Method for calculating spatial charge distribution based on tunneling effect and limited hole extraction

Abstract

The application discloses a space charge distribution calculation method based on tunneling effect and hole extraction restriction, and belongs to the technical field of electrical engineering, and comprises the following steps: a one-dimensional finite element structure model is established, taking polymer material as a main body and electrodes as boundaries; a periodic polarity reversal voltage is applied at the electrode boundary by coupling Maxwell electromagnetic field theory and Nernst-Planck theory, and a recombination carrier injection model and a hole extraction restricted boundary condition are introduced, so that a polymer material space charge distribution model based on tunneling effect and hole extraction restriction mechanism is obtained; and the polymer material space charge distribution model is solved to obtain the space charge distribution and electric field distortion law of the polymer material under the conditions of strong electric field and polarity reversal. The method solves the problem that the existing space charge model cannot effectively depict the tunneling emission and hole extraction restriction effect at the electrode interface, and provides important support for the polymer insulation aging research under the conditions of strong electric field and polarity reversal.

Description

Technical Field

[0001] This application relates to the field of electrical engineering technology, and in particular to a method for calculating space charge distribution based on tunneling effect and limited hole extraction. Background Technology

[0002] Polymer insulating materials are widely used in the main insulation systems of power equipment such as UHV transformer bushings, cable terminals, and gas-insulated switches (GIS) due to their excellent dielectric properties, thermal stability, and mechanical strength. Among them, thermosetting polymers, represented by epoxy resin, occupy a core position in composite insulation structures.

[0003] However, under the long-term influence of complex electrical stresses such as high temperature, strong electric field, and polarity reversal voltage, polymers are prone to carrier injection and capture processes, leading to significant space charge accumulation. This non-uniform charge distribution causes local electric field distortion, which in turn induces failure phenomena such as molecular chain degradation, electrical treeing, and dielectric breakdown, severely affecting the long-term reliability and operational safety of polymer insulating materials. In actual ultra-high voltage operating conditions, when the electric field strength reaches the order of hundreds of kilovolts per millimeter, electrons can be injected through the electrode barrier via quantum tunneling. Simultaneously, holes are significantly restricted during interface extraction due to barrier obstruction and differences in carrier concentration, resulting in restricted hole extraction. These two mechanisms together determine the rate of charge accumulation and dissipation in polymer materials, significantly affecting the electric field distribution.

[0004] Therefore, there is an urgent need to establish a strong electric field space charge simulation calculation method that can simultaneously consider the tunneling emission and hole extraction restriction mechanism, so as to accurately describe the charge dynamic behavior and electric field distortion law of polymer insulating materials under periodic polarity reversal conditions. Summary of the Invention

[0005] To address the aforementioned shortcomings in the prior art, this application provides a method for calculating space charge distribution based on tunneling effect and restricted hole extraction, which solves the problem that existing space charge models cannot effectively characterize the tunneling emission and restricted hole extraction effects at the electrode interface.

[0006] To achieve the aforementioned objectives, the technical solution adopted in this application is as follows:

[0007] This application provides a method for calculating space charge distribution based on tunneling effect and limited hole extraction, including:

[0008] S1: Establish a one-dimensional finite element structural model with polymer material as the main body and electrodes as the boundary;

[0009] S2: Based on a one-dimensional finite element structural model, coupled with Maxwell's electromagnetic field theory and Nernst-Planck theory, a periodic polarity reversal voltage is applied at the electrode boundary, and a composite carrier injection model and hole extraction restricted boundary conditions are introduced to obtain a space charge distribution model of polymer materials based on tunneling effect and hole extraction restricted mechanism.

[0010] S3: The space charge distribution model of polymer materials based on the tunneling effect and the hole extraction restriction mechanism is calculated by time stepping through a numerical solver to obtain the space charge distribution and electric field distortion law of polymer materials under strong electric field and polarity reversal conditions.

