WO2022237090A1 - Method for synchronously determining diode boundary electric fields and current densities - Google Patents

Method for synchronously determining diode boundary electric fields and current densities Download PDF

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WO2022237090A1
WO2022237090A1 PCT/CN2021/127813 CN2021127813W WO2022237090A1 WO 2022237090 A1 WO2022237090 A1 WO 2022237090A1 CN 2021127813 W CN2021127813 W CN 2021127813W WO 2022237090 A1 WO2022237090 A1 WO 2022237090A1
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electric field
current density
value
boundary
cathode
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PCT/CN2021/127813
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Chinese (zh)
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朱映彬
廖美焱
姚若河
耿魁伟
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华南理工大学
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/15—Correlation function computation including computation of convolution operations

Definitions

  • the present invention relates to the field of vacuum electronic devices, in particular to a method for synchronously determining the boundary electric field and current density of a diode.
  • the method includes a Single cycle and two iterations, the single cycle completes the adjustment of the electron current density, so that the cathode boundary electric field is zero; two iterations complete the update of the electron current density, making the cathode boundary electric field further tend to zero.
  • This technical document proposes There are some deficiencies in the method, first: this method is only suitable for solving the case where the boundary electric field is zero, for more general cases, this technical literature does not give a specific method; second: this method only involves a single boundary problem, When there are two kinds of charges in the diode, that is, the double boundary problem, the method proposed in this technical literature cannot be solved.
  • the Finite Difference Method (Finite Difference Method) is referred to as the difference method, and is widely used in the field of numerical analysis of electromagnetic fields.
  • the difference method is applied to electromagnetic fields
  • the calculation area is divided into many grid nodes first, and the differential quotient is approximated by the difference quotient.
  • the partial differential equation in the field is converted into a difference equation.
  • the difference equation and boundary conditions can be used Calculate the numerical solution of the potential of each discrete node.
  • the purpose of the present invention is to overcome the difficulties encountered in engineering calculations and provide a method for synchronously determining the boundary electric field and current density of a diode, which can simultaneously determine the boundary electric field and current density of a diode degree without solving complex nonlinear equations analytically.
  • the present invention is realized through at least one of the following technical solutions.
  • a method for synchronously determining the boundary electric field and current density of a diode comprising the following steps: Step 1, giving two initial values of electron current density and an initial value of ion current density; Step 2, setting the initial value of electron current density and an ion current Substitute the initial value of the density into the Poisson equation, and use the finite difference method for numerical iterative solution to obtain the corresponding cathode boundary electric field value; Step 3, based on the required cathode boundary conditions, and the obtained two cathode boundary electric field values and the corresponding Current density value, predicted electron current density value; Step 4.
  • step 2 Based on the predicted electron current density value, perform step 2 to obtain a new cathode boundary electric field value, and judge whether the new cathode boundary electric field value meets the required cathode boundary condition, if If it is not satisfied, use the latest two cathode boundary electric field values, and perform steps 3 to 4; if satisfied, calculate the anode boundary electric field value at this time; step 5, given a new ion current density value, perform steps 1 to Step 4, obtain the new anode boundary electric field value; Step 6, based on the required anode boundary conditions, two anode boundary electric field values and corresponding ion current density values, obtain the predicted ion current density value, the predicted ion current density value Will make the anode boundary electric field meet the given requirements; Step 7.
  • the iterations include double loops and three layers of iterations, and the three layers of iterations include internal iterations, intermediate iterations and external iterations.
  • the cathode potential of a given diode is zero, the anode potential is Vo, the cathode injection electron current density is 4(x), and the anode injection ion current density is (x), the potential at any point in space is 0(x, )0, then the Poisson equation in two dimensions is expressed as:
  • m,., Z, me, x, _y represent vacuum permittivity, ion mass, ion electric Charge number, electron mass, x-axis coordinates and The relative ratio of the current density to the electron current density;
  • the Poisson equation is discretized by the second-order finite difference method, and the following iterative formula is obtained:
  • ⁇ and ⁇ +1 represent the values of the ⁇ iteration and ⁇ +1 iteration respectively, ⁇ , / ⁇ , , , and % points
  • n and n+1 represent the value of the nth iteration and n+1 iteration respectively, is the cathode boundary electric field value, is the electron current density value at a certain point in space, / is according to Electron current density values predicted from the latest cathode boundary electric field values and electron current density values.
  • step 4 if the electrons are injected non-uniformly, it is judged that the new cathode boundary electric field value is
  • G and F are all unknown coefficients
  • the anode boundary condition is fl
  • the ion current density value is predicted according to the obtained two anode boundary electric field values and corresponding ion current density values, and the iterative formula is:
  • n and n+1 represent the value of the nth iteration and n+1 iteration respectively, and is the boundary electric field value of the anode, / course is the ion current at a certain point in space
  • the density value is the electron current density value predicted according to the newly obtained cathode boundary electric field value and the electron current density value.
  • the judgment condition is to calculate The absolute value of the cathode boundary electric field value obtained minus the required electric field value is less than the preset condition.
  • the judgment condition of whether the anode boundary electric field value of the judgment prediction meets the required anode boundary condition is the calculated anode boundary
  • the absolute value of the electric field value minus the required electric field value is less than the preset condition.
  • FIG. 1 is a flowchart of a method for synchronously determining the boundary electric field and current density of a diode according to an embodiment of the present invention map.
  • DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS In order to make the object, technical solution and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments.
  • a method for synchronously determining the electric field and current density at the boundary of a diode includes two loops and three iterations: internal iteration, intermediate iteration and external iteration. The two cycles complete the adjustment of the electron and ion current densities, so that the cathode and anode meet the given boundary conditions.
  • the internal iteration uses the finite difference method to numerically solve the Poisson equation to obtain the cathode electric field value; the intermediate iteration updates the electron current density according to the previously obtained cathode electric field value; the external iteration updates the ion current density according to the previously obtained anode electric field value.
  • the boundary electric field and current density of the diode can be determined synchronously without analytically solving complicated nonlinear equations.
  • Embodiment 1 In the case where electron and ion currents are both non-uniform in a limited emission region, the method for synchronously determining the boundary electric field and current density of a diode includes the following steps: Step 1.
  • Initial parameter settings are given two different An initial value vector of electron current density and an initial value vector of ion current density; Step 2, internal iteration If the cathode potential of a given diode is zero, the anode potential is Vo, the charge emission region is located in the center of the boundary, the width is r, and the cathode at X The injected electron current density is 4(x), the injected ion current density at the anode X is * /i(X), the potential at the space (X, )0 is 0(x, )0, then the Poise under the limited emission area
  • the loose equation can be expressed as:
  • « and «+1 represent the values of the nth iteration and the n+1 iteration respectively, is the cathode boundary electric field value, and is the electron current density at a certain point in space Value
  • / is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of / may be too large, so only take 0.1 times / when predicting the new current density.
  • a new initial value vector of electron current density can be obtained; Step 4.
  • step 2 Based on the new electron current density vector, perform step 2 to obtain the new cathode boundary electric field
  • the condition is that the absolute value of the calculated cathode boundary electric field value minus the required electric field value is less than the preset condition. Since electrons are injected non-uniformly, the electric field values of all cathode boundary grid points need to meet this condition.
  • the fitting function has three unknown parameters, so three electric field values closest to the anode are needed to calculate the electric field value at the boundary of the anode, where the electric field value is defined as the electric field value between two grid nodes; step five, given another ion current density Initial value vector, perform steps 1 to 4 to obtain another new anode boundary electric field vector; step 6, external iteration If the required anode boundary condition is fl , then according to the newly obtained two anode boundary electric field vectors and their corresponding The ion current density value of , can predict a new ion current density vector, the iterative formula is:
  • « and «+l represent the values of the nth iteration and «+1 iteration respectively, is the boundary electric field value of the anode, / is the ion current density at a certain point in space
  • the value of G is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value. Since the value of G is too large, only take 0.1 times /% when predicting the new current density Step 7. Based on the predicted ion current density vector, repeat steps 1 to 4 to obtain the predicted anode boundary electric field vector, and judge whether the predicted anode boundary electric field value meets the required anode boundary conditions.
  • the judgment condition is the calculated anode boundary
  • the absolute value of the electric field value minus the required electric field value is less than the preset condition, if it is not satisfied, use two new anode boundary values, perform step 6, and then perform step 7; repeat steps 1 to 4 as an intermediate iterative process , the purpose is to calculate the boundary electric field value of the anode.
  • Step 8. Output the final electron and ion current densities.
  • the method for synchronously determining the boundary electric field and current density of the diode includes the following steps: Step 1. Initial parameter setting given two different The initial value vector of the electron current density and an initial value vector of the ion current density, because the ion current is uniformly distributed, so in the emission region, the ion current density is equal to a fixed constant; Step 2, internal iteration If the cathode potential of the diode is given is zero, the anode potential is Vo, the charge emission region is located in the center of the boundary, and the width is R.
  • the injected electron current density at the cathode x is 4(X)
  • the injected ion current density at the anode x is /i(X)
  • the space (X , the potential at )0 is 0(x, )0
  • VV(x, _y) 0, for
  • ⁇ and ⁇ +1 represent the value of the ⁇ iteration and ⁇ +1 iteration respectively, ⁇ , / z, , , and % respectively represent the potential value at the space grid (/, j), grid Accuracy, the electron current density at the /th grid on the abscissa, and the q value at the /th grid on the abscissa.
  • the Poisson equation can be numerically solved through the above iterative formula.
  • « and «+1 represent the value of the nth iteration and the n+1 iteration respectively, is the cathode boundary electric field value, and is the electron current density value at a certain point in space
  • / is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of / may be too large, so only take 0.1 times / when predicting the new current density.
  • Step 4 based on the new electron current density vector, perform step 2 to obtain a new cathode boundary electric field
  • the condition is that the absolute value of the calculated cathode boundary electric field value minus the required electric field value is less than the preset condition. Since electrons are injected non-uniformly, the electric field values of all cathode boundary grid points need to meet this condition.
  • the fitting function is:
  • the fitting function has three unknown parameters, so three electric field values closest to the anode are needed to calculate the electric field value at the boundary of the anode, where the electric field value is defined as the electric field value between two grid nodes; Step 5. Given another initial value vector of ion current density, perform steps 1 to 4 to obtain another new anode boundary electric field vector; Step 6.
  • I n+l 0AI * + 0.9In the n formula, ⁇ and ⁇ +l respectively represent the value of the nth iteration and ⁇ +1 iteration, is the anode boundary electric field value, / is the ion current density value of a certain point in space , G is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of G may be too large, so only take 0.1 times /% when predicting the new current density needs It should be noted that since the ion current is uniform, the anode electric field always increases to zero at the middle position first, so it is only necessary to calculate the ion current density value at the center grid point of the anode boundary, and the ion current density values at other positions are equal to the current density value.
  • Step 7 Based on the predicted ion current density vector, repeat steps 1 to 4 to obtain the predicted anode boundary electric field vector, and judge whether the predicted electric field value at the center grid point of the anode boundary satisfies the required anode boundary condition.
