CN115510728A

CN115510728A - Plasma source simulation method and system, electronic equipment and storage medium

Info

Publication number: CN115510728A
Application number: CN202211156968.4A
Authority: CN
Inventors: 李京泽; 张钰如; 高飞; 王友年
Original assignee: Dalian University of Technology
Current assignee: Dalian University of Technology
Priority date: 2022-09-21
Filing date: 2022-09-21
Publication date: 2022-12-23

Abstract

The invention discloses a simulation method, a simulation system, electronic equipment and a storage medium of a plasma source, relating to the technical field of capacitively coupled plasma source simulation, wherein the method comprises the following steps: dividing the target area into a plurality of orthogonal grids connected with each other; determining the current-moment electron flow state parameter and the current-moment heavy particle flow state parameter of the capacitively coupled plasma source to be simulated, and determining the current-moment electron fluid equation set, the current-moment heavy particle fluid equation set and the current-moment discrete form electrostatic field equation; and when the preset termination condition is met, determining the flow state parameter, the potential and the electric field intensity in a second set time period as the simulation result of the capacitive coupling plasma source to be simulated. The invention improves the simulation efficiency of the capacitive coupling plasma source.

Description

Plasma source simulation method, system, electronic device and storage medium

Technical Field

The present invention relates to the field of capacitively coupled plasma source simulation technologies, and in particular, to a method and a system for simulating a plasma source, an electronic device, and a storage medium.

Background

The capacitively coupled plasma source is one of the most important plasma sources in industry, has the advantages of simple structure, low cost, capability of generating large-area uniform plasma and the like, is widely applied to the etching and deposition processes of materials, and plays an important role in the fields of semiconductor chip production and the like. Numerical simulation of the discharge process, which is essential for efficient optimization of the chamber structure and discharge conditions of the plasma source, is often performed using a fluid model. However, a series of difficulties exist in common simulation methods, such as a large amount of calculation for solving the poisson equation, poor stability of the electron-fluid equation, a slow convergence rate in some cases, and the like, so that the simulation of the capacitively coupled plasma is difficult to be performed efficiently.

Disclosure of Invention

The invention aims to provide a simulation method, a simulation system, electronic equipment and a storage medium of a plasma source, which improve the simulation efficiency of a capacitive coupling plasma source.

In order to achieve the purpose, the invention provides the following scheme:

a method of simulating a plasma source, the method comprising:

dividing the target area into a plurality of orthogonal grids connected with each other; the target region comprises a discharge chamber and a dielectric region of a capacitively coupled plasma source to be simulated;

determining the flow state parameters of electrons and heavy particles of the capacitively coupled plasma source to be simulated at the current moment; the flow state parameters of the electrons comprise density and temperature, and the flow state parameters of the heavy particles comprise density, flow velocity and temperature; the heavy particles include: ions and neutral particles;

determining an electronic fluid equation set at the current moment according to the flow state parameters of the electrons at the current moment; the system of electron fluid equations includes: an electron continuity equation and an electron energy equation;

determining a heavy particle fluid equation set at the current moment according to the flow state parameters of the heavy particles at the current moment; the set of heavy particle fluid equations includes: a heavy particle continuity equation, a heavy particle momentum equation and a heavy particle energy equation;

determining an electrostatic field equation at the current moment according to the flow state parameters of the electrons and the ions at the current moment;

obtaining an ultra-relaxation acceleration parameter at the current moment according to the flow state parameter of the electrons at the current moment, the flow state parameter of the electrons in the first set time period, the flow state parameter of the heavy particles at the current moment and the flow state parameter of the heavy particles in the first set time period; the first set time period is the time when the starting time is the first time, the deadline of the first set time period is the second time, the first time is the time when the step of dividing the target area into a plurality of mutually connected orthogonal grids begins, and the second time is the last time;

obtaining a source item of the electrons at the current moment, a source item of the heavy particles at the current moment, a transport coefficient of the electrons at the current moment and a transport coefficient of the heavy particles at the current moment by combining an ultra-relaxation acceleration method according to the flow state parameter of the electrons at the current moment, the flow state parameter of the heavy particles at the current moment and the ultra-relaxation acceleration parameter at the current moment;

dispersing the electrostatic field equation at the current moment by using a finite volume method and a semi-implicit format to obtain a discrete form electrostatic field equation at the current moment on the orthogonal grid; the semi-implicit format is a semi-implicit format for correcting the electron migration diffusion;