[0011] Further, S2 includes:

[0012] S201: Based on a one-dimensional finite element structural model and the relationship between charge injection and extraction, charge trapping and de-trapping, charge transport, charge neutralization and electric field, establish the relationship between electric field and the spatial charge dissipation behavior inside the medium.

[0013] S202: Based on the relationship between the electric field and the dissipation behavior of space charge inside the medium, add physical fields including electrostatic field and rare matter transfer field to the one-dimensional finite element structural model.

[0014] S203: Add physical parameters of materials to a one-dimensional finite element structural model and add boundary conditions to the physical field;

[0015] S204: By applying a periodic polarity reversal voltage at the boundary of the physical field and introducing a composite carrier injection model and a hole extraction restricted boundary condition, a space charge distribution model of polymer materials based on the tunneling effect and the hole extraction restricted mechanism is obtained.

[0016] Furthermore, the periodic polarity reversal voltage is represented by a globally defined time function, which includes a voltage holding segment and a cosine smooth transition segment.

[0017] The expression for the time function is:

[0018]

[0019] in, It is a time function. Voltage amplitude, For time variables, The duration of each half-cycle, This is the polarity reversal transition time. This is the floor function. To take the remainder.

[0020] Furthermore, the composite carrier injection model includes the Schottky emission mechanism and the Fowler-Nordheim tunneling emission mechanism;

[0021] The Fowler-Nordheim tunneling emission mechanism is triggered when the local electric field strength at the electrode interface exceeds a preset threshold.

[0022] The expression for the emission current density of the Fowler-Nordheim tunneling emission mechanism is as follows:

[0023]

[0024]

[0025]

[0026] in, For the injected current density, Injecting current density to the Schottky effect, The injection current density for the Fowler-Nordheim tunneling emission mechanism. For electric field strength, For location, For local electric field intensity, The threshold electric field strength, and The tunneling coefficient, Injecting a potential barrier into electrons, Richardson's constant, Thermodynamic temperature Boltzmann's constant, To create a potential barrier for electrons in a Schottky circuit. and These are the vacuum permittivity and the relative permittivity, respectively. The electric field strength at the cathode. This represents the amount of electric charge.

[0027] Furthermore, the restricted boundary conditions for hole extraction include:

[0028] The hole extraction restricted boundary condition uses a hole additional resistance factor to simulate the restricted hole extraction behavior at the electrode interface.

[0029] When electrons and holes migrate to the opposite electrode under the influence of an electric field and are extracted, the extraction of holes is suppressed by the additional resistance factor of holes due to the potential barrier effect at the electrode interface. The formula for calculating the extraction current density of holes and electrons is:

[0030]

[0031]

[0032] in, This represents the hole extraction current density. The electron extraction current density. Add an obstruction factor to the hole. and These are the extraction coefficients of holes and electrons at the electrodes, respectively. and These represent the mobilities of hole and electron carriers, respectively. and These represent the hole and electron carrier densities, respectively. for The electric field strength at the location.

[0033] Furthermore, the space charge distribution model of polymer materials based on the tunneling effect and the hole extraction restriction mechanism includes: Maxwell's electromagnetic field theory and Nernst-Planck theory;

[0034] Maxwell's electromagnetic field theory is used to calculate the potential distribution, which satisfies the Poisson equation:

[0035]

[0036] in, Let be the divergence of the vector field. For the potential gradient, This is the carrier occupancy rate conversion factor. Hole concentration Electron concentration;

[0037] The Nernst-Planck theory is used to describe the drift, diffusion, and recombination processes of electrons and holes, and the calculation formula is as follows:

[0038]

[0039]

[0040] in, For hole response terms, For electronic reaction terms, The hole diffusion coefficient is denoted as . The electron diffusion coefficient is... The driving force for hole diffusion, It is the driving force for electron diffusion.