  • the judgment condition is The absolute value of the calculated anode boundary electric field value minus the required electric field value is less than the preset condition, if not satisfied, use two new anode boundary values, perform step 6, and then perform step 7; step 8, output the final electron and ion current densities.
  • Embodiment 3 In the case where the electron and ion currents are uniform under the limited emission region, the method for synchronously determining the boundary electric field and current density of the diode includes the following steps: Step 1.
  • V 2 (f> ⁇ x,y ⁇ 0, for
  • ⁇ and ⁇ +1 represent the value of the ⁇ iteration and ⁇ +1 iteration respectively
  • ⁇ , / z, , , and % respectively represent the potential value at the space grid (/, j), grid Accuracy, the electron current density at the /th grid on the abscissa, and the q value at the /th grid on the abscissa.
  • the Poisson equation can be numerically solved through the above iterative formula. Substitute the given initial value of electron current density and ion current density into the Poisson equation, and use the finite difference method to numerically iteratively solve the Poisson equation to obtain the space potential distribution, and then use the fitting function to calculate the cathode boundary electric field.
  • the fitting function has three unknown parameters, so three electric field values closest to the cathode are needed to calculate the cathode boundary electric field value, where the electric field value is defined as the electric field value between two grid nodes. Step 3, intermediate iteration
  • « and «+1 represent the values of the nth iteration and the n+1 iteration respectively
  • is the cathode boundary electric field value and is the electron current density at a certain point in space Value
  • / is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of / may be too large, so only take 0.1 times / when predicting the new current density.
  • step 4 based on the new electron current density vector, perform step 2 to obtain the new cathode boundary electric field
  • the condition is that the absolute value of the calculated cathode boundary electric field value minus the required electric field value is less than the preset condition. Since electrons are injected non-uniformly, the electric field values of all cathode boundary grid points need to meet this condition.
  • the fitting function has three unknown parameters, so three electric field values closest to the anode are needed to calculate the electric field value at the boundary of the anode, where the electric field value is defined as the electric field value between two grid nodes; step five, given another ion current density Initial value vector, perform steps 1 to 4 to obtain another new anode boundary electric field vector; step 6, external iteration If the required anode boundary condition is fl , then according to the newly obtained two anode boundary electric field vectors and their corresponding The ion current density value of , can predict a new ion current density vector, the iterative formula is:
  • « and «+l represent the values of the nth iteration and «+1 iteration respectively, is the anode boundary electric field value, / is the ion current density value at a certain point in space, and is based on the latest cathode boundary electric field value and The electron current density value predicted by the electron current density value may be too large, so only take 0.1 times/% when predicting the new current density.
  • Step 7 based on the predicted ion current density vector, Repeat steps 1 to 4 to obtain the predicted electric field vector of the anode boundary, and judge whether the calculated electric field value of the center grid point of the anode boundary meets the required anode boundary conditions.
  • Example 4 Calculation of the confinement current density under the columnar structure, including the following steps: Step 1, the initial parameter setting of the algorithm Given two different initial values of the electron current density and an initial value of the ion current density; Step 2, the internal iteration if Given that the cathode potential of a cylindrical diode is zero, the anode potential is Vo, the cathode injection current density of radius is 4, and the anode injection current density of radius is , then the Poisson equation can be expressed as: Poisson’s equation is discretized by using the first-order finite difference method, and the following iterative formula can be obtained: In the formula, « and «+I represent the values of the «th iteration and «+1 iteration, respectively, and v ⁇ ra+1 represents the potential value at
  • « and «+1 represent the values of the nth iteration and the n+1 iteration respectively, is the cathode boundary electric field value, and is the electron current density value, / is according to The value of electron current density predicted by the newly obtained cathode boundary electric field value and electron current density value may be too large, so only 0.1 times / is used when predicting the new current density.
  • step 2 Based on the new electron current density vector, perform step 2 to obtain the new cathode boundary electric field
  • the fitting function has three unknown parameters, so three electric field values closest to the anode are needed to calculate the electric field value at the boundary of the anode, where the electric field value is defined as the electric field value between two grid nodes; step five, given another ion current density Initial value, execute steps 1 to 4 to get another new anode boundary electric field; step 6, external iteration
  • I n+l 0. ⁇ I * +0.9I n formula, " and "+1 respectively represent the value of the nth iteration and "+1 iteration, is the anode boundary electric field value, / is the ion at a certain point in space
  • the value of current density, i is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of u may be too large, so only take 0.1 times when predicting the new current density /% Step 7.
  • steps 1 to 4 Based on the predicted ion current density value, repeat steps 1 to 4 to obtain the predicted anode boundary electric field value, and judge whether the predicted anode boundary electric field value meets the required anode boundary conditions.
  • the judgment condition is calculated The absolute value of the anode boundary electric field value minus the required electric field value is less than the preset condition, if not satisfied, use two new anode boundary values, perform step six, and then perform step seven; step eight, output the final electron and ionic current density.

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Abstract

Disclosed in the present invention is a method for synchronously determining diode boundary electric fields and current densities, comprising the following steps: 1. giving two electronic current density initial values and an ion current density initial value; 2. substituting the electronic current density initial values and the ion current density initial value into Poisson's equation to obtain a corresponding cathode boundary electric field value; 3. predicting an electronic current density value; 4. obtaining a new cathode boundary electric field value on the basis of the predicted electronic current density value, and determining whether the new cathode boundary electric field value satisfies a required cathode boundary condition; 5. giving a new ion current density value, and performing steps 1-4 to obtain two anode boundary electric field values; 6. predicting an ion current density value; 7. predicting an anode boundary electric field value, and determining whether the predicted anode boundary electric field value satisfies a required anode boundary condition; and 8. outputting final electronic and ion current densities.

Description

一种 同步确 定二 极管边 界电 场与 电流密 度的 方法 技术 领域 本发 明涉及 真空电子 器件领 域， 特别涉及一 种同步 确定二极 管边界 电场与 电流 密度的方 法。 背景 技术 为了满 足日益 增长的 能源需求 ， 减少对传统 能源的依 赖， 开发高性能 热电 转换 技术引起 了世界 各国的 高度重视 。 热离子能量 转换器是 一种将 热能直接 转 换成 电能的 真空二极 管器件 。 但是在求解器 件允许 流过的最 大电流 密度时 ， 解 析求 解非线 性问题较 为困难 ， 特别在引入离子 之后 ， 因为电子和离子 电流密 度 都是 未知量 ， 并且是否随着 电子和 离子电流 密度的 增加， 阴极和阳极是否 能满 足所 要求的 边界条件 也是未知 的。文献 (Non-uniform space charge limited current injection into a nano contact solid" Sci. Rep. 5, 9173, 2015 ), 提出了一种计算边界 电场 值为零时 的电子 电流密度 值的方 法。 该方法包括 一个单循 环和两 次迭代 ， 该单 循环完成 对电子 电流密度 的调整 ， 使得阴极边界 电场为零 ； 两次迭代完成 对 电子电流密 度的更 新， 使得阴极边 界电场进 一步趋 向于零 。 该技术文献所 提 出 的方法存 在一些 不足， 第一： 该方法只 适合于求 解边界 电场为零 的情况 ， 对 于更 一般的 情况， 该技术文献并没 有给出具 体的方 法； 第二： 该方法只涉 及单 边界 问题， 当二极管内存在 两种电荷 时， 即双边界 问题， 该技术文献 所提出 的 方法 无法解 决。 有限 差分法 (Finite Difference Method) 简称差分 法， 广泛地应用 于电 磁场 数值分 析领域 。 差分法应用于 电磁场边 界问题求 解时， 先将计算 区域划 分 为许 多网格 的节点， 并用差商近似微 商， 然后， 将场域中 的偏微分 方程转 换成 差分 方程， 最后， 利用该差分方程 以及边界 条件便 可计算 出各离散 节点的 电位 的数 值解。 虽然解 析求解 非线性 问题较为 困难， 但是通过 差分法 以及结 合简单的 迭代 算法 可以较为 灵活地解 决工程 计算当 中所遇到 的非线性 以及双边 界问题 。 发明 内容 本发 明的 目的在于克 服工程 计算当 中所遇到 困难， 提供了一种同步 确定二 极管 边界电场 与电流 密度的 方法， 可以同步 地确定 二极管 的边界 电场与电流 密 度 ， 无需解析地求解 复杂的非 线性方程 。 本发 明至少通 过如下技 术方案之 一实现 。 一种 同步确定 二极管边 界电场与 电流密度 的方法 ， 包括以下步骤 ： 步骤一 、 给定两个电子 电流密度 初始值 以及一个 离子电流 密度初始 值； 步骤 二、 将电子电流 密度初始 值以及 一个离 子电流密 度初始 值代入 泊松方 程 ， 并采用有限差分 法进行数 值迭代求 解，获取对应的 阴极边界 电场值； 步骤三 、 基于所要求 的阴极 边界条件 ， 以及获得的两 个阴极 边界电场 值及 对应 的电子 电流密度值 ， 预测电子电流密 度值； 步骤 四、 基于预测的电子 电流密度 值， 执行步骤二 ，得到新的阴极 边界电场 值， 判断新的阴极边 界电场值 是否满 足所要求 的阴极 边界条件 ， 若不满足， 则 采用 最新的 两个阴极 边界 电场值， 执行步骤三 到步骤 四； 若满足， 则计算此 时 的 阳极边界 电场值； 步骤 五、 给定新的离 子电流密 度值， 执行步骤一到步 骤四， 得到新的阳极 边界 电场值； 步骤六 、 基于所要求 的阳极 边界条件 、 两个阳极边 界电场值 及对应 的离子 电流 密度值 ， 得到预测的离 子电流密 度值， 预测的离子电流 密度值将 使得阳 极 边界 电场满足 所给定的 要求； 步骤七 、 基于预测的 离子电流 密度值 ， 重复步骤一到 步骤四 ， 得到预测的 阳极 边界电场 值， 判断预测的阳极 边界电场 值是否满 足所要 求的阳极 边界条件 ， 若不 满足， 则利用最新计算 得到的两 个阳极 边界值 ， 执行步骤六， 然后执行 步 骤七 ； 步骤八 、 输出最终的电子 和离子 电流密度 。 优选 的， 所述迭代包 括两重 循环以及 三层迭 代， 所述三层迭 代包括 内部迭 代、 中间迭代和外部 迭代。 优选 的， 所述的步骤 二， 若给定二极 管的阴极 电势为零 ， 阳极电势为 Vo, 阴极 注入电子 电流密度 为 4(x),阳极注入离 子电流密 度为 (x)，空间任意一点 的 电势 为 0(x， )0， 则二维下的泊松方程表示为：
Figure imgf000004_0001
式中 ， m,.、 Z、 me、 x、 _y分别表不真空介电常数、 离子质量、 离子电 荷 数、电子质量、 x轴坐标和
Figure imgf000005_0001
Figure imgf000005_0002
电流密度 与电子 电流密度 的相对比值 ； 采 用二阶 有限差分 法将泊松 方程离 散化， 得到如下 迭代公式:
Figure imgf000005_0003
式 中， 《和《+1分别表不第 《次迭代 和《+1次迭 代的值 ， ＜、 / ^、 ，,和％分
Figure imgf000005_0004
网格精度 、 横坐标第 /个网格处 的电子 电流 密 度和横 坐标第 /个网格处的 q值， 通过上述迭代公式 求解泊松 方程， 得到空间 电势分布 。 优 选的， 所述阴极 边界电场利 用拟合 函数获取 ： f(y) = A + Byl/3 + Cy2/3
Figure imgf000005_0005
/(0) = 为阴极的边界 电场值， 因此需要 三个离 阴极最近 的电场值 来确定 阴极边界 电场值 ， 其中电场值定义 为两个 网格 节 点中间 的电场值 。 优 选的， 若所要求 的阴极边 界条件为 c， 根据获得的两个 阴极边界 电场值 及 对应的 电子电流 密度值， 预测电子电流密度 值， 迭代公式为 ：
Figure imgf000005_0006