substituting the current-moment electron flow state parameter, the current-moment ion flow state parameter and the current-moment electron transport coefficient into the current-moment discrete form electrostatic field equation and solving to obtain the current-moment potential on the orthogonal grid;

calculating the electric field intensity of the current moment on the orthogonal grid according to the electric potential of the current moment on the orthogonal grid;

dispersing the electronic fluid equation set at the current moment by using a finite volume method, a windward format and a multi-step length explicit Euler method to obtain a discrete form electronic fluid equation set at the current moment on the orthogonal grid;

substituting the flow state parameter of the electrons at the current moment, the source item of the electrons at the current moment, the transport coefficient of the electrons at the current moment, the potential at the current moment and the electric field strength at the current moment into the discrete form electronic fluid equation set at the current moment to obtain the flow state parameter of the electrons at the next moment;

dispersing the heavy particle fluid equation set at the current moment by using a finite volume method and a single-step length explicit Euler method to obtain a discrete form heavy particle fluid equation set at the current moment on the orthogonal grid;

substituting the flow state parameter of the heavy particles at the current moment, the source item of the heavy particles at the current moment, the transport coefficient of the heavy particles at the current moment, the potential at the current moment and the electric field strength at the current moment into the discrete form heavy particle fluid equation set at the current moment to obtain the flow state parameter of the heavy particles at the next moment;

judging whether the current moment meets a preset termination condition;

if yes, determining the flow state parameter of the electrons, the flow state parameter of the heavy particles, the electric potential and the electric field intensity in a second set time period as a simulation result of the capacitively coupled plasma source to be simulated; the starting time of the second set time period is the first time, the ending time of the second set time period is a third time, and the third time is the time meeting the preset termination condition;

and if the current time does not meet the flow state parameter, replacing the flow state parameter of the electrons at the current time with the flow state parameter of the electrons at the next time, replacing the flow state parameter of the heavy particles at the current time with the flow state parameter of the heavy particles at the next time, and returning to the step of determining the electron fluid equation set at the current time according to the flow state parameter of the electrons at the current time.

The present invention also provides a simulation system of a plasma source, the system comprising:

the grid division module is used for dividing the target area into a plurality of orthogonal grids connected with each other; the target region comprises a discharge chamber and a dielectric region of a capacitively coupled plasma source to be simulated;

the current-time flow state parameter determining module is used for determining the flow state parameters of electrons and heavy particles of the capacitive coupling plasma source to be simulated at the current time; the flow state parameters of the electrons comprise density and temperature, and the flow state parameters of the heavy particles comprise density, flow velocity and temperature; the heavy particles include: ions and neutral particles;

the electronic fluid equation set determining module is used for determining the electronic fluid equation set at the current moment according to the flow state parameters of the electrons at the current moment; the system of electron-fluid equations includes: an electron continuity equation and an electron energy equation;

the heavy particle fluid equation set determining module is used for determining the heavy particle fluid equation set at the current moment according to the flow state parameter of the heavy particles at the current moment; the set of heavy particle fluid equations includes: a heavy particle continuity equation, a heavy particle momentum equation and a heavy particle energy equation;

the electrostatic field equation determining module is used for determining an electrostatic field equation at the current moment according to the flow state parameters of the electrons and the ions at the current moment;

the ultra-relaxation acceleration parameter determining module is used for obtaining an ultra-relaxation acceleration parameter at the current moment according to the flow state parameter of the electrons at the current moment, the flow state parameter of the electrons in the first set time period, the flow state parameter of the heavy particles at the current moment and the flow state parameter of the heavy particles in the first set time period; the first set time period is the time when the starting time is the first time, the ending time of the first set time period is the second time, the first time is the time when the step of dividing the target area into a plurality of mutually connected orthogonal grids starts, and the second time is the last time;

the source item and transport coefficient determining module is used for obtaining a source item of the electrons at the current moment, a source item of the heavy particles at the current moment, a transport coefficient of the electrons at the current moment and a transport coefficient of the heavy particles at the current moment according to the flow state parameters of the electrons at the current moment, the flow state parameters of the heavy particles at the current moment and the ultra-relaxation acceleration parameters at the current moment by combining an ultra-relaxation acceleration method;

the first discrete module is used for dispersing the electrostatic field equation at the current moment by using a finite volume method and a semi-implicit format to obtain a discrete form electrostatic field equation at the current moment on the orthogonal grid; the semi-implicit format is a semi-implicit format for correcting the electron migration diffusion;

