the photodiode was in a transient state. For example, the arithmetic unit 30 can generate the transfer characteristics of the photodiode based on the data obtained when the photodiode was in a transient state.

[0083] In this modified example, the carriers are holes. This allows for the realization of transfer simulations in semiconductor devices where the carriers are holes.

[0084] [Differentiation C] In the above modified example B, the calculation device 30 determines the impurity density of the photodiode (donor density N). D and acceptor density N A Alternatively, the distribution may be applied to Poisson's equation as a spatially continuous distribution without spatial discretization. Poisson's equation can be expressed, for example, as equation (18).

[0085] In this modified version, the arithmetic unit 30 can omit setting the locations of the set number of trap sites when a number of trap sites corresponding to the trap concentration is set during transient calculation. The arithmetic unit 30 can set a trap level selected with uniform probability from among a plurality of trap levels uniformly set within the band gap of the photodiode as the trap level of the captured carrier. The arithmetic unit 30 can set a trap level for each captured carrier. The arithmetic unit 30 can store the trap levels for each captured carrier obtained in this way as captured carrier data 22 in the storage device 20. At this time, the captured carrier data 22 does not include data on the capture location of the captured carrier.

[0086] In this modified version, the computing device 30 is able to calculate a new electrostatic potential by reflecting the density changes due to the capture of residual carriers and the excitation of captured carriers across the entire computational domain within the photodiode in the Poisson equation. Specifically, the computing device 30 calculates the density N corresponding to one free carrier from the hole concentration p. t When the density N is uniformly distributed across the entire computational domain of the photodiode, tavg Subtracting this, the hole concentration p is equal to the density N, which corresponds to one capture carrier. t When the density N is uniformly distributed across the entire computational domain of the photodiode, tavg It is possible to add the density N, for example, as shown in equation (25), on the right-hand side of the Poisson equation from the free carrier density. tavg Subtracting this, the capture carrier density is obtained as density N tavg It is possible to add the density N to the hole concentration p, equal to the number of captured carriers included in the captured carrier data 22. tavg It is possible to perform addition and subtraction.

[0087]

number

[0088] In this modified example, the arithmetic unit 30 is capable of storing the number of excited carriers Z calculated using random numbers as excited carrier data 23 in the memory device 20. For example, as shown in equation (26), the arithmetic unit 30 adds the density N to the free carrier density on the right-hand side of the Poisson equation. tavg By adding these, it is possible to make the captured carrier density zero. The calculation device 30 calculates the density N to the hole concentration p for each excited carrier included in the excited carrier data 23. tavg It is possible to perform addition and subtraction.

[0089]

number

[0090] In this modified example, in step S209, the Poisson equation is used to calculate a new electrostatic potential by reflecting the density changes due to the capture of residual carriers and the excitation of captured carriers across the entire computational domain within the photodiode. Specifically, the impurity density (donor density N) of the photodiode is used. D and acceptor density N A The distribution is not spatially discrete, but rather a spatially continuous distribution that can be fitted to the Poisson equation. Furthermore, the trapping and excitation positions of free carriers when the photodiode is in a transient state are not set, and the addition and subtraction of densities corresponding to the trapping and excitation of free carriers is performed uniformly across the entire computational domain of the photodiode on the right-hand side of the Poisson equation. As a result, the computational load is reduced compared to the above embodiment because discretization is not required. Also, compared to the above embodiment, the computational stability is higher because there is no discreteness.

[0091] [Differentiation D] In the above embodiments and modifications A to C, the computing device 30 may be capable of setting multiple trap levels in the band gap of the photodiode according to the density of states (DOS) of the photodiode. In this modification, the density of states of the photodiode corresponds to one of the initial conditions. In step S303, the computing device 30 sets multiple trap levels in the band gap of the photodiode according to the density of states (DOS) of the photodiode. In this case, an excitation probability close to reality can be considered.

[0092] [Differentiation Example E] In the above embodiments and modifications A to D, the computing device 30 may be capable of setting the capture position of free carriers (residual carriers) according to the magnitude of the free carrier density in the photodiode. In step S303, the computing device 30 sets the capture position of free carriers (residual carriers) according to the magnitude of the free carrier density in the photodiode. In this case, it becomes possible to represent the capture phenomenon that takes into account the spatial distribution of the free carrier density.

[0093] [Modification F] In the above embodiments and modifications A to E, the trap concentration may have a distribution corresponding to the position within the photodiode. In this case, the calculation device 30 may be able to set the free carrier capture probability (capture range) according to the magnitude of the trap concentration. In step S301, the calculation device 30 sets the free carrier capture probability (capture range) according to the magnitude of the trap concentration. In this case, it becomes possible to represent the capture phenomenon considering both the spatial distribution of free carrier density and the spatial distribution of captured carrier density.