人 +1 = 0.1/ +0.9 式中， n和 n+1分别表示第 n次迭代和 n+1次迭代的 值， 为阴极边界 电场值， 为 空间某 一点的 电子电流 密度值 ， /是根据 最新得 到的阴 极边界 电场值和 电子 电流密度 值所预测 的电子 电流密度值 。 优 选的， 步骤四中， 若电子为非均 匀注入 ， 则判断新的 阴极边界 电场值是 TECHNICAL FIELD The present invention relates to the field of vacuum electronic devices, in particular to a method for synchronously determining the boundary electric field and current density of a diode. BACKGROUND OF THE INVENTION In order to meet the increasing energy demand and reduce the dependence on traditional energy sources, the development of high-performance thermoelectric conversion technology has attracted great attention from all over the world. A thermionic energy converter is a vacuum diode device that converts thermal energy directly into electrical energy. However, when solving the maximum current density allowed to flow through the device, it is difficult to analytically solve the nonlinear problem, especially after the introduction of ions, because both the electron and ion current densities are unknown, and whether with the increase of the electron and ion current density, Whether the cathode and anode can meet the required boundary conditions is also unknown. Literature (Non-uniform space charge limited current injection into a nano contact solid" Sci. Rep. 5, 9173, 2015 ), proposed a method for calculating the electron current density value when the boundary electric field value is zero. The method includes a Single cycle and two iterations, the single cycle completes the adjustment of the electron current density, so that the cathode boundary electric field is zero; two iterations complete the update of the electron current density, making the cathode boundary electric field further tend to zero. This technical document proposes There are some deficiencies in the method, first: this method is only suitable for solving the case where the boundary electric field is zero, for more general cases, this technical literature does not give a specific method; second: this method only involves a single boundary problem, When there are two kinds of charges in the diode, that is, the double boundary problem, the method proposed in this technical literature cannot be solved. The Finite Difference Method (Finite Difference Method) is referred to as the difference method, and is widely used in the field of numerical analysis of electromagnetic fields. The difference method is applied to electromagnetic fields When solving the boundary problem, the calculation area is divided into many grid nodes first, and the differential quotient is approximated by the difference quotient. Then, the partial differential equation in the field is converted into a difference equation. Finally, the difference equation and boundary conditions can be used Calculate the numerical solution of the potential of each discrete node. Although it is difficult to solve nonlinear problems analytically, the nonlinear and double boundary problems encountered in engineering calculations can be more flexibly solved through the difference method and a simple iterative algorithm. Invention Contents The purpose of the present invention is to overcome the difficulties encountered in engineering calculations and provide a method for synchronously determining the boundary electric field and current density of a diode, which can simultaneously determine the boundary electric field and current density of a diode degree without solving complex nonlinear equations analytically. The present invention is realized through at least one of the following technical solutions. A method for synchronously determining the boundary electric field and current density of a diode, comprising the following steps: Step 1, giving two initial values of electron current density and an initial value of ion current density; Step 2, setting the initial value of electron current density and an ion current Substitute the initial value of the density into the Poisson equation, and use the finite difference method for numerical iterative solution to obtain the corresponding cathode boundary electric field value; Step 3, based on the required cathode boundary conditions, and the obtained two cathode boundary electric field values and the corresponding Current density value, predicted electron current density value; Step 4. Based on the predicted electron current density value, perform step 2 to obtain a new cathode boundary electric field value, and judge whether the new cathode boundary electric field value meets the required cathode boundary condition, if If it is not satisfied, use the latest two cathode boundary electric field values, and perform steps 3 to 4; if satisfied, calculate the anode boundary electric field value at this time; step 5, given a new ion current density value, perform steps 1 to Step 4, obtain the new anode boundary electric field value; Step 6, based on the required anode boundary conditions, two anode boundary electric field values and corresponding ion current density values, obtain the predicted ion current density value, the predicted ion current density value Will make the anode boundary electric field meet the given requirements; Step 7. Based on the predicted ion current density value, repeat steps 1 to 4 to obtain the predicted anode boundary electric field value, and judge whether the predicted anode boundary electric field value meets the required If the anode boundary conditions are not satisfied, use the newly calculated two anode boundary values to perform step 6, and then perform step 7; step 8, output the final electron and ion current densities. Preferably, the iterations include double loops and three layers of iterations, and the three layers of iterations include internal iterations, intermediate iterations and external iterations. Preferably, in the second step, if the cathode potential of a given diode is zero, the anode potential is Vo, the cathode injection electron current density is 4(x), and the anode injection ion current density is (x), the potential at any point in space is 0(x, )0, then the Poisson equation in two dimensions is expressed as:
Figure imgf000004_0001
In the formula, m,., Z, me, x, _y represent vacuum permittivity, ion mass, ion electric Charge number, electron mass, x-axis coordinates and
Figure imgf000005_0001
Figure imgf000005_0002
The relative ratio of the current density to the electron current density; the Poisson equation is discretized by the second-order finite difference method, and the following iterative formula is obtained:
Figure imgf000005_0003
In the formula, < and <+1 represent the values of the << iteration and <+1 iteration respectively, <, / ^, , , and % points
Figure imgf000005_0004
The grid accuracy, the electron current density at the /th grid on the abscissa, and the q value at the /th grid on the abscissa, solve the Poisson equation through the above iterative formula, and obtain the space potential distribution. Preferably, the cathode boundary electric field is obtained using a fitting function: f(y) = A + By l/3 + Cy 2/3
Figure imgf000005_0005
/(0) = is the boundary electric field value of the cathode, so three electric field values closest to the cathode are needed to determine the boundary electric field value of the cathode, where the electric field value is defined as the electric field value between two grid nodes. Preferably, if the required cathode boundary condition is c , the electron current density value is predicted according to the obtained two cathode boundary electric field values and the corresponding electron current density values, and the iterative formula is:
Figure imgf000005_0006
People +1 = 0.1/ +0.9 In the formula, n and n+1 represent the value of the nth iteration and n+1 iteration respectively, is the cathode boundary electric field value, is the electron current density value at a certain point in space, / is according to Electron current density values predicted from the latest cathode boundary electric field values and electron current density values. Preferably, in step 4, if the electrons are injected non-uniformly, it is judged that the new cathode boundary electric field value is
3
Figure imgf000006_0001
阴极 边界条件£^ 条件 。 优选 的，所述的步骤 四，若阴极边界电场 值满足所 要求的 阴极边界 条件 则满 足则利用 拟合函数 来计算 阳极边界 电场， 拟合函数为 ： f (y) = D + Gyl/3 + Fy2/3 式中 参数 Z)、 G和 F均为未知的系数 ， /(0) =D为阳极的边界电场 值， 因此 需要 三个离 阳极最近 的电场 值来计算 阳极边 界电场值 ， 其中电场值定 义为两个 网格 节点中 间的电场值 。 优选 的， 所述阳极边界 条件为 fl， 根据获得的两个阳极边界 电场值及对 应 的离 子电流密 度值， 预测离子 电流密度 值， 迭代公式为 ：
Figure imgf000006_0002
3
Figure imgf000006_0001
Cathode boundary condition £^ condition. Preferably, in step 4, if the cathode boundary electric field value satisfies the required cathode boundary condition, then the fitting function is used to calculate the anode boundary electric field, and the fitting function is: f (y) = D + Gy l/3 + Fy 2/3 In the formula, parameters Z), G and F are all unknown coefficients, /(0)=D is the boundary electric field value of the anode, so three electric field values closest to the anode are needed to calculate the anode boundary electric field value, where the electric field value is defined as the electric field value between two grid nodes. Preferably, the anode boundary condition is fl , and the ion current density value is predicted according to the obtained two anode boundary electric field values and corresponding ion current density values, and the iterative formula is:
Figure imgf000006_0002
/„+1 = 0.ir +0.9/„ 式 中， n和 n+1分别表示第 n次迭 代和 n+1次迭代的值， 为阳极 边界电场 值， /„ 为空 间某一 点的离子 电流密 度值， 是 根据最 新得到 的阴极边 界电场值 和电子 电流 密度值所 预测的 电子电流密 度值。 优选 的， 所述判断新 的阴极 边界电场 值是否满 足所要 求的阴极 边界条 件， 判断 条件为 计算得到 的阴极 边界电场 值减去所 要求 的电场值 的绝对值 小于预设 条件 。 优选 的， 所述判断预 测的阳极 边界 电场值是否 满足所 要求的 阳极边界 条件 的判 断条件 为计算得 到的阳 极边界 电场值减 去所要求 的电场 值的绝对 值大小 小 于预 设条件 。 与现有 的技术 相比， 本发明的有 益效果 为： 本发 明通过差 分法以 及结合 简单的迭 代包括 两重循环 ， 以及三层迭代 ： 内 部迭 代、 中间迭代和 外部迭 代， 可以较为灵 活地解 决工程 计算当 中所遇到 的非 线性 以及双 边界问题 。 针对阴极有限 发射区域 的情况 ， 以及电子和离 子为均 匀 与非 均匀发 射的情况 ， 由于边缘效应 ， 在迭代过程 中可以清 晰地看 到电流密 度 分布 的非均 匀特性。 附图 说明 图 1 为本发明实施 例一种同 步确定二 极管边 界电场与 电流密度 的方法 的流 程 图。 具体 实施方式 为了 使本发 明的目的 、 技术方案以及 优点更 加清楚 明白， 以下结合附图及 实施 例， 对本发明进 行进一步 的详细 说明。 应当理解 ， 此处所描述的 具体实施 例仅 用于解释 本发明 ， 并不限于本发明 。 如图 1 所示的一种 同步确定 二极管 边界电场 与电流密 度的方法 ， 该迭代算 法包 括两重 循环， 以及三层迭代： 内部迭代、 中间迭代和 外部迭代 。 所述两重 循环 完成对 电子和离 子电流密 度的调 整， 使得阴极和 阳极满 足所给定 的边界 条 件 。 所述的内部迭代 采用有 限差分 法数值 求解泊松 方程， 得到阴极电场值 ； 中 间迭 代根据 之前得到 阴极电 场值更新 电子电流 密度； 外部迭代根据之 前得到 阳 极 电场值更新 离子 电流密度 。 通过运行本申 请所提供 的算法 ， 可以同步地 确定 二极 管的边界 电场与 电流密度 ， 无需解析地求解 复杂的非 线性方程 。 