the potential calculation module is used for substituting the current-moment electron flow state parameters, the current-moment ion flow state parameters and the current-moment electron transport coefficients into the current-moment discrete form electrostatic field equation and solving to obtain the potential of the current moment on the orthogonal grid;

the electric field intensity calculation module is used for calculating the electric field intensity of the current moment on the orthogonal grid according to the electric potential of the current moment on the orthogonal grid;

the second discrete module is used for utilizing a finite volume method, a windward format and a multi-step length explicit Euler method to disperse the electronic fluid equation set at the current moment to obtain a discrete form electronic fluid equation set at the current moment on the orthogonal grid;

the electron flow state parameter calculation module is used for substituting the flow state parameter of the electron at the current moment, the source item of the electron at the current moment, the transport coefficient of the electron at the current moment, the electric potential at the current moment and the electric field strength at the current moment into the discrete form electron fluid equation set at the current moment to obtain the flow state parameter of the electron at the next moment;

the third discrete module is used for utilizing a finite volume method and a single-step length explicit Euler method to disperse the heavy particle fluid equation set at the current moment to obtain a discrete form heavy particle fluid equation set at the current moment on the orthogonal grid;

the heavy particle flow state parameter calculation module is used for substituting the flow state parameter of the heavy particles at the current moment, the source item of the heavy particles at the current moment, the transport coefficient of the heavy particles at the current moment, the potential at the current moment and the electric field strength at the current moment into the discrete form heavy particle fluid equation set at the current moment to obtain the flow state parameter of the heavy particles at the next moment;

the judging module is used for judging whether the current moment meets a preset termination condition;

a first execution module, configured to determine, if a current time meets a preset termination condition, a flow state parameter of the electrons, a flow state parameter of the heavy particles, the electric potential, and the electric field strength in a second set time period as a simulation result of the to-be-simulated capacitively coupled plasma source; the starting time of the second set time period is the first moment, the ending time of the second set time period is a third moment, and the third moment is a moment meeting the preset termination condition;

and the second execution module is used for replacing the flow state parameter of the electrons at the current moment with the flow state parameter of the electrons at the next moment, replacing the flow state parameter of the heavy particles at the current moment with the flow state parameter of the heavy particles at the next moment and returning to the electronic fluid equation set determination module if the current moment does not meet the preset termination condition.

The present invention also provides an electronic device, comprising:

one or more processors;

a storage device having one or more programs stored thereon;

the one or more programs, when executed by the one or more processors, cause the one or more processors to implement the methods described above.

The invention also provides a storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the method described above.

According to the specific embodiment provided by the invention, the invention discloses the following technical effects:

the invention discloses a simulation method, a simulation system, electronic equipment and a storage medium of a plasma source, wherein the method comprises the following steps: dividing a target area into a plurality of orthogonal grids connected with each other; determining the current-moment electron flow state parameter and the current-moment heavy particle flow state parameter of the capacitively coupled plasma source to be simulated, and determining the current-moment electron fluid equation set, the current-moment heavy particle fluid equation set and the current-moment discrete form electrostatic field equation; and when the preset termination condition is met, determining the flow state parameter, the potential and the electric field intensity in a second set time period as the simulation result of the capacitive coupling plasma source to be simulated. The invention improves the simulation efficiency of the capacitive coupling plasma source.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.

FIG. 1 is a flow chart of a simulation method for a plasma source according to an embodiment of the present invention;

fig. 2 is a block diagram of a simulation system of a plasma source according to an embodiment of the present invention.

Detailed Description

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.

The invention aims to provide a simulation method, a simulation system, electronic equipment and a storage medium of a plasma source, aims to improve the simulation efficiency of a capacitive coupling plasma source, and can be applied to the technical field of simulation of the capacitive coupling plasma source.

In order to make the aforementioned objects, features and advantages of the present invention more comprehensible, the present invention is described in detail with reference to the accompanying drawings and the detailed description thereof.

A rapid simulation method is provided for a capacitive coupling plasma source which is common in the industry.