[0041] The beneficial effects of this application are:

[0042] This application provides a method for calculating space charge distribution based on tunneling effect and hole extraction restriction. This method establishes a space charge distribution model for polymer materials that incorporates electric field distribution and carrier transport, simulating charge injection, trapping, detrapping, and recombination behavior under strong electric fields and periodic polarity reversal voltages. Furthermore, a tunneling emission model and hole extraction restriction boundary conditions are introduced into the electrode boundary conditions, and periodic voltage reversal is achieved through a time function to accurately characterize the charge migration and spatial distribution characteristics under polarity reversal. In addition, the model effectively reflects the synergistic influence of tunneling emission and extraction restriction mechanisms on the evolution of interface potential and space charge, providing important theoretical and simulation support for the aging mechanism analysis, insulation design, and lifetime prediction of polymer materials under strong electric fields and polarity reversal conditions. Attached Figure Description

[0043] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other embodiments can be obtained based on these drawings.

[0044] Figure 1 This is a flowchart illustrating a method for calculating space charge distribution based on tunneling effect and limited hole extraction, provided in an embodiment of this application.

[0045] Figure 2 This is a schematic diagram of a polarity reversal voltage provided in an embodiment of this application.

[0046] Figure 3 The experimental results are shown in the figure provided for the embodiments of this application.

[0047] Figure 4 The simulation results are shown in the embodiments of this application.

[0048] Figure 5 This is a schematic diagram illustrating the electric field distortion of an epoxy sample under a polarity reversal voltage at 100°C and an electric field strength of Epr = 100 kV / mm at the electrode, as provided in an embodiment of this application.

[0049] Figure 6 This is a schematic diagram illustrating the electric field distortion of an epoxy sample under a polarity reversal voltage at 100°C and an electric field strength of Epr = 20 kV / mm at the electrode, as provided in an embodiment of this application. Detailed Implementation

[0050] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art based on this application are within the scope of protection of this application.

[0051] This application provides a method for calculating space charge distribution based on tunneling effect and restricted hole extraction. This method can be found in [reference needed]. Figure 1 , Figure 1 The diagram shown is a flowchart illustrating a method for calculating space charge distribution based on tunneling effect and restricted hole extraction, as provided in an embodiment of this application. The method includes:

[0052] S1: A one-dimensional finite element structural model is established using Maxwell's electromagnetic field theory and Nernst-Planck's theory.

[0053] In one embodiment of this application, a one-dimensional finite element structural model is first established using Maxwell's electromagnetic field theory and Nernst-Planck theory. The following steps are designed in this embodiment:

[0054] Taking the calculation of space charge distribution in epoxy materials as an example, this numerical simulation, based on multi-physics coupling, simultaneously calls the electrostatic field module and the rare matter transport module for joint solution to achieve dynamic coupling calculation of electric field distribution and carrier migration. The simulation domain adopts a one-dimensional structural model with epoxy resin as the dielectric material, with a thickness of 0.24 mm. Ideal electrodes are set on both sides, defined as the anode and cathode, respectively. The mesh is adaptively refined along the thickness direction, with a minimum element length of 0.2 μm to ensure high-precision analysis of the electric field gradient and space charge density.

[0055] S2: Construct a space charge distribution model for polymer materials based on the tunneling effect and the hole extraction restriction mechanism.

[0056] Specifically, it includes:

[0057] S201: Based on a one-dimensional finite element structural model and the relationship between charge injection and extraction, charge trapping and detachment, charge transport, charge neutralization and electric field, establish the relationship between electric field and the spatial charge dissipation behavior inside the medium.

[0058] S202: Based on the relationship between the electric field and the dissipation behavior of space charge inside the medium, physical fields including electrostatic field and rare matter transfer field are added to the one-dimensional finite element structural model.

[0059] S203: Add physical parameters of materials to a one-dimensional finite element structural model and add boundary conditions to the physical field.

[0060] S204: By applying a periodic polarity reversal voltage at the boundary of the physical field and introducing a composite carrier injection model and a hole extraction restricted boundary condition, a space charge distribution model of polymer materials based on the tunneling effect and the hole extraction restricted mechanism is obtained.