[0094] [Differentiation G] In the above embodiments and modified examples A to F, the arithmetic unit 30 may make the time step (dt) for advancing transient calculations and the time step (dt) for determining carrier capture and excitation different from each other. The time step (dt) for advancing transient calculations corresponds to the time step used for temporal discretization in the current continuity equation. The time step (dt) for determining carrier capture and excitation corresponds to the time step used for temporal discretization in the Poisson equation. The arithmetic unit 30 is capable of making the time step (dt) for determining carrier capture and excitation smaller than the time step (dt) for advancing transient calculations. In steps S205 and S207, the arithmetic unit 30 makes the time step (dt) for determining carrier capture and excitation smaller than the time step (dt) for advancing transient calculations.

[0095] In this modified example, the arithmetic unit 30 may be capable of solving the Poisson equation (equation (16) or equation (23)) only when a capture event or an excitation event occurs during the capture and excitation calculations. In step S209, the arithmetic unit 30 solves the Poisson equation (equation (16) or equation (23)) only when a capture event or an excitation event occurs. In this case, the amount of computation required to solve the Poisson equation (equation (16) or equation (23)) can be reduced.

[0096] In this modified example, the arithmetic unit 30 may be able to adjust the time step (dt) for determining carrier capture and excitation to match the time step (dt) for advancing the transient calculation as the photodiode approaches a steady state during transient calculations. In steps S205 and S207, the arithmetic unit 30 adjusts the time step (dt) for determining carrier capture and excitation to match the time step (dt) for advancing the transient calculation as the photodiode approaches a steady state. In this case, the amount of computation required to determine carrier capture and excitation can be reduced.

[0097] [Modification H] The above embodiment is also applicable to the analysis of the charge injection process into the floating gate of a flash memory and the charge discharge process from the floating gate of a flash memory. In this case, the computing unit 30 is capable of performing the following processes. In this modified example, the flash memory corresponds to a specific example of the "semiconductor device" in one embodiment of the present disclosure.

[0098] If the semiconductor device is a flash memory, the input device 10 determines the impurity density (donor density N) of the floating gate of the flash memory. D and acceptor density N A The system can accept inputs of the structure (dimensions and electrode position), the trapping cross-sectional area, thermal velocity and trap concentration of the trapping site of the floating gate of the flash memory, and the external bias (gate voltage, etc.) applied to the floating gate of the flash memory as set values.

[0099] The output device 40 is an information processing device having a GUI for visualizing data on the electrical characteristics of the flash memory, which is the result of calculations performed by the arithmetic unit 30. The output device 40 is an information processing device having a GUI for visualizing, for example, the time evolution of the number of free carriers, carrier density, and electrostatic potential of the floating gate of the flash memory, as well as the time evolution of the density distribution of captured carriers and excited carriers of the floating gate of the flash memory, as transient data of the flash memory. The output device 40 is capable of acquiring data on the electrical characteristics of the flash memory, which is the result of calculations performed by the arithmetic unit 30, from the arithmetic unit 30. The output device 40 is also capable of acquiring the above transient data from the arithmetic unit 30. The output device 40 is capable of outputting (for example, displaying) the data acquired from the arithmetic unit 30. The output device 40 may, for example, display the data acquired from the arithmetic unit 30 as a graph on the screen. The output device 40 may, for example, generate the data acquired from the arithmetic unit 30 as a data file and store it in a specified memory.

[0100] When the semiconductor device is a flash memory, the transient state includes the process of charge injection into the floating gate and the process of charge release from the floating gate. The computing unit 30 is capable of calculating the phenomenon of carriers being trapped at trap sites in the tunnel oxide film and the phenomenon of the trapped carriers being excited.

[0101] The calculation unit 30 can calculate the number of remaining carriers in the floating gate by spatially integrating the free carrier density in the floating gate during the transient state. The calculation unit 30 can determine the trapping state at the trap site in the tunnel oxide film for each remaining carrier in the floating gate. The calculation unit 30 can determine whether or not each trapped carrier is excited. The calculation unit 30 can calculate a new free carrier density that takes into account the trapping of remaining carriers at the trap site and the excitation of trapped carriers. The calculation unit 30 can calculate a new electrostatic potential by performing calculations on carrier trapping and excitation when the floating gate is in the transient state.

[0102] The arithmetic unit 30 can repeatedly calculate the number of remaining carriers in the floating gate based on the new free carrier density, and determine whether the remaining carriers are captured at the trap site and whether the captured carriers are excited, until the number of remaining carriers in the floating gate converges to a predetermined threshold or less. When the number of free carriers remaining in the floating gate converges to a predetermined threshold or less, the arithmetic unit 30 can calculate data about the electrical characteristics of the flash memory based on the data obtained when the floating gate was in a transient state. For example, the arithmetic unit 30 can calculate the write characteristics and retention characteristics of the flash memory based on the data obtained when the floating gate was in a transient state.