The simulation method is based on a fluid mechanics model of capacitively coupled plasma. The space-time distribution of the macroscopic states (density, flow velocity and temperature) of each particle is obtained by solving an electrostatic field Poisson equation and a continuity equation, a momentum conservation equation and an energy conservation equation of each particle (including electrons, various ions and neutral particles, and the ions and the neutral particles are collectively called heavy particles). The spatial and temporal distribution of these states can be used to predict the performance of a capacitively coupled plasma source during etching, deposition, etc. processes (e.g., the magnitude of the density and spatial distribution of a particular particle component can reflect the efficiency and uniformity of the process). Through multiple simulations under different chamber structures, gas components and process conditions, the influence of various factors on the performance of the capacitively coupled plasma source can be analyzed, so that the design of the capacitively coupled plasma source is optimized.

Example 1

Fig. 1 is a flowchart of a simulation method of a plasma source according to an embodiment of the present invention. As shown in fig. 1, the simulation method of the plasma source in the present embodiment includes:

step 101: dividing the target area into a plurality of orthogonal grids connected with each other; the target region includes a discharge chamber and a dielectric region of a capacitively coupled plasma source to be simulated.

Step 102: and determining the flow state parameters of the capacitively coupled plasma source to be simulated at the current moment.

Specifically, step 102 includes:

step 1021: determining the flow state parameters of electrons and heavy particles of the capacitive coupling plasma source to be simulated at the current moment; the flow state parameters of the electrons comprise density and temperature, and the flow state parameters of the heavy particles comprise density, flow velocity and temperature; the heavy particles include: ions and neutral particles.

Step 103: and determining an equation set and a discrete form electrostatic field equation at the current moment according to the flow state parameter at the current moment.

Specifically, step 103 includes:

step 1031: determining an electronic fluid equation set at the current moment according to the flow state parameters of the electrons at the current moment; the system of electron fluid equations includes: an electron continuity equation and an electron energy equation.

Step 1032: determining a heavy particle fluid equation set at the current moment according to the flow state parameters of the heavy particles at the current moment; the system of heavy particle flow equations includes: a heavy particle continuity equation, a heavy particle momentum equation, and a heavy particle energy equation.

Step 1033: and determining the electrostatic field equation of the current moment according to the flow state parameters of the electrons and the ions at the current moment.

Step 1034: obtaining an ultra-relaxation acceleration parameter at the current moment according to the flow state parameter of the electrons at the current moment, the flow state parameter of the electrons in the first set time period, the flow state parameter of the heavy particles at the current moment and the flow state parameter of the heavy particles in the first set time period; the first set time period is a time at which the step of dividing the target area into a plurality of orthogonal grids connected to each other starts, and the second set time period is a previous time.

Step 1035: and combining a super-relaxation acceleration method, and obtaining a source item of the electrons at the current moment, a source item of the heavy particles at the current moment, a transport coefficient of the electrons at the current moment and a transport coefficient of the heavy particles at the current moment according to the flow state parameters of the electrons at the current moment, the flow state parameters of the heavy particles at the current moment and the super-relaxation acceleration parameters at the current moment.

Step 1036: dispersing the electrostatic field equation at the current moment by using a finite volume method and a semi-implicit format to obtain a discrete form electrostatic field equation at the current moment on the orthogonal grid; the semi-implicit format is a semi-implicit format in which electromigration diffusion is corrected.

Step 104: the potential and electric field strength on the orthogonal grid at the present moment are determined.

Specifically, step 104 includes:

step 1041: and substituting the current-time electron flow state parameter, the current-time ion flow state parameter and the current-time electron transport coefficient into the current-time discrete form electrostatic field equation and solving to obtain the potential of the current time on the orthogonal grid.

Step 1042: and calculating the electric field intensity on the orthogonal grid at the current moment according to the electric potential on the orthogonal grid at the current moment.

Step 105: and determining the flow state parameter at the next moment according to the equation set at the current moment.

Specifically, step 105 includes:

step 1051: and (3) discretizing the electronic fluid equation set at the current moment by using a finite volume method, a windward format and a multi-step length explicit Euler method to obtain the discrete form electronic fluid equation set at the current moment on the orthogonal grid.

Step 1052: and substituting the flow state parameter of the electron at the current moment, the source item of the electron at the current moment, the transport coefficient of the electron at the current moment, the potential at the current moment and the electric field strength at the current moment into the discrete form electron fluid equation set at the current moment to obtain the flow state parameter of the electron at the next moment.

Step 1053: and (3) dispersing the heavy particle fluid equation set at the current moment by using a finite volume method and a single-step length explicit Euler method to obtain the discrete form heavy particle fluid equation set at the current moment on the orthogonal grid.