实施例 1: 有限发射区域 下电子与 离子电流 都为非均 匀的情 况， 所述一种同 步确 定二极 管边界电场 与电流密 度的方 法， 包括以下步 骤： 步骤一 、 初始参数设置 给定两 个不 同的电子 电流密度 初始值 向量以 及一个 离子电流 密度初 始值向 量 ； 步骤二 、 内部迭代 若给定 二极管 的阴极 电势为零， 阳极电势为 Vo， 电荷发射区域位于边界 中 心 ， 宽度为 r， 阴极 X处的注入电子电流密 度为 4(x), 阳极 X处的注入离子电流 密度 为 */i(X)， 空间 (X， )0处的电势为 0(x， )0， 则有限发射区域下的泊松方程可以 表示 为： /„ +1 = 0.ir +0.9/„ In the formula, n and n+1 represent the value of the nth iteration and n+1 iteration respectively, and is the boundary electric field value of the anode, /„ is the ion current at a certain point in space The density value is the electron current density value predicted according to the newly obtained cathode boundary electric field value and the electron current density value.Preferably, whether the new cathode boundary electric field value of the judgment meets the required cathode boundary condition, the judgment condition is to calculate The absolute value of the cathode boundary electric field value obtained minus the required electric field value is less than the preset condition.Preferably, the judgment condition of whether the anode boundary electric field value of the judgment prediction meets the required anode boundary condition is the calculated anode boundary The absolute value of the electric field value minus the required electric field value is less than the preset condition.Compared with the prior art, the beneficial effects of the present invention are: the present invention comprises two loops through the differential method and combines simple iterations, and three Layer iteration: internal iteration, intermediate iteration and external iteration can more flexibly solve the nonlinear and double boundary problems encountered in engineering calculations. For the case of the limited emission area of the cathode, and the uniform and non-uniform emission of electrons and ions Due to the edge effect, the non-uniform characteristics of the current density distribution can be clearly seen in the iterative process. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 is a flowchart of a method for synchronously determining the boundary electric field and current density of a diode according to an embodiment of the present invention map. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS In order to make the object, technical solution and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described here are only used to explain the present invention, but not to limit the present invention. As shown in Figure 1, a method for synchronously determining the electric field and current density at the boundary of a diode, the iterative algorithm includes two loops and three iterations: internal iteration, intermediate iteration and external iteration. The two cycles complete the adjustment of the electron and ion current densities, so that the cathode and anode meet the given boundary conditions. The internal iteration uses the finite difference method to numerically solve the Poisson equation to obtain the cathode electric field value; the intermediate iteration updates the electron current density according to the previously obtained cathode electric field value; the external iteration updates the ion current density according to the previously obtained anode electric field value. By running the algorithm provided in this application, the boundary electric field and current density of the diode can be determined synchronously without analytically solving complicated nonlinear equations. Embodiment 1: In the case where electron and ion currents are both non-uniform in a limited emission region, the method for synchronously determining the boundary electric field and current density of a diode includes the following steps: Step 1. Initial parameter settings are given two different An initial value vector of electron current density and an initial value vector of ion current density; Step 2, internal iteration If the cathode potential of a given diode is zero, the anode potential is Vo, the charge emission region is located in the center of the boundary, the width is r, and the cathode at X The injected electron current density is 4(x), the injected ion current density at the anode X is * /i(X), the potential at the space (X, )0 is 0(x, )0, then the Poise under the limited emission area The loose equation can be expressed as:
▽V(x, _y) = 0， for |x| > W/2 for Ixl < W/2
Figure imgf000007_0001
式中 ， 、
Figure imgf000007_0002
离子 质量、 离子电 荷数 、电子质量、 x轴坐标和
Figure imgf000007_0003
q(x) = Jt (x)/ Je (x)(m./Zme)1/2； j{x, y)表示空间 (x， y)处的电势, ▽V(x, _y) = 0, for |x| > W/2 for Ixl < W/2
Figure imgf000007_0001
In the formula, ,
Figure imgf000007_0002
Ion mass, ion charge number, electron mass, x-axis coordinates and
Figure imgf000007_0003
q(x) = J t (x)/ J e (x)(m . /Zm e ) 1/2 ; j{x, y) represents the electric potential at the space (x, y),
5 流密 度与电 子电流密 度的相 对比值 ， 采用二阶有限 差分法将 泊松方 程离散 化， 可得 到如下迭 代公式 ：
Figure imgf000008_0001
式中 ，《和《+1分别表 不第 《次迭代和 《+1次迭代 的值， ＜、 / z、 ，,和％分 别表 示空间 网格 (/， j)处的电势值、 网格精度、 横坐标第 /个网格处的电子 电流 密度 和横坐标 第 /个网格处 的 q值， 通过上述迭代 公式求解 泊松方程 ， 得到空间 电势 分布。 将给 定的电子 电流密 度初始值 以及离 子电流 密度代入 泊松方 程， 采用有限 差分 法对泊 松方程进 行数值 迭代求解 之后， 可以得到空间 电势分布 ， 然后利用 拟合 函数来计 算阴极边 界电场 ， 拟合函数为： f(y) = A + Byl/3 + Cy2/3
Figure imgf000008_0002
/(0) =A为阴极的边 界电场值 。 拟合函数 有三 个未知 参数， 因此需要三个离 阴极最近 的电场 值来计算 阴极边界 电场值 ， 其 中电场值定 义为两个 网格节点 中间的 电场值。 步骤三 、 中间迭代
Figure imgf000008_0003
及其 对应的 电子电流密 度值，可以预 测一个新 的电子 电流密度 值，迭代公式 为：
Figure imgf000008_0004
jn+l = 0.\f +0.9jn 式中 ， 《和《+1分别表示第 n次迭代 和 n+1次迭代 的值， 为阴极边界 电场 值， 为空间某一 点的电子 电流密 度值， /是 根据最新 得到的 阴极边界 电场值和 电子 电流密度 值所预 测的电子 电流密度 值， 由于 /的值可能过大 ， 因此在预测 新 的电流密度 时只取 0. 1倍的 /。 显然根据上面 的迭代 公式， 可得到一个 新的 电子 电流密度 初始值 向量； 步骤 四、 基于新的电子 电流密度 向量， 执行步骤二 ，得到新的阴极 边界电场
Figure imgf000009_0001
件为 计算得 到的阴极 边界电场 值减去 所要求 的电场值 的绝对值 小于预设 条件， 由于 电子为非 均匀注入 ， 因此所有阴极边 界网格 点的电场 值都需要 满足该条 件， 若不 满足， 则采用最新得到 的两个 阴极边界 电场向量 ， 执行步骤三到 步骤四 ； 若满 足， 则满足则利用 拟合函数 来计算 阳极边界 电场， 拟合函数为 ： f (y) = D + Gyl/3 + Fy2/3 式中参 数 Z)、 G和 F均为未 知的系数 ， / ⑼ =/)为阳极的边界电场值。 拟合 函数 有三个 未知参数 ， 因此需要三 个离阳 极最近 的电场值来 计算阳 极边界 电场 值， 其中电场值定义为 两个网格 节点中 间的电场值 ； 步骤 五、 给定另外一 个离子 电流密度 初始值 向量， 执行步骤 一到步 骤四， 得到 另外一个 新的阳 极边界 电场向量 ； 步骤六 、 外部迭代 若所 要求的阳 极边界条 件为 fl， 那么根据最新获得 的两个阳 极边界 电场向 量及 其对应 的离子 电流密度 值， 可以预测一个 新的离 子电流 密度向量 ， 迭代公 式为 ：
Figure imgf000009_0002
5 The relative ratio of the current density to the electron current density, using the second-order finite difference method to discretize the Poisson equation, the following iterative formula can be obtained:
Figure imgf000008_0001
In the formula, « and «+1 represent the values of the «th iteration and «+1 iteration respectively, <, / z, , , and % respectively represent the potential value at the spatial grid (/, j), grid Accuracy, the electron current density at the /th grid on the abscissa and the q value at the /th grid on the abscissa, the Poisson equation is solved by the above iterative formula, and the space potential distribution is obtained. Substitute the given initial value of electron current density and ion current density into the Poisson equation, and use the finite difference method to numerically iteratively solve the Poisson equation to obtain the space potential distribution, and then use the fitting function to calculate the cathode boundary electric field. The composite function is: f(y) = A + By l/3 + Cy 2/3
Figure imgf000008_0002
/(0) =A is the boundary electric field value of the cathode. The fitting function has three unknown parameters, so three electric field values closest to the cathode are needed to calculate the cathode boundary electric field value, where the electric field value is defined as the electric field value between two grid nodes. Step 3, intermediate iteration
Figure imgf000008_0003
and its corresponding electron current density value, a new electron current density value can be predicted, and the iterative formula is:
Figure imgf000008_0004
j n+l = 0.\f +0.9j n In the formula, « and «+1 represent the values of the nth iteration and the n+1 iteration respectively, is the cathode boundary electric field value, and is the electron current density at a certain point in space Value, / is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of / may be too large, so only take 0.1 times / when predicting the new current density. Obviously, according to the above iterative formula, a new initial value vector of electron current density can be obtained; Step 4. Based on the new electron current density vector, perform step 2 to obtain the new cathode boundary electric field
Figure imgf000009_0001
The condition is that the absolute value of the calculated cathode boundary electric field value minus the required electric field value is less than the preset condition. Since electrons are injected non-uniformly, the electric field values of all cathode boundary grid points need to meet this condition. If not , then use the newly obtained two cathode boundary electric field vectors, and perform steps 3 to 4; if satisfied, use the fitting function to calculate the anode boundary electric field, and the fitting function is: f (y) = D + Gy l /3 + Fy 2/3 In the formula, the parameters Z), G and F are all unknown coefficients, and / ⑼ =/) is the boundary electric field value of the anode. The fitting function has three unknown parameters, so three electric field values closest to the anode are needed to calculate the electric field value at the boundary of the anode, where the electric field value is defined as the electric field value between two grid nodes; step five, given another ion current density Initial value vector, perform steps 1 to 4 to obtain another new anode boundary electric field vector; step 6, external iteration If the required anode boundary condition is fl , then according to the newly obtained two anode boundary electric field vectors and their corresponding The ion current density value of , can predict a new ion current density vector, the iterative formula is:
Figure imgf000009_0002
/„+1 = 0.1/* + 0.9/„ 式中 ， 《和《+l分别表示第 n次迭代和 《+1次迭 代的值， 为阳极 边界电场 值， /为空间某 一点的离 子电流密 度值， 广是根据最新得到 的阴极边 界电场值 和 电子 电流密度 值所预 测的电子 电流密 度值， 由于广的值可能 过大， 因此在预测 新 的电流密度 时只取 0. 1倍的 /% 步骤七 、 基于预测的 离子电流 密度 向量， 重复步骤一 到步骤 四， 得到预测 的 阳极边界 电场向量 ， 判断预测的阳 极边界 电场值是 否满足所 要求的 阳极边 界 条件 ， 判断条件为计 算得到 的阳极 边界电场 值减去所 要求的 电场值 的绝对值 小 于预 设条件 ， 若不满足， 则利用两个 新的阳 极边界 值， 执行步骤六 ， 然后执行 步骤 七； 重复步骤一 到步骤 四为中间 迭代过程 ， 目的是计算 得到阳极 边界电 场 值。 步骤八 、 输出最终的电子 和离子 电流密度 。 /„ +1 = 0.1/ * + 0.9/„ In the formula, « and «+l represent the values of the nth iteration and «+1 iteration respectively, is the boundary electric field value of the anode, / is the ion current density at a certain point in space The value of G is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value. Since the value of G is too large, only take 0.1 times /% when predicting the new current density Step 7. Based on the predicted ion current density vector, repeat steps 1 to 4 to obtain the predicted anode boundary electric field vector, and judge whether the predicted anode boundary electric field value meets the required anode boundary conditions. The judgment condition is the calculated anode boundary The absolute value of the electric field value minus the required electric field value is less than the preset condition, if it is not satisfied, use two new anode boundary values, perform step 6, and then perform step 7; repeat steps 1 to 4 as an intermediate iterative process , the purpose is to calculate the boundary electric field value of the anode. Step 8. Output the final electron and ion current densities.