[0061] In one embodiment of this application, the polarity reversal voltage is as follows: Figure 2 As shown, the periodic polarity reversal voltage is represented by a globally defined time function. This time function is set according to the polarity reversal voltage experiment procedure and is a piecewise function, including a voltage holding segment and a cosine smooth transition segment. It is used to simulate the actual polarity reversal voltage waveform, thereby achieving a smooth switching of voltage amplitude between positive and negative polarities. The formula is as follows:

[0062]

[0063] in, It is a time function. Voltage amplitude, The duration of each half-cycle, The polarity reversal transition time ( Smooth transition area). This is the floor function. To take the remainder, It is a time variable used to simulate charge behavior under long-term polarity reversal conditions.

[0064] The process of defining polarity reversal voltage includes:

[0065] Define the anode: Apply voltage: V(t) = pw⁴(t);

[0066] The time function is defined as:

[0067] pw4(t) = piecewise(

[0068] 0 <= t < 10800, 24000

[0069] 10800<=t<10860, 24000*cos(pi*(t-10800) / 60),

[0070] 10860 <= t < 21600, -24000

[0071] 21600<=t<21660,-24000*cos(pi*(t-21600) / 60),

[0072] 21660<=t<32400, 24000)

[0073] Where pi is the mathematical constant pi, used to simulate a smooth transition, and * represents multiplication.

[0074] The function describes a period of approximately 3 × 10⁻⁶. 4 The polarity reversal voltage of s has a peak value of 24kV, corresponding to a field strength of 100kV / mm.

[0075] Define the cathode potential as fixed at 0V and set a restricted boundary for hole extraction:

[0076]

[0077] in, This represents the hole extraction current density. This is the hole extraction coefficient at the electrode. The mobility of hole carriers. The hole carrier density, for The electric field strength at the location.

[0078] The extraction flux is only valid when the electric field direction satisfies the condition that the anode voltage is less than 0, to reflect the limited extraction characteristics. All other boundaries are set to be electrically insulated.

[0079] In one embodiment of this application, during the process of charge injection, extraction and setting interface boundary conditions, this embodiment focuses on setting the thermally excited emission mechanism, the quantum tunneling emission mechanism and the hole extraction restricted boundary conditions at the electrode interface.

[0080] Based on the injection of electrons and holes, electrons are injected through two electrodes during pressurization: Schottky emission and tunneling. Schottky emission is present throughout the simulation, but tunneling requires a critical electric field strength to trigger. Holes are generated during pressurization according to Schottky emission injection.

[0081] According to Schottky's injection law, the formulas for injecting electrons and holes are as follows:

[0082]

[0083]

[0084] in, The injection current density for electrons. The injection current density of holes. Richardson's constant, Thermodynamic temperature Boltzmann's constant, For charge quantity, To create a potential barrier for electrons in a Schottky circuit. To inject a potential barrier into a hole's Schottky barrier. The electric field strength at the cathode. The electric field strength at the anode is... and These are the vacuum permittivity and the relative permittivity, respectively. The location is the thickness of the sample.

[0085] When the electric field strength exceeds a critical value, the electron injection current density adopts a superposition model of two mechanisms: Schottky emission and field-induced quantum tunneling emission based on the Fowler-Nordheim equation. The tunneling effect is triggered when the local electric field strength exceeds a preset threshold, specifically expressed as follows:

[0086]

[0087]

[0088]

[0089] in, For the injected current density, Injecting current density to the Schottky effect, The injection current density for the Fowler-Nordheim tunneling emission mechanism. For electric field strength, For location, For local electric field intensity, The threshold electric field strength, and The tunneling coefficient, Injecting a potential barrier into electrons.