[0103] This modified version can reproduce the effects of traps during the charge injection process into the floating gate of flash memory and the charge release process from the floating gate of flash memory. This makes it possible to accurately predict the write speed delay and data retention characteristic degradation in actual flash memory.

[0104] [Modification I] The above embodiment can also be applied to the analysis of the refresh time of DRAM. In this case, the arithmetic unit 30 can perform the following processing. In this modified example, the DRAM corresponds to a specific example of the "semiconductor device" in one embodiment of the present disclosure.

[0105] If the semiconductor device is a DRAM, the input device 10 has an impurity density (donor density N) of the DRAM capacitor. D and acceptor density N A The system can accept inputs of the following as set values: the structure (dimensions and electrode positions), the trapping cross-sectional area of ​​the DRAM capacitor's trapping site, thermal velocity and trap concentration, and the external bias applied to the DRAM capacitor (gate voltage, etc.).

[0106] The output device 40 is an information processing device having a GUI for visualizing data on the electrical characteristics of the DRAM, which is the result of calculations performed by the arithmetic unit 30. The output device 40 is an information processing device having a GUI for visualizing, for example, the time evolution of the number of free carriers, carrier density, and electrostatic potential of the DRAM capacitor, as well as the time evolution of the density distribution of captured carriers and excited carriers of the DRAM capacitor, as transient data of the flash memory. The output device 40 is capable of acquiring data on the electrical characteristics of the DRAM, which is the result of calculations performed by the arithmetic unit 30, from the arithmetic unit 30. The output device 40 is also capable of acquiring the above transient data from the arithmetic unit 30. The output device 40 is capable of outputting (for example, displaying) the data acquired from the arithmetic unit 30. The output device 40 may, for example, display the data acquired from the arithmetic unit 30 as a graph on the screen. The output device 40 may, for example, generate the data acquired from the arithmetic unit 30 as a data file and store it in a specified memory.

[0107] When the semiconductor device is a DRAM, the transient state includes the process by which the charge stored in the DRAM's capacitor leaks over time. The arithmetic unit 30 is capable of calculating the phenomenon of carriers being trapped in the DRAM's capacitor and at trap sites in the transistor region, and the phenomenon of these trapped carriers being excited.

[0108] The computing device 30 can calculate the number of remaining carriers in the capacitor by spatially integrating the free carrier density in the capacitor during the transient state. The computing device 30 can determine the trapping state of each remaining carrier in the capacitor at its trap site. The computing device 30 can determine whether or not each trapped carrier is excited. The computing device 30 can calculate a new free carrier density that takes into account the trapping of remaining carriers at their trap sites and the excitation of trapped carriers. The computing device 30 can calculate a new electrostatic potential by performing calculations on carrier trapping and excitation when the capacitor is in a transient state.

[0109] The computing unit 30 can repeatedly calculate the number of remaining carriers in the capacitor based on the new free carrier density, determine the capture of the remaining carriers at the trap site, and determine the excitation of the captured carriers, until the number of remaining carriers in the capacitor converges to a predetermined threshold or less. When the number of free carriers remaining in the capacitor converges to a predetermined threshold or less, the computing unit 30 can generate data on the electrical characteristics of the DRAM based on the data obtained when the capacitor was in a transient state. For example, the computing unit 30 can calculate the refresh time of the DRAM based on the data obtained when the capacitor was in a transient state. The computing unit 30 can calculate the refresh time considering temperature dependence and trap density dependence.

[0110] This modified example can reproduce the effects of traps that affect the refresh time of DRAM. This makes it possible to accurately predict the temperature dependence and manufacturing variations of data retention time in actual DRAM.

[0111] [Modification J] The above embodiment is also applicable to the analysis of threshold voltage shift and mobility degradation of a MOSFET. In this case, the computing device 30 can perform the following processing. In this modified example, the MOSFET corresponds to a specific example of the "semiconductor device" in one embodiment of the present disclosure.

[0112] If the semiconductor device is a MOSFET, the input device 10 determines the impurity density (donor density N) of the channel region of the MOSFET. D and acceptor density N A The system can accept inputs of the MOSFET's structure (dimensions and electrode positions), the trapping cross-sectional area of ​​the trap site in the MOSFET's channel region, thermal velocity and trap concentration, and an external bias (gate voltage, etc.) applied to the MOSFET's channel region as set values.

[0113] The output device 40 is an information processing device having a GUI for visualizing data on the electrical characteristics of a MOSFET, which is the result of calculations performed by the arithmetic device 30. The output device 40 is an information processing device having a GUI for visualizing, for example, the time evolution of the number of free carriers in the channel region of the MOSFET, the carrier density and the electrostatic potential, as well as the time evolution of the density distribution of trapped carriers and excited carriers in the channel region of the MOSFET, as transient data of the MOSFET. The output device 40 is capable of acquiring data on the electrical characteristics of the MOSFET, which is the result of calculations performed by the arithmetic device 30, from the arithmetic device 30. The output device 40 is also capable of acquiring the above transient data from the arithmetic device 30. The output device 40 is capable of outputting (for example, displaying) the data acquired from the arithmetic device 30. The output device 40 may, for example, display the data acquired from the arithmetic device 30 as a graph on the screen. The output device 40 may, for example, generate the data acquired from the arithmetic device 30 as a data file and store it in a specified memory.