7 实施 例 2: 有限发射区域 下电子 电流非均匀 但是离 子电流为 均匀的情 况， 所述一 种同 步确定二 极管边 界电场与 电流密度 的方法 ， 包括以下步骤： 步骤一 、 初始参数设置 给定两 个不 同的电子 电流密度 初始值 向量以及 一个离 子电流 密度初 始值向 量， 由于离子电流 为均匀分 布， 因此在发射区 域， 离子电流密度 都等于一 个固定 的常 数； 步骤二 、 内部迭代 若给定 二极管 的阴极 电势为零， 阳极电势为 Vo， 电荷发射区域位于边界 中 心 ， 宽度为 R 阴极 x处的注入电子 电流密度 为 4(X), 阳极 x处的注入离子 电流 密度 为/i(X)， 空间 (X， )0处的电势为 0(x， )0， 则有限发射区域下的泊松方程可以 表示 为： 7 Embodiment 2: In the case where the electron current is non-uniform but the ion current is uniform under the limited emission area, the method for synchronously determining the boundary electric field and current density of the diode includes the following steps: Step 1. Initial parameter setting given two different The initial value vector of the electron current density and an initial value vector of the ion current density, because the ion current is uniformly distributed, so in the emission region, the ion current density is equal to a fixed constant; Step 2, internal iteration If the cathode potential of the diode is given is zero, the anode potential is Vo, the charge emission region is located in the center of the boundary, and the width is R. The injected electron current density at the cathode x is 4(X), the injected ion current density at the anode x is /i(X), and the space (X , the potential at )0 is 0(x, )0, then the Poisson equation under the limited emission area can be expressed as:
VV(x, _y) = 0， for |x| > W/2 for Ixl < W/2
Figure imgf000010_0001
式中 ， 、
Figure imgf000010_0002
离子 质量、 离子电 荷数 、电子质量、 x轴坐标和； y轴坐标，其中
Figure imgf000010_0003
Figure imgf000010_0004
流密 度与电 子电流密 度的相 对比值 ， 采用二阶有限 差分法将 泊松方 程离散 化， 可得 到如下迭 代公式 ：
Figure imgf000010_0005
式中 ，《和《+1分别表 不第 《次迭代和 《+1次迭代 的值， < 、 / z、 ，,和％分 别表 示空间 网格 (/， j)处的电势值、 网格精度、 横坐标第 /个网格处的电子 电流 密度 和横坐标 第 /个网格处的 q值。通过上述迭代公 式就可 以数值求解 泊松方 程。 VV(x, _y) = 0, for |x| > W/2 for Ixl < W/2
Figure imgf000010_0001
In the formula, ,
Figure imgf000010_0002
Ion mass, ion charge number, electron mass, x-axis coordinates and; y-axis coordinates, where
Figure imgf000010_0003
Figure imgf000010_0004
The relative ratio of the current density to the electron current density, using the second-order finite difference method to discretize the Poisson equation, the following iterative formula can be obtained:
Figure imgf000010_0005
In the formula, < and <+1 represent the value of the << iteration and <+1 iteration respectively, < , / z, , , and % respectively represent the potential value at the space grid (/, j), grid Accuracy, the electron current density at the /th grid on the abscissa, and the q value at the /th grid on the abscissa. The Poisson equation can be numerically solved through the above iterative formula.
8 将给 定的电子 电流密 度初始 值以及离 子电流 密度代入 泊松方 程， 采用有限 差分 法对 泊松方程 进行数 值迭代 求解之后 ， 可以得到空 间电势分 布， 然后利用拟 合 函数 来计算 阴极边界 电场， 拟合函数为 ： f(y) = A + Byy3 + Cy2/3
Figure imgf000011_0001
/(0) =A为阴极的 边界电场 值。 拟合函数 有三 个未知 参数， 因此需要三个离 阴极最近 的电场 值来计算 阴极边界 电场值 ， 其 中电场值定 义为两个 网格节点 中间的 电场值。 步骤三 、 中间迭代
Figure imgf000011_0002
及其 对应的 电子电流密 度值，可以预 测一个新 的电子 电流密度 值，迭代公式 为：
Figure imgf000011_0003
jn+l = 0.lf +0.9jn 式中 ， 《和《+1分别表示第 n次迭代 和 n+1次迭代 的值， 为阴极边界 电场 值， 为空间某一 点的电 子电流密 度值， /是 根据最新 得到的 阴极边界 电场值和 电子 电流密度 值所预 测的电子 电流密度 值， 由于 /的值可能过大 ， 因此在预测 新 的电流密度 时只取 0. 1倍的 /。 显然根据上面 的迭代 公式， 可得到一个 新的 电子 电流密度 初始值 向量； 步骤 四、 基于新的电子 电流密度 向量， 执行步骤二 ，得到新的阴极 边界电场
Figure imgf000011_0004
件为 计算得 到的阴极 边界电场 值减去 所要求 的电场值 的绝对值 小于预设 条件， 由于 电子为非 均匀注入 ， 因此所有阴极边 界网格 点的电场 值都需要 满足该条 件， 若不 满足， 则采用最新得到 的两个 阴极边界 电场向量 ， 执行步骤三到 步骤四 ； 若满 足， 则满足则利用 拟合函数 来计算 阳极边界 电场， 拟合函数为 ：
Figure imgf000011_0005
式中参 数 Z)、 G和 F均为未 知的系数 ， / ⑼ =/)为阳极的边界电场值。 拟合 函数 有三个 未知参数 ， 因此需要三 个离阳 极最近 的电场值来 计算阳 极边界 电场 值， 其中电场值定义为 两个网格 节点中 间的电场值 ； 步骤五 、 给定另外一个 离子 电流密度 初始值 向量， 执行步骤 一到步骤 四， 得到 另外一个 新的阳 极边界 电场向量； 步骤六 、 外部迭代 若所要 求的阳 极边界条 件为 fl， 那么根据最新获得 的两个阳 极边界电 场向 量及 其对应 的离子 电流密度 值， 可以预测一个 新的离 子电流 密度向量 ， 迭代公 式为 ：
Figure imgf000012_0001
8 Substitute the given initial value of electron current density and ion current density into the Poisson equation, and use the finite difference method to numerically iteratively solve the Poisson equation to obtain the space potential distribution, and then use the fitting function to calculate the cathode boundary electric field. The combined function is: f(y) = A + By y3 + Cy 2/3
Figure imgf000011_0001
/(0) =A is the boundary electric field value of the cathode. The fitting function has three unknown parameters, so three electric field values closest to the cathode are needed to calculate the cathode boundary electric field value, where the electric field value is defined as the electric field value between two grid nodes. Step 3, intermediate iteration
Figure imgf000011_0002
and its corresponding electron current density value, a new electron current density value can be predicted, and the iterative formula is:
Figure imgf000011_0003
j n+l = 0.lf +0.9j n In the formula, « and «+1 represent the value of the nth iteration and the n+1 iteration respectively, is the cathode boundary electric field value, and is the electron current density value at a certain point in space , / is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of / may be too large, so only take 0.1 times / when predicting the new current density. Obviously, according to the above iterative formula, a new initial value vector of electron current density can be obtained; Step 4, based on the new electron current density vector, perform step 2 to obtain a new cathode boundary electric field
Figure imgf000011_0004
The condition is that the absolute value of the calculated cathode boundary electric field value minus the required electric field value is less than the preset condition. Since electrons are injected non-uniformly, the electric field values of all cathode boundary grid points need to meet this condition. If not , then use the newly obtained two cathode boundary electric field vectors, and perform steps 3 to 4; if satisfied, then use the fitting function to calculate the anode boundary electric field, and the fitting function is:
Figure imgf000011_0005
In the formula, the parameters Z), G and F are all unknown coefficients, / ⑼ =/) is the boundary electric field value of the anode. The fitting function has three unknown parameters, so three electric field values closest to the anode are needed to calculate the electric field value at the boundary of the anode, where the electric field value is defined as the electric field value between two grid nodes; Step 5. Given another initial value vector of ion current density, perform steps 1 to 4 to obtain another new anode boundary electric field vector; Step 6. External iteration If the required anode boundary condition is fl , then according to the latest obtained The two anode boundary electric field vectors and their corresponding ion current density values can predict a new ion current density vector, and the iterative formula is:
Figure imgf000012_0001
In+l = 0AI* +0.9In 式中 ， 《和《+l分别表示第 n次迭代和 《+1次迭 代的值， 为阳极 边界电场 值， /为空间某 一点的离 子电流密 度值， 广是根据最新得到 的阴极边 界电场值 和 电子 电流密度 值所预 测的电子 电流密度 值， 由于广的值可能 过大， 因此在预 测 新的 电流密度 时只取 0. 1倍的 /% 需要注意的是， 由于离子电流为均 匀， 阳极 电场 总是在 中间位置 最早增加 到零 ， 因此只需要计算 阳极边 界中心 网格点 的离 子 电流密度值 ， 其余位置的离 子电流密 度值都等 于该电流 密度值 。 步骤七 、 基于预测的 离子电流 密度向 量， 重复步骤一 到步骤 四， 得到预测 的阳 极边界 电场向量 ， 判断预测的阳 极边界 中心网格 点的 电场值是否 满足所 要 求的 阳极边 界条件 ， 判断条件为计算 得到的 阳极边 界电场值 减去所要 求的 电场 值的 绝对值小 于预设条 件， 若不满足 ， 则利用两个新 的阳极边 界值， 执行步骤 六， 然后执行步骤七 ； 步骤八 、 输出最终的电子 和离子 电流密度 。 实施 例 3: 有限发射区域 下电子和 离子电流 为均匀 的情况， 所述一种同 步确定二 极管 边界电场 与电流密 度的方法 ， 包括以下步骤 ： 步骤一 、 初始参数设置 给定两 个不 同的电子 电流密度 初始值 向量以及 一个离 子电流 密度初始 值向 量， 由于电子和离 子电流为 均匀分布 ， 因此在发射区域 ， 电子和离子电流 密度都 等于 一个固定 的常数 ； 步骤二 、 内部迭代 若给定 二极管 的阴极电 势为零， 阳极电势为 Vo， 电荷发射区域位于边界 中 心， 宽度为 R 阴极 x处的注入电子 电流密度 为 4(X), 阳极 x处的注入离子 电流 I n+l =0AI * + 0.9In the n formula, 《 and 》+l respectively represent the value of the nth iteration and 《+1 iteration, is the anode boundary electric field value, / is the ion current density value of a certain point in space , G is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of G may be too large, so only take 0.1 times /% when predicting the new current density needs It should be noted that since the ion current is uniform, the anode electric field always increases to zero at the middle position first, so it is only necessary to calculate the ion current density value at the center grid point of the anode boundary, and the ion current density values at other positions are equal to the current density value. Step 7. Based on the predicted ion current density vector, repeat steps 1 to 4 to obtain the predicted anode boundary electric field vector, and judge whether the predicted electric field value at the center grid point of the anode boundary satisfies the required anode boundary condition. The judgment condition is The absolute value of the calculated anode boundary electric field value minus the required electric field value is less than the preset condition, if not satisfied, use two new anode boundary values, perform step 6, and then perform step 7; step 8, output the final electron and ion current densities. Embodiment 3: In the case where the electron and ion currents are uniform under the limited emission region, the method for synchronously determining the boundary electric field and current density of the diode includes the following steps: Step 1. Initial parameter setting Given two different electron currents Density initial value vector and an ion current density initial value vector, because the electron and ion current are uniformly distributed, so in the emission region, the electron and ion current density are equal to a fixed constant; Step 2, internal iteration If the cathode of the diode is given The potential is zero, the anode potential is Vo, the charge emission region is located in the center of the boundary, and the width is R. The injected electron current density at the cathode x is 4(X), and the injected ion current at the anode x