Step 1054: and substituting the flow state parameter of the heavy particles at the current moment, the source item of the heavy particles at the current moment, the transport coefficient of the heavy particles at the current moment, the potential at the current moment and the electric field intensity at the current moment into the discrete form heavy particle fluid equation set at the current moment to obtain the flow state parameter of the heavy particles at the next moment.

Step 106: and judging whether the current moment meets a preset termination condition.

If yes, go to step 107: the flow state parameters, the electric potential and the electric field strength of the second set period of time were determined as simulation results.

Specifically, step 107 specifically includes: determining the flow state parameters of electrons, the flow state parameters of heavy particles, the electric potential and the electric field intensity in a second set time period as the simulation result of the capacitively coupled plasma source to be simulated; the starting time of the second set time period is the first time, the ending time of the second set time period is the third time, and the third time is the time meeting the preset termination condition.

If not, go to step 108: the flow state parameter at the current time is replaced with the flow state parameter at the next time, and the process returns to step 103.

Specifically, step 108 includes:

step 1081: and replacing the flow state parameter of the electron at the current moment with the flow state parameter of the electron at the next moment.

Step 1082: and replacing the flow state parameter of the heavy particle at the current moment with the flow state parameter of the heavy particle at the next moment.

As an optional implementation manner, step 1031 specifically includes:

and determining an electronic continuity equation of the current moment according to the density of the electrons of the current moment.

And determining an electron energy equation at the current moment according to the temperature of the electrons at the current moment.

And determining the electron continuity equation at the current moment and the electron energy equation at the current moment as the electron fluid equation set at the current moment.

As an optional implementation manner, step 1032 specifically includes:

and determining a heavy particle continuity equation of the current moment according to the density of the heavy particles of the current moment.

And determining a momentum equation of the heavy particles at the current moment according to the flow velocity of the heavy particles at the current moment.

And determining the energy equation of the heavy particles at the current moment according to the temperature of the heavy particles at the current moment.

And determining the heavy particle continuity equation at the current moment, the heavy particle momentum equation at the current moment and the heavy particle energy equation at the current moment as the heavy particle fluid equation set at the current moment.

Specifically, the simulation method is based on a finite volume method solved by partial differential equations, and the specific steps of carrying out one-time simulation are as follows:

step 1: and (5) grid division.

(1) Dividing a capacitively coupled plasma source to be simulated, which comprises a discharge chamber region (in which plasma in the capacitively coupled plasma source will be generated and evolve), and an adjacent dielectric region, into orthogonal grids connected with each other, thereby obtaining grid topology information and geometric information required to be used in subsequent calculation.

Mesh topology information typically includes: the number of grid cells, cell surfaces, and vertices; the surface and the vertex of each unit are formed; the vertices that make up each surface; cells on both sides of each surface, etc. The mesh geometry information typically includes: the location of each cell, cell surface and vertex; the volume of each cell; the area of each surface; the distance of each surface to the center of gravity of the two side units, etc. The mesh topology information and geometry information are used in step 4, step 5 and step 6 to construct a spatially discrete format of differential equations under the finite volume method.

The partition method can adopt a conventional partition method of orthogonal grids under a finite volume method.

Compared with a non-orthogonal grid, the orthogonal grid can greatly simplify the numerical format and improve the efficiency and the precision of simulation.

Step 2: and (5) initializing.

(1) And determining the type of the physical quantity of which the initial value needs to be set according to the discharge gas composition of the capacitive coupling plasma source to be simulated and the simulation requirement.

The physical quantities for which the initial values are to be set usually need to include at least: density of electrons n _e Electron temperature T _e And the density n of the heavy particles. May also optionally include: the flow velocity u of the heavy particles (if one wishes to take into account the momentum equation for the corresponding component in the simulation), the temperature T of the heavy particles (if one wishes to take into account the energy equation for the corresponding component in the simulation), etc.

Simulation requirements, which typically include simulation efficiency, accuracy, correctness, and concerns about a physical process, determine whether it is necessary to solve momentum equations or energy equations for a heavy particle component.

(2) According to the grid division result of the step 1, for each physical quantity, the discrete value of each physical quantity on each grid is used for describing the spatial distribution of the physical quantity in the simulation area. On the basis, the physical quantity (discrete value on the grid) which needs to be set with the initial value is assigned with the initial value.

Each physical quantity has a spatial distribution only over its domain of definition. The domain of the physical quantities n, u, T of the flowing state of each component is the inside of the discharge chamber.