[0114] When the semiconductor device is a MOSFET, the transient state includes the inversion layer formation process in the channel region when a gate voltage is applied, and the carrier capture and excitation processes near the gate oxide interface. The computing unit 30 is capable of calculating the phenomenon of carriers being captured at trap sites in the gate oxide interface and gate oxide, and the phenomenon of the captured carriers being excited.

[0115] The calculation unit 30 can calculate the number of residual carriers in the channel region of the MOSFET by spatially integrating the free carrier density in the channel region of the MOSFET during the transient state. The calculation unit 30 can determine the trapping state of each residual carrier at the trap site at the gate oxide interface. The calculation unit 30 can determine whether or not each trapped carrier is excited. The calculation unit 30 can calculate a new free carrier density that takes into account the trapping of residual carriers at the trap site and the excitation of trapped carriers. The calculation unit 30 can calculate a new electrostatic potential by performing calculations on carrier trapping and excitation when the channel region of the MOSFET is in a transient state.

[0116] The computing unit 30 can repeatedly calculate the number of remaining carriers in the channel region of the MOSFET based on the new free carrier density, and determine whether the remaining carriers are trapped at trap sites and whether the trapped carriers are excited, until the number of remaining carriers in the channel region of the MOSFET converges to a predetermined threshold. When the number of free carriers remaining in the channel region of the MOSFET converges to a predetermined threshold, the computing unit 30 can generate data on the electrical characteristics of the MOSFET based on the data obtained when the channel region of the MOSFET is in a transient state. For example, the computing unit 30 can calculate the threshold voltage shift amount of the MOSFET, taking into account the change in interface charge due to carriers trapped at trap sites, based on the data obtained when the channel region of the MOSFET is in a transient state. The computing unit 30 can further calculate the decrease in effective carrier mobility due to interface trapping, for example. For example, under bias temperature instability (BTI) test conditions, the computing unit 30 can calculate the threshold voltage shift and mobility degradation over time, using gate voltage and temperature as parameters. The computing unit 30 can predict the time-dependent changes in the electrical characteristics of the MOSFET, for example, as a function of gate voltage application time, temperature, and trap density.

[0117] This modified example can reproduce the effects of traps that influence the threshold voltage shift and mobility degradation of MOSFETs. This makes it possible to accurately predict BTI degradation and hot carrier degradation in actual MOSFETs. In particular, it allows for the evaluation of the effects of interface traps, which are significant in advanced FinFET and GAA-FET processes, at the design stage.

[0118] [Differentiation K] The above embodiment is also applicable to the analysis of the photoelectric conversion efficiency and carrier lifetime of solar cells. In this case, the computing device 30 is capable of performing the following processes. In this modified example, the solar cell corresponds to a specific example of the "semiconductor device" in one embodiment of the present disclosure.

[0119] If the semiconductor device is a solar cell, the input device 10 determines the impurity density (donor density N) of the semiconductor substrate of the solar cell. D and acceptor density N A The system can accept inputs of the structure (dimensions and electrode positions), the trapping cross-sectional area of ​​the trapping sites on the semiconductor substrate of the solar cell, the thermal velocity and trap concentration, and the external bias applied to the semiconductor substrate of the solar cell as set values.

[0120] The output device 40 is an information processing device having a GUI for visualizing data on the electrical characteristics of a solar cell, which is the result of calculations performed by the arithmetic device 30. The output device 40 is an information processing device having a GUI for visualizing, for example, the time evolution of the number of free carriers, carrier density, and electrostatic potential of the semiconductor substrate of the solar cell, as well as the time evolution of the density distribution of captured carriers and excited carriers of the semiconductor substrate of the solar cell, as transient data of the solar cell. The output device 40 is capable of acquiring data on the electrical characteristics of the solar cell, which is the result of calculations performed by the arithmetic device 30, from the arithmetic device 30. The output device 40 is also capable of acquiring the above transient data from the arithmetic device 30. The output device 40 is capable of outputting (for example, displaying) the data acquired from the arithmetic device 30. The output device 40 may, for example, display the data acquired from the arithmetic device 30 as a graph on the screen. The output device 40 may, for example, generate the data acquired from the arithmetic device 30 as a data file and store it in a specified memory.

[0121] When the semiconductor device is a solar cell, the transient state includes the process by which a pair of electrons and holes generated by light irradiation are extracted to an external circuit via a pn junction. The computing unit 30 is capable of calculating the phenomenon of carriers being trapped at trap sites within the semiconductor substrate of the solar cell and near the pn junction interface, and the phenomenon of the trapped carriers being excited.