[0090] In one embodiment of this application, when electrons and holes migrate to the opposite electrode under the influence of an electric field, the blocking effect of the electrode interface needs to overcome the potential barrier; establishing a hole extraction constraint model, the steps include:

[0091] When electrons and holes migrate to the opposite electrode under the influence of an electric field and are extracted, the hole extraction is subject to an additional hindering factor due to the potential barrier at the electrode interface. The suppression of holes and electrons is determined by the following formulas:

[0092]

[0093]

[0094] in, This represents the hole extraction current density. The electron extraction current density. and These are the extraction coefficients of holes and electrons at the electrodes, respectively. and These represent the mobilities of hole and electron carriers, respectively. and These represent the hole and electron carrier densities, respectively. for The electric field strength at the location, For the range of values ​​within The hole extraction resistance factor is used to characterize the inhibitory effect of the interface barrier on hole extraction.

[0095] In one embodiment of this application, Maxwell's electromagnetic field theory is coupled with Nernst-Planck's theory through space charge density. and potential gradient Achieving strong coupling allows for accurate simulation of the dynamic changes in space charge accumulation, release, and electric field distortion within epoxy materials under a polarity-reversed electric field. The coupling model, which is a space charge distribution model for polymer materials based on the tunneling effect and the hole extraction restriction mechanism, includes Maxwell's electromagnetic field theory and Nernst-Planck theory.

[0096] Maxwell's electromagnetic field theory is used to calculate the electric potential distribution. It satisfies the Poisson equation:

[0097]

[0098] in, Let be the divergence of the vector field. For the potential gradient, This is the carrier occupancy rate conversion factor. Hole concentration This refers to the electron concentration.

[0099] The Nernst-Planck theory is used to describe the drift, diffusion, and recombination processes of electrons and holes. The calculation formula is as follows:

[0100]

[0101]

[0102] In the formula, For hole response terms, For electronic reaction terms, The hole diffusion coefficient is denoted as . The electron diffusion coefficient is... The driving force for hole diffusion, It is the driving force for electron diffusion.

[0103] The diffusion coefficient is determined by the Einstein relation:

[0104]

[0105] In one embodiment of this application, after electrons and holes are injected into the polymer material from the electrode, the formula for establishing the transport process of charge carriers in the epoxy material is as follows:

[0106]

[0107]

[0108]

[0109] in, Let be the apparent mobility of charge. The diffusion coefficient is expressed in m² / s. The free charge carrier density, in C / m³. Carrier density, in C / m³. This refers to the conduction current density, expressed in A / m². For source terms, The applied voltage is interpreted as the potential distribution. Net charge density, in units of C / m³.

[0110] , , , The changes in charge number of free electrons, free holes, trapped electrons, and trapped hole carriers due to recombination, trapping, and detrapping, respectively, are expressed by the following formulas:

[0111]

[0112]

[0113]

[0114]

[0115] in, The density of the trap holes, The density of trapped electrons, and These represent the hole and electron carrier densities, respectively. The electron trap density concentration, The trap density concentration of holes. , , and This represents the combined effect between the four types of charges. and Here, are the trapping coefficients, representing the ability of a deep trap to capture holes and electrons, respectively. and , where represents the escape coefficient, and represents the escape capability of holes and electrons, respectively.

[0116] In one embodiment of this application, a periodic polarity reversal voltage is applied to the electrode boundary. The waveform of this voltage is defined as a function of time, with an amplitude of ±24 kV, corresponding to an average electric field strength of approximately 100 kV / mm, and a polarity reversal period of 10800 s. The model operating temperature is set to 373.15 K, and the relative permittivity is... Carrier diffusion coefficient , Both are 2×10 - ¹²m² / s. The tunneling emission formula and hole extraction constraint coefficient are introduced into the electrode boundary conditions. Electron extraction coefficient The carrier transport equations include source terms for drift, diffusion, trapping, and escape, and the trapping coefficient... , rate of detachment .