10 密度 为 */i(X)， 空间 (X， )0处的电势为 0(x， )0， 则有限发射区域下的泊松方程可以 表示 为： 10 The density is * /i(X), the electric potential at the space (X, )0 is 0(x, )0, then the Poisson equation under the limited emission area can be expressed as:
V2(f>^x,y^ = 0, for |x| > W/2 for Ixl < W/2
Figure imgf000013_0001
式中 ， 、
Figure imgf000013_0002
离子 质量、 离子电 荷数 、电子质量、 x轴坐标和； y轴坐标，其中
Figure imgf000013_0003
Figure imgf000013_0004
流密 度与电 子电流密 度的相 对比值 ， 采用二阶有限 差分法将 泊松方 程离散 化， 可得 到如下迭 代公式 ：
Figure imgf000013_0005
式中 ，《和《+1分别表 不第 《次迭代和 《+1次迭代 的值， < 、 / z、 ，,和％分 别表 示空间 网格 (/， j)处的电势值、 网格精度、 横坐标第 /个网格处的电子 电流 密度 和横坐标 第 /个网格处的 q值。通过上述迭代公 式就可 以数值求解 泊松方 程。 将给 定的电子 电流密 度初始 值以及离 子电流 密度代入 泊松方 程， 采用有限差 分 法对 泊松方程 进行数 值迭代 求解之后 ， 可以得到空 间电势分 布， 然后利用拟 合 函数 来计算 阴极边界 电场， 拟合函数为: f(y) = A + By1/3 + Cy2/3
Figure imgf000013_0006
/(0) =A为阴极的边 界电场值 。 拟合函数 有三 个未知 参数， 因此需要三个离 阴极最近 的电场 值来计算 阴极边界 电场值 ， 其 中电场值定 义为两个 网格节点 中间的 电场值。 步骤三 、 中间迭代 V 2 (f>^x,y^ = 0, for |x| > W/2 for Ixl < W/2
Figure imgf000013_0001
In the formula, ,
Figure imgf000013_0002
Ion mass, ion charge number, electron mass, x-axis coordinates and; y-axis coordinates, where
Figure imgf000013_0003
Figure imgf000013_0004
The relative ratio of the current density to the electron current density, using the second-order finite difference method to discretize the Poisson equation, the following iterative formula can be obtained:
Figure imgf000013_0005
In the formula, < and <+1 represent the value of the << iteration and <+1 iteration respectively, < , / z, , , and % respectively represent the potential value at the space grid (/, j), grid Accuracy, the electron current density at the /th grid on the abscissa, and the q value at the /th grid on the abscissa. The Poisson equation can be numerically solved through the above iterative formula. Substitute the given initial value of electron current density and ion current density into the Poisson equation, and use the finite difference method to numerically iteratively solve the Poisson equation to obtain the space potential distribution, and then use the fitting function to calculate the cathode boundary electric field. The combined function is: f(y) = A + By 1/3 + Cy 2/3
Figure imgf000013_0006
/(0) =A is the boundary electric field value of the cathode. The fitting function has three unknown parameters, so three electric field values closest to the cathode are needed to calculate the cathode boundary electric field value, where the electric field value is defined as the electric field value between two grid nodes. Step 3, intermediate iteration
11 若所 要求的阴 极边界条 件为 那么根 据最新 获得的两 个阴极边 界电场值 及其 对应的 电子电流密 度值，可以预 测一个新 的电子 电流密度 值，迭代公式 为：
Figure imgf000014_0001
jn+l = 0.\f +0.9jn 式中 ， 《和《+1分别表示第 n次迭代 和 n+1次迭代 的值， 为阴极边界 电场 值， 为空间某一 点的电子 电流密 度值， /是 根据最新 得到的 阴极边界 电场值和 电子 电流密度 值所预 测的电子 电流密度 值， 由于 /的值可能过大 ， 因此在预测 新 的电流密度 时只取 0. 1倍的 /。 需要注意的是 ， 由于电子电流 为均匀， 阴极 电场 总是在 中间位置 最早增加 到零 ， 因此只需要计算 阴极边 界中心 网格点 的电 子 电流密度值 ， 其余位置的 电子电流密 度值都等 于该电流 密度值 ； 步骤 四、 基于新的电子 电流密度 向量， 执行步骤二 ，得到新的阴极 边界电场
Figure imgf000014_0002
件为 计算得 到的阴极 边界电场 值减去 所要求 的电场值 的绝对值 小于预设 条件， 由于 电子为非 均匀注入 ， 因此所有阴极边 界网格 点的电场 值都需要 满足该条 件， 若不 满足， 则采用最新得到 的两个 阴极边界 电场向量 ， 执行步骤三到 步骤四 ； 若满 足， 则满足则利用 拟合函数 来计算 阳极边界 电场， 拟合函数为 ： f (y) = D + Gyl/3 + Fy2/3 式中参 数 Z)、 G和 F均为未 知的系数 ， / ⑼ =/)为阳极的边界电场值。 拟合 函数 有三个 未知参数 ， 因此需要三 个离阳 极最近 的电场值来 计算阳 极边界 电场 值， 其中电场值定义为 两个网格 节点中 间的电场值 ； 步骤 五、 给定另外一 个离子 电流密度 初始值 向量， 执行步骤 一到步骤 四， 得到 另外一个 新的阳 极边界 电场向量； 步骤六 、 外部迭代 若所 要求的阳 极边界条 件为 fl， 那么根据最新获得 的两个阳 极边界 电场向 量及 其对应 的离子 电流密度 值， 可以预测一个 新的离 子电流 密度向量 ， 迭代公 式为 ：
Figure imgf000014_0003
11 If the required cathode boundary condition is then a new electron current density value can be predicted according to the newly obtained two cathode boundary electric field values and their corresponding electron current density values, the iterative formula is:
Figure imgf000014_0001
j n+l = 0.\f +0.9j n In the formula, « and «+1 represent the values of the nth iteration and the n+1 iteration respectively, is the cathode boundary electric field value, and is the electron current density at a certain point in space Value, / is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of / may be too large, so only take 0.1 times / when predicting the new current density. It should be noted that since the electron current is uniform, the electric field of the cathode always increases to zero at the middle position first, so it is only necessary to calculate the electron current density value at the center grid point of the cathode boundary, and the electron current density values at other positions are equal to the current Density value; step 4, based on the new electron current density vector, perform step 2 to obtain the new cathode boundary electric field
Figure imgf000014_0002
The condition is that the absolute value of the calculated cathode boundary electric field value minus the required electric field value is less than the preset condition. Since electrons are injected non-uniformly, the electric field values of all cathode boundary grid points need to meet this condition. If not , then use the newly obtained two cathode boundary electric field vectors, and perform steps 3 to 4; if satisfied, use the fitting function to calculate the anode boundary electric field, and the fitting function is: f (y) = D + Gy l /3 + Fy 2/3 In the formula, the parameters Z), G and F are all unknown coefficients, and / ⑼ =/) is the boundary electric field value of the anode. The fitting function has three unknown parameters, so three electric field values closest to the anode are needed to calculate the electric field value at the boundary of the anode, where the electric field value is defined as the electric field value between two grid nodes; step five, given another ion current density Initial value vector, perform steps 1 to 4 to obtain another new anode boundary electric field vector; step 6, external iteration If the required anode boundary condition is fl , then according to the newly obtained two anode boundary electric field vectors and their corresponding The ion current density value of , can predict a new ion current density vector, the iterative formula is:
Figure imgf000014_0003
/„ =o.ir+o.9/, / „ =o.ir+o.9/,
12 式中 ， 《和《+l分别表示第 n次迭代和 《+1次迭 代的值， 为阳极 边界电场 值， /为空间某 一点的离 子电流密 度值， 是 根据最新 得到的 阴极边界 电场值和 电子 电流密度 值所预 测的电子 电流密 度值， 由于 的值可能过大 ， 因此在预测 新 的电流密度 时只取 0. 1倍的 /% 需要注意的是 ， 由于离子电流为 均匀， 阳极 电场 总是在 中间位置 最早增加 到零 ， 因此只需要计算 阳极边 界中心 网格点 的离 子 电流密度值 ， 其余位置的离 子电流密 度值都等 于该电流 密度值 ； 步骤七 、 基于预测的 离子电流 密度向 量， 重复步骤一 到步骤 四， 得到预测 的 阳极边界 电场向量 ， 判断计算得到 的阳极 边界中 心网格点 的电场 值是否满 足 所要 求的阳 极边界条 件， 判断条件 为计算得 到的阳 极边界 中心网格 点的电 场值 减去 所要求 的电场值 的绝对值 小于预 设条件 ， 若不满足， 则利用两个 新的阳 极 边界 值， 执行步骤六 ， 然后执行步骤七 ； 步骤八 、 输出最终的电子 和离子 电流密度 。 实施 例 4: 柱形结构下 的限制 电流密度 计算， 包括以下步 骤： 步骤一 、 算法初始参数 设置 给定两 个不同 的电子 电流密度初 始值以及 一个离 子电流密 度初始值 ； 步骤二 、 内部迭代 若给定 柱形二 极管的 阴极电势为 零， 阳极电势为 Vo, 半径为 的阴极 注入 电子 电流密度 为 4,半径为 的 阳极注入 离子电流 密度为 则泊松方 程可以表 示为 ：
Figure imgf000015_0001
的 电势值和离 子电流 密度与 电子电流 密度的 相对比值 ， 采用 阶有限差 分法将 泊松 方程离散 化， 可得到如下 迭代公 式：
Figure imgf000015_0002
式中 ， 《和《+I分别表示第 《次迭代 和《+1次迭 代的值 ， v^ra+1表示在《+1次 迭代 中第 /个网格处的电势 值。通过上述 迭代公 式就可 以数值求解 泊松方 程。将 12 In the formula, « and «+l represent the values of the nth iteration and «+1 iteration respectively, is the anode boundary electric field value, / is the ion current density value at a certain point in space, and is based on the latest cathode boundary electric field value and The electron current density value predicted by the electron current density value may be too large, so only take 0.1 times/% when predicting the new current density. It should be noted that since the ion current is uniform, the total electric field of the anode is the earliest increase to zero at the middle position, so it is only necessary to calculate the ion current density value of the center grid point of the anode boundary, and the ion current density values at other positions are equal to the current density value; Step 7, based on the predicted ion current density vector, Repeat steps 1 to 4 to obtain the predicted electric field vector of the anode boundary, and judge whether the calculated electric field value of the center grid point of the anode boundary meets the required anode boundary conditions. The absolute value of the electric field value minus the required electric field value is less than the preset condition, if not satisfied, use two new anode boundary values, perform step six, and then perform step seven; step eight, output the final electron and ion current density. Example 4: Calculation of the confinement current density under the columnar structure, including the following steps: Step 1, the initial parameter setting of the algorithm Given two different initial values of the electron current density and an initial value of the ion current density; Step 2, the internal iteration if Given that the cathode potential of a cylindrical diode is zero, the anode potential is Vo, the cathode injection current density of radius is 4, and the anode injection current density of radius is , then the Poisson equation can be expressed as:
Figure imgf000015_0001
Poisson’s equation is discretized by using the first-order finite difference method, and the following iterative formula can be obtained:
Figure imgf000015_0002
In the formula, « and «+I represent the values of the «th iteration and «+1 iteration, respectively, and v^ ra+1 represents the potential value at the /th grid in the «+1 iteration. The Poisson equation can be numerically solved through the above iterative formula. Will
13 给定 的电子 电流密度 初始值 以及离子 电流密 度代入 泊松方程 ， 采用有限差分 法 对泊 松方程进 行数值 迭代求 解之后 ， 可以得到空间 电势分布 ， 然后利用拟合 函 数来 计算阴极 边界电场 ， 拟合函数为： f(y) = A + Byl/3 + Cy2/3
Figure imgf000016_0001
/(0) =A为阴极的 边界电场 值。 拟合函数 有三 个未知 参数， 因此需要三个离 阴极最近 的电场 值来计算 阴极边界 电场值 ， 其 中电场值定 义为两个 网格节点 中间的 电场值。 步骤三 、 中间迭代
Figure imgf000016_0002
及其 对应的 电子电流密 度值，可以预 测一个新 的电子 电流密度 值，迭代公式 为：
Figure imgf000016_0003
jn+l = 0.lf +0.9jn 式中 ， 《和《+1分别表示第 n次迭代 和 n+1次迭代 的值， 为阴极边界 电场 值， 为电子电流 密度值 ， /是根据 最新得到 的阴极边 界电场值 和电子 电流密度 值所 预测的 电子电流密 度值， 由于 /的值可能 过大， 因此在预测新的 电流密度 时只 取 0. 1倍的 /。 步骤 四、 基于新的电子 电流密度 向量， 执行步骤二 ，得到新的阴极 边界电场
Figure imgf000016_0004
件为 计算得 到的阴极 边界电场 值减去 所要求 的电场值 的绝对值 小于预设 条件， 若不 满足， 则采用最新得到 的两个 阴极边界 电场值 ， 执行步骤三到步 骤四； 若 满足 ， 则满足则利用拟 合函数来 计算阳极 边界电场 ， 拟合函数为： f (y) = D + Gyl/3 + Fy2/3 式中参 数 Z)、 G和 F均为未 知的系数 ， / ⑼ =/)为阳极的边界电场值。 拟合 函数 有三个 未知参数 ， 因此需要三 个离阳 极最近 的电场值来 计算阳 极边界 电场 值， 其中电场值定义为 两个网格 节点中 间的电场值 ； 步骤 五、 给定另外一 个离子 电流密度 初始值 ， 执行步骤一到 步骤四 ， 得到 另外 一个新 的阳极边 界电场； 步骤六 、 外部迭代 13 The given initial value of electron current density and ion current density are substituted into the Poisson equation, and the Poisson equation is numerically iteratively solved by using the finite difference method, the space potential distribution can be obtained, and then the cathode boundary electric field is calculated by using the fitting function, fitting The function is: f(y) = A + By l/3 + Cy 2/3
Figure imgf000016_0001
/(0) =A is the boundary electric field value of the cathode. The fitting function has three unknown parameters, so three electric field values closest to the cathode are needed to calculate the cathode boundary electric field value, where the electric field value is defined as the electric field value between two grid nodes. Step 3, intermediate iteration
Figure imgf000016_0002
and its corresponding electron current density value, a new electron current density value can be predicted, and the iterative formula is:
Figure imgf000016_0003
j n+l = 0.lf +0.9j n In the formula, « and «+1 represent the values of the nth iteration and the n+1 iteration respectively, is the cathode boundary electric field value, and is the electron current density value, / is according to The value of electron current density predicted by the newly obtained cathode boundary electric field value and electron current density value may be too large, so only 0.1 times / is used when predicting the new current density. Step 4. Based on the new electron current density vector, perform step 2 to obtain the new cathode boundary electric field
Figure imgf000016_0004
The condition is that the absolute value of the calculated cathode boundary electric field value minus the required electric field value is less than the preset condition, if not satisfied, use the latest two cathode boundary electric field values, and perform steps 3 to 4; if satisfied, If it is satisfied, the fitting function is used to calculate the anode boundary electric field, and the fitting function is: f (y) = D + Gy l/3 + Fy 2/3 where parameters Z), G and F are unknown coefficients, / ⑼ =/) is the boundary electric field value of the anode. The fitting function has three unknown parameters, so three electric field values closest to the anode are needed to calculate the electric field value at the boundary of the anode, where the electric field value is defined as the electric field value between two grid nodes; step five, given another ion current density Initial value, execute steps 1 to 4 to get another new anode boundary electric field; step 6, external iteration
14 若所 要求的阳 极边界条 件为 fl， 那么根据最新获得 的两个阳 极边界 电场值 及其 对应的离 子电流密 度值， 可以预测 一个新的 离子电流 密度， 迭代公式为：
Figure imgf000017_0001
14 If the required anode boundary condition is fl , then a new ion current density can be predicted according to the newly obtained two anode boundary electric field values and their corresponding ion current density values, and the iterative formula is:
Figure imgf000017_0001
In+l = 0.\I* +0.9In 式中 ， 《和《+1分别表示第 n次迭代和 《+1次迭 代的值， 为阳极 边界电场 值， /为空间某 一点的离 子电流密 度值， 广是根据最新得到 的阴极边 界电场值 和 电子 电流密度 值所预 测的电子 电流密 度值， 由于广的值可能 过大， 因此在预测 新 的电流密度 时只取 0. 1倍的 /% 步骤七 、 基于预测的 离子电流 密度值 ， 重复步骤一到 步骤四 ， 得到预测的 阳极 边界电场 值， 判断预测的阳极 边界电场 值是否满 足所要 求的阳极 边界条件 ， 判断 条件为 计算得到 的阳极 边界电场 值减去 所要求 的电场值 的绝对值 小于预设 条件 ， 若不满足， 则利用两个新的 阳极边界 值， 执行步骤六， 然后执行步骤 七； 步骤八 、 输出最终的电子 和离子 电流密度 。 以上 所述实施 例的描 述较为具 体和详 细， 但并不能 因此而理 解为对本 发明 范 围的限制 。 应当指出的是 ， 对于本领域的 普通技 术人员来 说， 在不脱离本 发 明构 思的前提 下，还可以做 出若干 变形和改 进，这些都属 于本发 明的保护 范围。 因此 ， 本发明的保护 范围应以权 利要求 所述为准 。 I n+l = 0.\I * +0.9I n formula, " and "+1 respectively represent the value of the nth iteration and "+1 iteration, is the anode boundary electric field value, / is the ion at a certain point in space The value of current density, i is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value, because the value of u may be too large, so only take 0.1 times when predicting the new current density /% Step 7. Based on the predicted ion current density value, repeat steps 1 to 4 to obtain the predicted anode boundary electric field value, and judge whether the predicted anode boundary electric field value meets the required anode boundary conditions. The judgment condition is calculated The absolute value of the anode boundary electric field value minus the required electric field value is less than the preset condition, if not satisfied, use two new anode boundary values, perform step six, and then perform step seven; step eight, output the final electron and ionic current density. The descriptions of the above-mentioned embodiments are relatively specific and detailed, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make several modifications and improvements without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the claims.
15 15

Claims

权利要求 Rights request
1、 一种同步确 定二极管 边界电场 与电流密 度的方 法， 其特征在于 ， 包括以 下步 骤： 步骤 一、 给定两个电子 电流密度 初始值 以及一个 离子电流 密度初 始值； 步骤 二、 将电子电流 密度初 始值以及 一个离 子电流 密度初始 值代入 泊松方 程 ， 并采用有限差分 法进行数 值迭代 求解，获取对应的 阴极边界 电场值； 步骤 三、 基于所要求 的阴极 边界条件 ， 以及获得的 两个阴极 边界 电场值及 对应 的电子 电流密度值 ， 预测电子电流密 度值； 步骤 四、 基于预测的 电子电流密 度值， 执行步骤二 ，得到新的阴极 边界电场 值 ， 判断新的阴极边 界电场 值是否满 足所要 求的阴极 边界条 件， 若不满足， 则 采用 最新的 两个阴 极边界 电场值， 执行步骤 三到步骤 四； 若满足， 则计算此时 的 阳极边界 电场值； 步骤 五、 给定新的离 子电流 密度值 ， 执行步骤一到步 骤四， 得到新的阳极 边界 电场值； 步骤 六、 基于所要求 的阳极 边界条件 、 两个阳极边 界电场值 及对应 的离子 电流 密度值 ， 得到预测的离 子电流密 度值， 预测的离子电流 密度值将 使得阳 极 边界 电场满足 所给定 的要求； 步骤 七、 基于预测的 离子电流 密度值 ， 重复步骤一到 步骤四 ， 得到预测的 阳极 边界电场 值， 判断预测的阳 极边界 电场值是否 满足所要 求的阳 极边界条 件， 若不 满足， 则利用最新计算 得到的 两个阳极 边界值 ， 执行步骤六， 然后执行步 骤七 ； 步骤八 、 输出最终的 电子和离子 电流密度 。 1. A method for synchronously determining the boundary electric field and current density of a diode, characterized in that it comprises the following steps: Step 1, giving two initial values of the electron current density and an initial value of the ion current density; Step 2, setting the electron current density The initial value and an initial value of the ion current density are substituted into the Poisson equation, and the numerical iterative solution is performed using the finite difference method to obtain the corresponding cathode boundary electric field value; Step 3. Based on the required cathode boundary conditions and the obtained two cathode boundaries The electric field value and the corresponding electron current density value, predict the electron current density value; step 4, based on the predicted electron current density value, perform step 2 to obtain a new cathode boundary electric field value, and judge whether the new cathode boundary electric field value meets the requirements The cathode boundary conditions of , if not satisfied, use the latest two cathode boundary electric field values, and perform steps 3 to 4; if satisfied, calculate the anode boundary electric field value at this time; step 5, given a new ion current density value, perform steps 1 to 4 to obtain a new anode boundary electric field value; step 6, based on the required anode boundary conditions, two anode boundary electric field values and corresponding ion current density values, obtain the predicted ion current density value, The predicted ion current density value will make the anode boundary electric field meet the given requirements; step seven, based on the predicted ion current density value, repeat steps one to four to obtain the predicted anode boundary electric field value, and judge the predicted anode boundary electric field Whether the value meets the required anode boundary conditions, if not, use the newly calculated two anode boundary values, perform step 6, and then perform step 7; step 8, output the final electron and ion current densities.