[0122] The calculation unit 30 can set the carriers generated by light irradiation as initial conditions. The calculation unit 30 can calculate the number of residual carriers in the semiconductor substrate of the solar cell by spatially integrating the free carrier density in the semiconductor substrate of the solar cell during the transient state. The calculation unit 30 can determine the trapping state of each residual carrier in the semiconductor substrate of the solar cell at trap sites such as bulk crystal defects and interface states. The calculation unit 30 can determine whether or not each trapped carrier is excited. The calculation unit 30 can calculate a new free carrier density that takes into account the trapping of residual carriers at trap sites and the excitation of trapped carriers. The calculation unit 30 can calculate a new electrostatic potential by performing calculations on carrier trapping and excitation when the semiconductor substrate of the solar cell is in the transient state.

[0123] The computing device 30 can repeatedly calculate the number of remaining carriers in the semiconductor substrate of the solar cell based on the new free carrier density, and determine the capture of these remaining carriers at trap sites and the excitation of the captured carriers, until the number of remaining carriers in the semiconductor substrate of the solar cell converges to a predetermined threshold. When the number of remaining free carriers in the semiconductor substrate of the solar cell converges to a predetermined threshold, the computing device 30 can generate data on the electrical characteristics of the solar cell based on data obtained when the semiconductor substrate of the solar cell is in a transient state. For example, the computing device 30 can calculate the recombination rate due to carrier capture at trap sites and the effective carrier lifetime based on data obtained when the semiconductor substrate of the solar cell is in a transient state. The computing device 30 can further calculate the open-circuit voltage (Voc), short-circuit current (Isc), curve factor (FF), and photoelectric conversion efficiency of the solar cell based on the calculated carrier lifetime. For example, the computing device 30 can calculate the current-voltage characteristics (IV characteristics) of the solar cell using light irradiation intensity, temperature, and trap density as parameters. The computing device 30 can predict the degradation characteristics of a solar cell over time, for example, by taking into account the increase in trap density due to light-induced degradation (LID) under long-term light irradiation conditions.

[0124] This modified model can reproduce the effects of traps that affect the photoelectric conversion efficiency of solar cells. This makes it possible to accurately predict the decrease in carrier lifetime and the decrease in efficiency due to photodegradation in actual solar cells. In particular, it can evaluate the effects of traps according to the material system, such as defects like oxygen precipitates and interstitial iron in crystalline silicon solar cells, and trap formation due to ion transfer in perovskite solar cells.

[0125] [Modified version L] In the above modifications H to K, the methods described in the above modifications A to G may be applied. In this case, the effects described in the above modifications A to G can be obtained in the above modifications H to K.

[0126] The effects described herein are for illustrative purposes only. The effects of this disclosure are not limited to those described herein. This disclosure may have effects other than those described herein.