[0117] S3: The space charge distribution model of polymer materials based on the tunneling effect and the hole extraction restriction mechanism is calculated by time stepping through a numerical solver to obtain the space charge distribution and electric field distortion law of polymer materials under strong electric field and polarity reversal conditions.

[0118] In one embodiment of this application, a time step is used. Total calculation time The TimeDependent solver, with a relative tolerance set to 1×10⁻⁶. - ³, to ensure the convergence of the strongly nonlinear coupled equations. The main parameters are shown in Table 1.

[0119] Table 1 Main Parameters

[0120]

[0121] like Figure 3 , Figure 4 As shown, the simulated charge distribution peak and decay time constant agree well with the experimental results when compared with those obtained from pulsed electroacoustic (PEA) measurements. Consistent with the experimental results, the model also exhibits a clear positive charge accumulation characteristic on the cathode side, i.e., the red band in the lower left gradually expands into the sample over time and dissipates rapidly after polarity reversal. Simultaneously, a significant negative charge accumulation region also appears near the upper and lower electrodes. These phenomena indicate that after holes are injected at the anode, they are restricted by the interface barrier upon reaching the cathode interface and are difficult to extract effectively, thus gradually accumulating to form a high-density positive charge region near the cathode.

[0122] like Figure 5 and Figure 6 As shown, the left figure illustrates the evolution of the electric field distortion rate, and the right figure illustrates the evolution of the location of the maximum electric field. Figure 5 and Figure 6 As can be seen in the left figure, the evolution law of the simulated electric field distortion rate is highly consistent with the experimental results: in each polarization stage, the electric field distortion rate exhibits a typical trend of "first rapidly increasing, then gradually decreasing". Figure 5 and Figure 6 As can be seen in the right figure, the location of the maximum electric field also exhibits obvious temporal migration characteristics, that is, initially it moves from the electrode interface to the middle of the sample, and after reaching the middle, it alternates between the lower electrode and the middle of the sample.

[0123] The simulation results are highly consistent with experimental observations, verifying the accurate description capability of the proposed model for the space charge evolution of epoxy materials under strong electric fields and polarity reversal conditions. This method can be extended to other insulating material systems; by adjusting the relative permittivity and trap density parameters, it can be used for space charge analysis of different temperature gradients or multilayer structures.

[0124] This application provides a method for calculating space charge distribution based on tunneling effect and restricted hole extraction. A composite carrier injection model is established at the electrode-polymer interface, incorporating both thermally excited barrier crossing (Schottky effect) and barrier transmission (quantum tunneling) emission channels. Under strong electric fields, the tunneling channel dominates, enabling electrons to enter the polymer bulk phase at a higher injection rate. This significantly increases the carrier density and space charge accumulation level within the material, and has a crucial impact on subsequent electric field distortion, charge transport, and polarization response processes. Furthermore, this method overcomes the limitations of traditional Schottky models in describing electron injection behavior under high field strengths, achieving a multi-physics coupled description of interface barrier modulation, electron tunneling injection, and space charge evolution. This provides a new theoretical foundation and computational framework for modeling the charge behavior of polymer insulating materials under ultra-high voltage electric fields.

[0125] It should be noted that those skilled in the art will recognize that the embodiments described herein are for the purpose of helping readers understand the principles of this application, and should be understood as not limiting the scope of protection of this application to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this application without departing from the essence of this application, and these modifications and combinations are still within the scope of protection of this application.