And step 3: and calculating collision reaction source items and transport coefficients.

(1) And calculating a source item and a transport coefficient in a fluid equation set of each particle according to discharge gas components of the capacitive coupling plasma source to be simulated, simulation requirements and the particle macro flow state value (n, u, T) at the current moment.

Wherein, the source items needing to be calculated at least need to include: particle source term for electrons

Energy source term

Particle number source term for each heavy particle

May also optionally include: each kind of heavy grainMomentum source term of son

(if it is desired to consider the momentum equations for the corresponding components in the simulation) and the energy source term(s) (ii)

Or

) (if it is desired to consider the energy equations for the corresponding components in the simulation). Wherein t is time, K _e Is the average total kinetic energy of electrons, m is the mass of the heavy particles, K is the average total kinetic energy of the heavy particles, C _v Is the constant volumetric heat capacity of heavy particles.

The transport coefficients that need to be calculated optionally include the mobility μ of each particle (if a migration diffusion approximation is used in the simulation instead of the momentum equation for the corresponding component), the diffusion coefficient D (if a (migration) diffusion approximation is used in the simulation instead of the momentum equation for the corresponding component), the viscosity coefficient η (if a complete momentum equation containing a viscosity term is desired in the simulation), the thermal conductivity κ (if an energy equation for the corresponding component is desired to be considered in the simulation), etc.

The components of the discharge gas determine which collision reactions occur in the system, thereby determining a calculation formula for calculating a source term and a transport coefficient to a certain extent.

Simulation requirements typically include simulation efficiency, accuracy, correctness, and concerns about a physical process. These requirements determine whether it is necessary to solve viscous terms, momentum equations or energy equations for a certain heavy particle component, and thus which source terms and transport coefficients need to be calculated. These requirements also determine whether it is necessary to consider the influence of a certain collision reaction occurring in the system on the number of particles, momentum, and energy, and thus to some extent, the calculation formula for calculating the source term and the transport coefficient.

The types of heavy particle source items and the types of heavy particle transport coefficients to be calculated depend on the specific adopted heavy particle fluid equation model. The specific calculation formula of the source term and the transport coefficient depends on the user.

Common methods for calculating transport coefficients include: mobility ratio

Coefficient of diffusion

Coefficient of shear viscosity

Thermal conductivity

Where e is the unit charge and T' is the temperature of the particle (including the electron temperature T) _e And temperature T) of heavy particles, n' being the density of the particles (including the density of electrons n) _e And a heavy particle density n), m being the mass of the particles (including the mass of electrons and the mass of heavy particles), m _r V is the elastic collision frequency of the particles with the background gas particles (n ', T' are known quantities, m) _r And v is an input parameter).

If the convergence simulation result is expected to be obtained more quickly in the periodic steady-state problem, the reaction coefficient or source term of the important collision reaction can be properly scaled by adopting the idea of super-relaxation acceleration on the basis of the conventional method. For example, if an ionization reaction plays an important role in the convergence of the simulation, the relative change in plasma density over a period may be recorded first

(n _p1 ，n _p2 Representing the plasma density before and after one cycle, respectively), and then selecting an appropriate function f (e.g., f (x) = min (a, max (b, cx + 1)), where abc is an empirically determined constant), the contribution of the collision reaction to the population source term

Is modified into

The measurement mode of the relative change of the plasma density, the correction function form, the source item type needing to be corrected and the like are determined by experience.

And 4, step 4: and solving the electrostatic field.

(1) And performing semi-implicit correction on a charge density term in the electrostatic field Poisson equation.

Poisson's equation is a differential equation describing the electrostatic field, which in the region of the discharge chamber takes the form

In the form of a dielectric region

Where D represents the electric displacement vector, E represents the electric field strength,

represents an electric potential, Z _i Represents the charge amount of the particle component i,. Epsilon ₀ And epsilon represents the dielectric constant in vacuum and in the medium respectively,

is a differential operator symbol; e is unit charge, n _i Is the density of the ith particle.