[0127] Furthermore, for example, this disclosure can take the following configuration. <1> The number of residual carriers in the semiconductor device is calculated by spatially integrating the free carrier density in the semiconductor device during a transient state, For each of the remaining carriers, it is determined whether or not the remaining carrier will be captured at the trap site. For each capture carrier captured at the trap site, it is determined whether or not the capture carrier is excited. Using Poisson's equation, the electron current continuity equation, and the hole current continuity equation, a new free carrier density is calculated that takes into account the capture of the remaining carriers at the trap site and the excitation of the captured carriers. Based on the new free carrier density, the number of residual carriers in the semiconductor device is calculated, and the capture of the residual carriers at the trap site and the excitation of the captured carriers are repeatedly performed until the number of residual carriers in the semiconductor device converges. including Simulation method. <2> In calculating the new free carrier density, the Poisson equation is used to calculate a new electrostatic potential that takes into account the capture of residual carriers at the trap site and the excitation of the captured carriers. Then, the new free carrier density is calculated by solving the electron current continuity equation and the hole current continuity equation, which are obtained by substituting the calculated new electrostatic potential with the free carrier density before spatial integration. <1> The simulation method described above. <3> In the Poisson equation, the new electrostatic potential is calculated taking into account the trapping positions of the remaining carriers and the excitation positions of the trapped carriers. <2> The simulation method described above. <4> In the Poisson equation, the new electrostatic potential is calculated by reflecting the density changes due to the capture of residual carriers and the excitation of the captured carriers across the entire computational domain within the semiconductor device. <2> The simulation method described above. <5> In determining whether the remaining carriers are captured by the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured by the trap site. In determining the excitation of the capture carrier, a random number is issued for each capture carrier, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. <1> or <4> One of the simulation methods described below. <6> The excitation range is determined by the trap level corresponding to the density of states of the device. <5> The simulation method described above. <7> In determining whether the remaining carriers are captured by the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured by the trap site. In determining the excitation of the capture carrier, a random number is issued for each capture carrier, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. <3> The simulation method described above. <8> In determining whether the remaining carriers are captured at the trap site, the capture location of the remaining carriers is set according to the magnitude of the free carrier density. <7> The simulation method described above. <9> The trap concentration has a distribution that corresponds to the location within the device. In determining whether the remaining carriers are captured at the trap site, the capture range is set according to the magnitude of the trap concentration. <7> The simulation method described above. <10> In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and the Poisson equation is solved only when a capture event or excitation event occurs. <2> or <4> One of the simulation methods described below. <11> In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and as the semiconductor device approaches a steady state, the first time step is brought closer to the second time step. <2> or <4> One of the simulation methods described below. <12> The number of residual carriers in the semiconductor device is calculated by spatially integrating the free carrier density in the semiconductor device during a transient state, For each of the remaining carriers, it is determined whether or not the remaining carrier will be captured at the trap site. For each capture carrier captured at the trap site, it is determined whether or not the capture carrier is excited. Using Poisson's equation, the electron current continuity equation, and the hole current continuity equation, a new free carrier density is calculated that takes into account the capture of the remaining carriers at the trap site and the excitation of the captured carriers. Based on the new free carrier density, the number of residual carriers in the semiconductor device is calculated, and the capture of the residual carriers at the trap site and the excitation of the captured carriers are determined, until the number of residual carriers in the semiconductor device converges. Make the computer execute it. Simulation program. <13> In calculating the new free carrier density, the Poisson equation is used to calculate a new electrostatic potential that takes into account the capture of residual carriers at the trap site and the excitation of the captured carriers. Then, the new free carrier density is calculated by solving the electron current continuity equation and the hole current continuity equation, which are obtained by substituting the calculated new electrostatic potential with the free carrier density before spatial integration. <12> The simulation program described below. <14> In the Poisson equation, the new electrostatic potential is calculated taking into account the trapping positions of the remaining carriers and the excitation positions of the trapped carriers. <13> The simulation program described below. <15> In the Poisson equation, the new electrostatic potential is calculated by reflecting the density changes due to the capture of residual carriers and the excitation of the captured carriers across the entire computational domain within the semiconductor device. <13> The simulation program described below. <16> In determining whether the remaining carriers are captured by the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured by the trap site. In determining the excitation of the capture carrier, a random number is issued for each capture carrier, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. <12> or <15> A simulation program described in one of the following. <17> The excitation range is determined by the trap level corresponding to the density of states of the device. <16> The simulation program described below. <18> In determining whether the remaining carriers are captured by the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured by the trap site. In determining the excitation of the capture carrier, a random number is issued for each capture carrier, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. <14> The simulation program described below. <19> In determining whether the remaining carriers are captured at the trap site, the capture location of the remaining carriers is set according to the magnitude of the free carrier density. <18> The simulation program described below. <20> The trap concentration has a distribution that corresponds to the location within the device. In determining whether the remaining carriers are captured at the trap site, the capture range is set according to the magnitude of the trap concentration. <18> The simulation program described below. <21> In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and the Poisson equation is solved only when a capture event or excitation event occurs. <13> or <15> A simulation program described in one of the following. <22> In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and as the semiconductor device approaches a steady state, the first time step is brought closer to the second time step. <13> or <15> A simulation program described in one of the following. <23> An input device capable of acquiring boundary conditions, A computing device capable of calculating the transient state of a semiconductor device using the boundary conditions obtained via the input device, and calculating the electrical characteristics of the semiconductor device based on the calculated transient state, An output device capable of outputting the electrical characteristics of the semiconductor device obtained by the calculation device, Equipped with, The aforementioned computing device is The number of remaining carriers in the semiconductor device is calculated by spatially integrating the free carrier density in the semiconductor device during the transient state. For each of the remaining carriers, it is determined whether or not the remaining carrier will be captured at the trap site. For each capture carrier captured at the trap site, it is determined whether or not the capture carrier is excited. Using Poisson's equation, the electron current continuity equation, and the hole current continuity equation, a new free carrier density is calculated that takes into account the capture of the remaining carriers at the trap site and the excitation of the captured carriers. Based on the new free carrier density, the number of remaining carriers in the semiconductor device is calculated, and the capture of the remaining carriers at the trap site and the excitation of the captured carriers are repeatedly performed until the number of remaining carriers in the semiconductor device converges. Simulation system. <24> In calculating the new free carrier density, the Poisson equation is used to calculate a new electrostatic potential that takes into account the capture of residual carriers at the trap site and the excitation of the captured carriers. Then, the new free carrier density is calculated by solving the electron current continuity equation and the hole current continuity equation, which are obtained by substituting the calculated new electrostatic potential with the free carrier density before spatial integration. <23> The simulation system described below. <25> In the Poisson equation, the new electrostatic potential is calculated taking into account the trapping positions of the remaining carriers and the excitation positions of the trapped carriers. <24> The simulation system described below. <26> In the Poisson equation, the new electrostatic potential is calculated by reflecting the density changes due to the capture of residual carriers and the excitation of the captured carriers across the entire computational domain within the semiconductor device. <24> The simulation system described below. <27> In determining whether the remaining carriers are captured by the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured by the trap site. In determining the excitation of the capture carrier, a random number is issued for each capture carrier, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. <23> or <26> A simulation system described in any one of the following. <28> The excitation range is determined by the trap level corresponding to the density of states of the device. <27> The simulation system described below. <29> In determining whether the remaining carriers are captured by the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured by the trap site. In determining the excitation of the capture carrier, a random number is issued for each capture carrier, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. <25> The simulation system described below. <30> In determining whether the remaining carriers are captured at the trap site, the capture location of the remaining carriers is set according to the magnitude of the free carrier density. <29> The simulation system described below. <31> The trap concentration has a distribution that corresponds to the location within the device. In determining whether the remaining carriers are captured at the trap site, the capture range is set according to the magnitude of the trap concentration. <29> The simulation system described below. <32> In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and the Poisson equation is solved only when a capture event or excitation event occurs. <24> or <26> A simulation system described in any one of the following. <33> In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and as the semiconductor device approaches a steady state, the first time step is brought closer to the second time step. <24> or <26> A simulation system described in any one of the following. <34> An input device capable of accepting the impurity density and structure of a semiconductor device, the trapping cross-sectional area of ​​the trap site of the semiconductor device, the thermal velocity and trap concentration, and the input of an external bias applied to the semiconductor device as set values, An output device capable of outputting the time evolution of the number of carriers, carrier density, and electrostatic potential, as well as the time evolution of the density distribution of captured carriers and excited carriers, calculated using the set values ​​obtained via the input device, as transient data of the semiconductor device. Equipped with Information processing device. [Explanation of Symbols]