2、根据权利要 求 1所述的 一种同步 确定二 极管边界 电场与电 流密度的 方法， 其特 征在于 ， 所述迭代包括 两重循 环以及 三层迭代 ， 所述三层迭代 包括 内部迭 代 、 中间迭代和外部 迭代。 2. A method for synchronously determining the boundary electric field and current density of a diode according to claim 1, characterized in that, the iterations include double loops and three-layer iterations, and the three-layer iterations include internal iterations, intermediate iterations, and external iteration.
3、根据权利要 求 2所述的 一种同步 确定二 极管边界 电场与电 流密度的 方法， 其特 征在于 ， 所述步骤二的求 解具体 包括： 若给 定二极管 的阴极 电势为零 ， 阳极电势为 Vo, 阴极注入电子电流 密度为 Ux), 阳极注入离 子电流密 度为 0, 空间任意一点的 电势为 x， y)， 则二维下 的泊 松方程 表示为： 3. A method for synchronously determining the boundary electric field and current density of a diode according to claim 2, characterized in that the solution of step 2 specifically includes: if the cathode potential of a given diode is zero, and the anode potential is Vo, The electron current density injected into the cathode is Ux), the ion current density injected into the anode is 0, and the potential at any point in space is x, y), then the Poisson equation in two dimensions is expressed as:
16
Figure imgf000019_0001
式中 ，
Figure imgf000019_0002
x、 y分别表示 真空介 电常数、 离子质量、 离子电 荷数 、电子质量、 x轴坐标和
Figure imgf000019_0003
q(x) = (x)/ Je (x)(m./Zme)1/2；
Figure imgf000019_0004
y)表示空间 (x， y)处的电势，
Figure imgf000019_0005
电流 密度与 电子电流密 度的相 对比值； 采用 二阶有限 差分法将 泊松方 程离散化 ， 得到如下迭代 公式:
Figure imgf000019_0006
式中 ，《和《+1分别表 不第 《次迭代和 《+1次迭代 的值， ^ 、 h、 ，,和％分 别表 示空间 网格 (/， y)处的电势值、 网格精度、 横坐标第 /个网格处的电子 电流 密度 和横坐标 第 /个网格处 的 a值， 通过上述迭代 公式求解 泊松方程 ， 得到空间 电势 分布。 16
Figure imgf000019_0001
In the formula,
Figure imgf000019_0002
x, y represent vacuum permittivity, ion mass, ion charge number, electron mass, x-axis coordinates and
Figure imgf000019_0003
q(x) = (x)/ J e (x)(m . /Zm e ) 1/2 ;
Figure imgf000019_0004
y) represents the electric potential at the space (x, y),
Figure imgf000019_0005
The relative ratio of the current density to the electron current density; the Poisson equation is discretized by the second-order finite difference method, and the following iterative formula is obtained:
Figure imgf000019_0006
In the formula, « and «+1 represent the values of the «th iteration and «+1 iteration respectively, ^ , h, , , and % respectively represent the potential value at the space grid (/, y), the grid precision , the electron current density at the /th grid on the abscissa and the a value at the /th grid on the abscissa, the Poisson equation is solved by the above iterative formula, and the space potential distribution is obtained.
4、根据权利要 求 3所述的 一种同步 确定二极 管边界 电场与电流 密度的 方法， 其特 征在于 ， 所述阴极边界 电场利用拟 合函数获 取： f(y) = A + Byl/3 + Cy2/3
Figure imgf000019_0007
/(0) = 为阴极的边 界电场值 ， 因此需要 三个 离阴极 最近的 电场值来 确定阴极 边界电场 值， 其中电场值 定义为两 个网格 节点 中间的 电场值。 4. A method for synchronously determining the boundary electric field and current density of a diode according to claim 3, wherein the cathode boundary electric field is obtained using a fitting function: f(y) = A + By l/3 + Cy 2/3
Figure imgf000019_0007
/(0) = is the boundary electric field value of the cathode, so three electric field values closest to the cathode are needed to determine the boundary electric field value of the cathode, where the electric field value is defined as the electric field value between two grid nodes.
5、根据权利要 求 4所述的 一种同步 确定二极 管边界 电场与电流 密度的 方法， 其特 征在于 ， 若所要
Figure imgf000019_0008
根据 获得的两 个阴极边 界电场值 及对 应的电子 电流密度 值， 预测电子电流 密度值 ， 迭代公式为： 5. A method for synchronously determining the boundary electric field and current density of a diode according to claim 4, characterized in that, if desired
Figure imgf000019_0008
According to the obtained two cathode boundary electric field values and the corresponding electron current density values, the electron current density value is predicted, and the iterative formula is:
17
Figure imgf000020_0001
jn+l = 0.lf +0.9jn 式 中， n和 n+1分别表示第 n次迭 代和 n+1次迭代的值， 为阴极 边界电场值 ， 为空 间某一 点的电子 电流密度 值， /是根据最 新得到 的阴极边 界电场值 和电子 电流 密度值所 预测的 电子电流密 度值。 17
Figure imgf000020_0001
j n+l = 0.lf +0.9j n In the formula, n and n+1 represent the value of the nth iteration and the n+1 iteration respectively, is the cathode boundary electric field value, and is the electron current density value at a certain point in space , / is the predicted electron current density value based on the latest cathode boundary electric field value and electron current density value.
6、根据权利要 求 5所述的 一种同步 确定二极 管边界 电场与电流 密度的 方法， 其特 征在于 ， 步骤四中， 若电子为非 均匀注 入， 则判断新 的阴极边 界电场值 是
Figure imgf000020_0002
阴极 边界条件£^ 条件 。 6. A method for synchronously determining the boundary electric field and current density of a diode according to claim 5, characterized in that, in step 4, if electrons are injected non-uniformly, it is judged that the new cathode boundary electric field value is
Figure imgf000020_0002
Cathode boundary condition £^ condition.
7、根据权利要 求 6所述的 一种同步 确定二极 管边界 电场与电流 密度的 方法， 其特 征在于 ，所述的步骤四 ，若阴极边界电场值 满足所要 求的阴 极边界条 件 则满 足则利用 拟合函数 来计算 阳极边界 电场， 拟合函数为 ： f (y) = D + Gyl/3 + Fy2/3 式中 参数 Z)、 G和 F均为未知的系数 ， /(0) =D为阳极的边界电场 值， 因此 需要 三个离 阳极最近 的电场 值来计算 阳极边 界电场值 ， 其中电场值定 义为两个 网格 节点中 间的电场值 。 7. A method for synchronously determining the boundary electric field and current density of a diode according to claim 6, characterized in that in step 4, if the cathode boundary electric field value satisfies the required cathode boundary condition, then use the fitting function to calculate the anode boundary electric field, the fitting function is: f (y) = D + Gy l/3 + Fy 2/3 where parameters Z), G and F are unknown coefficients, /(0)=D is The boundary electric field value of the anode, so three electric field values closest to the anode are needed to calculate the boundary electric field value of the anode, where the electric field value is defined as the electric field value between two grid nodes.
8、根据权利要 求 7所述的 一种同步 确定二极 管边界 电场与电流 密度的 方法， 其特 征在于 ， 所述阳极边界条 件为 fl， 根据获得的两个阳极 边界电场 值及对应 的离 子电流密 度值， 预测离子 电流密度 值， 迭代公式为 ：
Figure imgf000020_0003
8. A method for synchronously determining the boundary electric field and current density of a diode according to claim 7, characterized in that, the anode boundary condition is fl , and according to the obtained two anode boundary electric field values and corresponding ion current density values , to predict the ion current density value, the iterative formula is:
Figure imgf000020_0003
/„+1 = 0.ir +0.9/„ 式 中， n和 n+1分别表示第 n次迭 代和 n+1次迭代的值， 为阳极 边界电场 值， /„ 为空 间某一 点的离子 电流密 度值， 广是根据最新得 到的阴极 边界 电场值和 电子 电流 密度值所 预测的 电子电流密 度值。 /„ +1 = 0.ir +0.9/„ In the formula, n and n+1 represent the value of the nth iteration and n+1 iteration respectively, and is the boundary electric field value of the anode, /„ is the ion current at a certain point in space Density value, is the electron current density value predicted according to the latest cathode boundary electric field value and electron current density value.
9、根据权利要 求 8所述的 一种同步 确定二极 管边界 电场与电流 密度的 方法， 其特 征在于 ， 所述判断新 的阴极边 界电场值 是否满足 所要求 的阴极边 界条件 ， 判断 条件为 计算得到 的阴极 边界电场 值减去所 要求 的电场值 的绝对值 小于预设 9. A method for synchronously determining the boundary electric field and current density of a diode according to claim 8, characterized in that, for judging whether the new cathode boundary electric field value satisfies the required cathode boundary condition, the judgment condition is calculated The absolute value of the cathode boundary electric field value minus the required electric field value is less than the preset
18 条件 。 18 condition.
10、 根据权利要 求 9所述的一 种同步 确定二极 管边界 电场与电流 密度的方 法， 其特征在于， 所述判断 预测的 阳极边界 电场值是 否满足 所要求 的阳极边 界 条件 的判断 条件为计 算得到 的阳极 边界电场 值减去所 要求的 电场值 的绝对值大 小小 于预设条 件。 10. A method for synchronously determining the boundary electric field and current density of a diode according to claim 9, wherein the judgment condition for judging whether the predicted anode boundary electric field value satisfies the required anode boundary condition is calculated The absolute value of the anode boundary electric field value minus the required electric field value is smaller than the preset condition.
19 19
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