The semi-implicit correction is to correct the electron density at the current time in the charge density term to the electron density at the future time, and the specific operation method is as follows: 1) First, the Poisson's equation is applied to the vacuum dielectric constant ε in the fluid region ₀ Corrected to predict the vacuum dielectric constant for electromigration,

wherein the content of the first and second substances,

to predict the vacuum dielectric constant of electron transport,. DELTA.t is the calculated interval of the Poisson equation,. Mu. _e Is the mobility of the electrons. 2) The electron density n is then measured in the charge density term of the Poisson equation _e Correction to the electron density that predicted electron diffusion:

wherein the content of the first and second substances,

to predict the electron density of electron diffusion, D _e Is the diffusion coefficient of electrons, T _e Is the temperature of the electrons. 3) Finally, the modified poisson equation in the fluid region is obtained as:

the semi-implicit correction of the poisson equation has the effect of eliminating strong coupling of an electric field and electrons, so that the poisson equation can adopt a longer calculation interval than an electron-fluid equation set, the solving times of the poisson equation are greatly reduced, and the calculation efficiency of simulation is remarkably improved.

(2) Discretizing the modified Poisson equation into electrical potentials using a finite volume method

And the discrete values satisfy a linear equation set.

The discrete method is the same as the conventional finite volume method. The specific discrete method comprises the following steps: the electrostatic field poisson equation is integrated on each unit (the grid information obtained in step 1 needs to be used), and the integration result on each unit forms one equation in the linear equation system.

Note 2: the input of this step includes the input obtained in step 4 (1)

And

density n of each ionic component _i And a series of input parameters (Z) _i ε, etc.). The output of this step is a system of linear equations (the solution to the system of linear equations is the potential)

Discrete values of (a).

(3) Solving the linear equation system obtained by dispersion to obtain the potential

Discrete spatial distribution of (a).

The linear equation system can adopt a conventional solving method such as a Krylov subspace series method and the like.

(4) Using numerical differentiation under finite volume method to calculate and calculate the potential

The corresponding electric field strength E distribution is distributed. Is calculated by the formula

The numerical differential calculation method adopts a conventional method for calculating a spatial derivative under a finite volume method.

And 5: and solving the electron fluid equation set.

(1) Electric field E, electron temperature T in reference simulation region _e And (3) analyzing the stability of the electron fluid equation set at different spatial positions according to the spatial distribution of the physical quantity, and setting different (electron fluid equation set) time step lengths delta t for different grids.

It is necessary to set a longer time step for a grid with high stability and a shorter time step for a grid with low stability. The different time steps should be integer multiples of each other and the time step on the surface should be the maximum of the unit time steps on both sides.

The specific method of distinguishing the grid into different time steps is empirical. Experience shows that a specific judgment method with a good effect is as follows: if the electric field E on a grid is greater than a threshold, or electricitySub-temperature T _e If the grid is larger than a threshold (optionally belonging to the input information, which may be spatially varying), or if the grid is statically judged to be of low stability (belonging to the input information), the grid is set to a short time step, otherwise the grid is set to a long time step.

(2) Under the finite volume method, the electron fluid equation set is spatially discretized using an upwind format.

Wherein the electron fluid equation set is in the form of

Wherein J _e Is flux of electron particles, Q _e Is flux of energy, p _e ＝n _e T _e Representing the electron pressure.

The discrete method comprises the following steps: 1) Firstly, integrating the equation system in each unit according to a conventional finite volume method to obtain:

where V and a denote the volume of the grid cell and the area of the cell surface, respectively, and the subscripts c and s denote the surface s of the grid cell c and grid cell c, respectively. 2) The migration term in the flux of the unit surface is then processed using the windward format, namely:

wherein the subscript c ₁ 、c ₂ Respectively two cells on either side of the surface s.

Compared with other space discrete formats, the windward format has the advantages of simple realization, small calculation amount and high stability.

(3) Under the space discrete form of the electron fluid equation obtained in the step 5 (1) and obtained in the step 5 (2), a multi-step explicit Euler method is used for the electron density n _e And electron temperature T _e Performing time advance to obtain the electricity at the next momentThe electron density and the electron temperature.

The formula for the explicit euler method to advance the physical quantity g from time t to t + Δ t is:

in this step g is the electron density n _e And electron energy density 1.5n _e T _e 。

In the multi-step explicit eulerian method, for every 1 advance of the long-step unit, the short-step unit needs to perform several advances (the number of advances is the ratio of long to short-time steps). The specific implementation mode is as follows: 1) First, the flux over the entire surface is calculated, thereby advancing the entire unit one (long or short) time step; 2) Then, refreshing the flux on the surface of the short step, thereby advancing the short step unit by a short time step; 3) Repeating the previous step until the sum of the cumulated advanced short steps equals the long step.