[0128] 1…Simulation system, 10…Input device, 20…Storage device, 21…Calculation program, 22…Capture carrier data, 23…Excitation carrier data, 30…Calculation unit, 40…Output device.

Claims

1. The number of residual carriers in the semiconductor device is calculated by spatially integrating the free carrier density in the semiconductor device during a transient state, For each of the remaining carriers, it is determined whether or not the remaining carrier will be captured at the trap site. For each capture carrier captured at the trap site, it is determined whether or not the capture carrier is excited. Using Poisson's equation, the electron current continuity equation, and the hole current continuity equation, a new free carrier density is calculated that takes into account the capture of the remaining carriers at the trap site and the excitation of the captured carriers. Based on the new free carrier density, the number of residual carriers in the semiconductor device is calculated, and the capture of the residual carriers at the trap site and the excitation of the captured carriers are determined, until the number of residual carriers in the semiconductor device converges. Includes, In calculating the new free carrier density, the Poisson equation is used to calculate a new electrostatic potential that takes into account the capture of residual carriers at the trap site and the excitation of the captured carriers. Then, the new free carrier density is calculated by solving the electron current continuity equation and the hole current continuity equation, which are obtained by substituting the calculated new electrostatic potential with the free carrier density before spatial integration. In the Poisson equation, the new electrostatic potential is calculated taking into account the trapping positions of the remaining carriers and the excitation positions of the trapped carriers. In determining whether the remaining carriers are captured by the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured by the trap site. In determining the excitation of the capture carrier, a random number is issued for each capture carrier, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. Simulation method.

2. In the Poisson equation, the new electrostatic potential is calculated by reflecting the density changes due to the capture of residual carriers and the excitation of the captured carriers across the entire computational domain within the semiconductor device. The simulation method according to claim 1.

3. The excitation range is determined by the trap level corresponding to the density of states of the device. The simulation method according to claim 1.

4. In determining whether the remaining carriers are captured at the trap site, the capture location of the remaining carriers is set according to the magnitude of the free carrier density. The simulation method according to claim 1.

5. The trap concentration has a distribution that depends on the location within the device. In determining whether the remaining carriers are captured at the trap site, the capture range is set according to the magnitude of the trap concentration. The simulation method according to claim 1.

6. In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and the Poisson equation is solved only when a capture event or excitation event occurs. The simulation method according to claim 1.

7. In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and as the semiconductor device approaches a steady state, the first time step is brought closer to the second time step. The simulation method according to claim 1.