Claims

1. A method for calculating space charge distribution based on tunneling effect and restricted hole extraction, characterized in that, include: S1: Establish a one-dimensional finite element structural model with polymer material as the main body and electrodes as the boundary; S2: Based on a one-dimensional finite element structural model, coupled with Maxwell's electromagnetic field theory and Nernst-Planck theory, a periodic polarity reversal voltage is applied at the electrode boundary, and a composite carrier injection model and hole extraction restricted boundary conditions are introduced to obtain a space charge distribution model of polymer materials based on tunneling effect and hole extraction restricted mechanism. S3: The space charge distribution model of polymer materials based on tunneling effect and hole extraction restriction mechanism is calculated by time stepping through numerical solver to obtain the space charge distribution and electric field distortion law of polymer materials under strong electric field and polarity reversal conditions. S2 includes: S201: Based on a one-dimensional finite element structural model and the relationship between charge injection and extraction, charge trapping and de-trapping, charge transport, charge neutralization and electric field, establish the relationship between electric field and the spatial charge dissipation behavior inside the medium. S202: Based on the relationship between the electric field and the dissipation behavior of space charge inside the medium, add physical fields including electrostatic field and rare matter transfer field to the one-dimensional finite element structural model. S203: Add physical parameters of materials to a one-dimensional finite element structural model and add boundary conditions to the physical field; S204: Apply a periodic polarity reversal voltage at the boundary of the physical field, and introduce a composite carrier injection model and a hole extraction restricted boundary condition to obtain a space charge distribution model of polymer materials based on the tunneling effect and the hole extraction restricted mechanism. The periodic polarity reversal voltage is represented by a globally defined time function, which includes a voltage holding segment and a cosine smooth transition segment. The expression for the time function is: in, It is a time function. Voltage amplitude, For time variables, The duration of each half-cycle, This is the polarity reversal transition time. This is the floor function. To take the remainder.

2. The method for calculating space charge distribution based on tunneling effect and restricted hole extraction as described in claim 1, characterized in that, The composite carrier injection model includes the Schottky emission mechanism and the Fowler-Nordheim tunneling emission mechanism; The Fowler-Nordheim tunneling emission mechanism is triggered when the local electric field strength at the electrode interface exceeds a preset threshold. The expression for the emission current density of the Fowler-Nordheim tunneling emission mechanism is as follows: in, For the injected current density, Injecting current density to the Schottky effect, The injection current density for the Fowler-Nordheim tunneling emission mechanism. For electric field strength, For location, For local electric field intensity, The threshold electric field strength, and The tunneling coefficient, Injecting a potential barrier into electrons, Richardson's constant, Thermodynamic temperature Boltzmann's constant, To create a potential barrier for electrons in a Schottky circuit. and These are the vacuum permittivity and the relative permittivity, respectively. The electric field strength at the cathode. This represents the amount of electric charge.

3. The method for calculating space charge distribution based on tunneling effect and restricted hole extraction as described in claim 2, characterized in that, The restricted boundary conditions for hole extraction include: The hole extraction restricted boundary condition uses a hole additional resistance factor to simulate the restricted hole extraction behavior at the electrode interface. When electrons and holes migrate to the opposite electrode under the influence of an electric field and are extracted, the extraction of holes is suppressed by the additional resistance factor of holes due to the potential barrier effect at the electrode interface. The formula for calculating the extraction current density of holes and electrons is: in, This represents the hole extraction current density. The electron extraction current density. Add an obstruction factor to the hole. and These are the extraction coefficients of holes and electrons at the electrodes, respectively. and These represent the mobilities of hole and electron carriers, respectively. and These represent the hole and electron carrier densities, respectively. for The electric field strength at the location.

4. The method for calculating space charge distribution based on tunneling effect and restricted hole extraction as described in claim 3, characterized in that, The space charge distribution model of polymer materials based on tunneling effect and hole extraction restriction mechanism includes: Maxwell's electromagnetic field theory and Nernst-Planck theory; Maxwell's electromagnetic field theory is used to calculate the potential distribution, which satisfies the Poisson equation: in, Let be the divergence of the vector field. For the potential gradient, This is the carrier occupancy rate conversion factor. Hole concentration Electron concentration; The Nernst-Planck theory is used to describe the drift, diffusion, and recombination processes of electrons and holes, and the calculation formula is as follows: in, For hole response terms, For electronic reaction terms, The hole diffusion coefficient is denoted as . The electron diffusion coefficient is... The driving force for hole diffusion, It is the driving force for electron diffusion.

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