8. The number of residual carriers in the semiconductor device is calculated by spatially integrating the free carrier density in the semiconductor device during a transient state, For each of the remaining carriers, it is determined whether or not the remaining carrier will be captured at the trap site. For each capture carrier captured at the trap site, it is determined whether or not the capture carrier is excited. Using Poisson's equation, the electron current continuity equation, and the hole current continuity equation, a new free carrier density is calculated that takes into account the capture of the remaining carriers at the trap site and the excitation of the captured carriers. Based on the new free carrier density, the number of residual carriers in the semiconductor device is calculated, and the capture of the residual carriers at the trap site and the excitation of the captured carriers are determined, until the number of residual carriers in the semiconductor device converges. In calculating the new free carrier density, the new electrostatic potential is calculated using the Poisson equation, taking into account the capture of residual carriers at the trap site and the excitation of the captured carriers. Then, the new free carrier density is calculated by solving the electron current continuity equation and the hole current continuity equation, which are obtained by substituting the calculated new electrostatic potential with the free carrier density before spatial integration. In the Poisson equation, the new electrostatic potential is calculated taking into account the trapping positions of the remaining carriers and the excitation positions of the trapped carriers. In determining whether the remaining carriers are captured at the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured at the trap site. In determining the excitation of the aforementioned capture carrier, a random number is issued for each of the capture carriers, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. Make the computer execute it. Simulation program.

9. In the Poisson equation, the new electrostatic potential is calculated by reflecting the density changes due to the capture of residual carriers and the excitation of the captured carriers across the entire computational domain within the semiconductor device. The simulation program according to claim 8.

10. The excitation range is determined by the trap level corresponding to the density of states of the device. The simulation program according to claim 8.

11. In determining whether the remaining carriers are captured at the trap site, the capture location of the remaining carriers is set according to the magnitude of the free carrier density. The simulation program according to claim 8.

12. The trap concentration has a distribution that depends on the location within the device. In determining whether the remaining carriers are captured at the trap site, the capture range is set according to the magnitude of the trap concentration. The simulation program according to claim 8.

13. In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and the Poisson equation is solved only when a capture event or excitation event occurs. The simulation program according to claim 8.

14. In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and as the semiconductor device approaches a steady state, the first time step is brought closer to the second time step. The simulation program according to claim 8.

15. An input device capable of acquiring boundary conditions, A computing device capable of calculating the transient state of a semiconductor device using the boundary conditions obtained via the input device, and calculating the electrical characteristics of the semiconductor device based on the calculated transient state, An output device capable of outputting the electrical characteristics of the semiconductor device obtained by the calculation device, Equipped with, The aforementioned computing device is The number of remaining carriers in the semiconductor device is calculated by spatially integrating the free carrier density in the semiconductor device during the transient state. For each of the remaining carriers, it is determined whether or not the remaining carrier will be captured at the trap site. For each capture carrier captured at the trap site, it is determined whether or not the capture carrier is excited. Using Poisson's equation, the electron current continuity equation, and the hole current continuity equation, a new free carrier density is calculated that takes into account the capture of the remaining carriers at the trap site and the excitation of the captured carriers. Based on the new free carrier density, the number of remaining carriers in the semiconductor device is calculated, and the capture of the remaining carriers at the trap site and the excitation of the captured carriers are repeatedly performed until the number of remaining carriers in the semiconductor device converges. In calculating the new free carrier density, the Poisson equation is used to calculate a new electrostatic potential that takes into account the capture of residual carriers at the trap site and the excitation of the captured carriers. Then, the new free carrier density is calculated by solving the electron current continuity equation and the hole current continuity equation, which are obtained by substituting the calculated new electrostatic potential with the free carrier density before spatial integration. In the Poisson equation, the new electrostatic potential is calculated taking into account the trapping positions of the remaining carriers and the excitation positions of the trapped carriers. In determining whether the remaining carriers are captured by the trap site, a random number is issued for each remaining carrier, and if the issued random number is included in the capture range, it is determined that the remaining carriers are captured by the trap site. In determining the excitation of the capture carrier, a random number is issued for each capture carrier, and if the issued random number falls within the excitation range, it is determined that the capture carrier is excited. Simulation system.

16. In the Poisson equation, the new electrostatic potential is calculated by reflecting the density changes due to the capture of residual carriers and the excitation of the captured carriers across the entire computational domain within the semiconductor device. The simulation system according to claim 15.

17. The excitation range is determined by the trap level corresponding to the density of states of the device. The simulation system according to claim 15.

18. In determining whether the remaining carriers are captured at the trap site, the capture location of the remaining carriers is set according to the magnitude of the free carrier density. The simulation system according to claim 15.

19. The trap concentration has a distribution that depends on the location within the device. In determining whether the remaining carriers are captured at the trap site, the capture range is set according to the magnitude of the trap concentration. The simulation system according to claim 15.

20. In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and the Poisson equation is solved only when a capture event or excitation event occurs. The simulation system according to claim 15.

21. In calculating the new free carrier density, the first time step for calculating the new electrostatic potential is made smaller than the second time step for calculating the new free carrier density, and as the semiconductor device approaches a steady state, the first time step is brought closer to the second time step. The simulation system according to claim 15.

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