This step often requires a continuous (cyclic) push for a number of long time steps due to the low stability of the e-fluid equation set. The sum of the time steps that need to be pushed in total depends on the particular problem, and typically takes a value of one two hundredths of the power cycle.

Compared with the implicit method, the explicit Euler method can effectively reduce the single step calculation amount under the condition of keeping higher stability, thereby effectively improving the calculation efficiency of the simulation method. Compared with a single-step rectangular method, the multi-step rectangular method can effectively overcome the influence caused by poor stability of the electron fluid equation set in the areas near the sheath layer and the electrode tip, and the like, and can also effectively improve the calculation efficiency of the simulation method.

Step 6: solving a system of heavy particle flow equations. (step 6 has no dependency on step 5, and may be exchanged).

(1) And establishing a proper heavy particle fluid equation set according to the discharge gas composition and the simulation requirement.

Specifically, 1) first, the kind of heavy particles present during discharge is determined from the discharge gas composition; 2) Then, based on the balance between the simulation efficiency and the correctness of the simulation result, simplification is carried out on the basis of a complete fluid mechanics equation set of all heavy particles, so that a part of unimportant particles and equations which do not need to be solved are removed.

A complete fluid equation set form contains the continuity equation, momentum equation and energy equation:

wherein p is,

q represents the pressure, viscosity tensor and heat conduction flux of the heavy particles, m, Z, C _v Respectively represent the mass, the charge amount (neutral particles are 0) and the equivalent capacitance heat capacity of the heavy particles, eta', kappa respectively represent the shear viscosity coefficient, the volume viscosity coefficient and the thermal conductivity in the transport coefficient of the heavy particles,

representing the unit tensor.

There are various ways to simplify the complete fluid equation set, for example, 1) ignoring the viscosity term:

simplified to

) Ignoring the energy equation:

simplified to T = T ₀ (ii) a Wherein, T in the above formula ₀ Is the temperature of the background gas. 3) The momentum equation is replaced by a migration diffusion approximation:

simplified to

Besides, it can alsoCombining the simplified methods, or using a modified migration diffusion approximation, etc. Note 4: this step only establishes equations and does not involve actual calculations.

(2) The system of fluid mechanics equations for the heavy particles is spatially discretized using a finite volume method.

The spatially discrete approach to the finite volume method is to integrate the equation over each cell (and process the flux across the cell surface in a suitable format).

(3) In a discrete form of the system of heavy particle fluid mechanics equations, a single step explicit euler method is used to time advance the density of the heavy particles, n, the flow velocity of the heavy particles, u, and the temperature of the heavy particles, T.

Only a conventional single step explicit euler method (here, single step means that the step size does not vary spatially) needs to be used. The formula for the explicit euler method to advance the physical quantity g from time t to t + Δ t is:

in this step g is the heavy particle density n, the momentum density mnu and the thermal energy density nC _v T。

Because the stability of the heavy particle fluid equation set is high, the step can be selected according to the situation whether the time step which needs to be pushed in total is split into a plurality of small time steps for cycle pushing (the fewer the splits are, the better, but the maximum length of each small time step is limited by the stability). The total time step to be advanced is determined according to the specific problem. The number of time step splits for different particles need not be the same.

If it is desired to obtain a more rapid convergence simulation in the periodic steady-state problem, the time step advance for some neutral particle components that are not responsive to the periodic electric field can be suitably amplified, for example, tens to hundreds of Δ t for neutral particle advance when charged particles advance Δ t, in which case the total time step advance for different particles need not be the same.

And 7: cyclically repeat the above calculations

(1) And (6) repeating the calculation work of the steps 3 to 6 until a judgment condition of completing the simulation is reached.

The commonly used judgment conditions are: the calculation reaches a specified time (i.e., the cumulative time to advance is greater than a threshold) or a specified convergence accuracy (the relative change of some physical quantity over a certain time to advance is less than a threshold).

The specific cycle is similar to: calculating source terms and transport coefficients → solving electrostatic field → advancing electron flow state (solving electron fluid equation set) → advancing flow state of all heavy particles (solving heavy particle fluid equation set) → judging whether simulation is completed or not → (if simulation is not completed) → calculating source terms and transport coefficients → 8230. Or: step 3 → step 4 → step 5 → step 6 → judge whether simulation is completed → (if simulation is not completed) step 3 → \8230;.

(2) And outputting the calculation results of the physical quantities.

Each output physical quantity may be its discrete spatiotemporal distribution. The discrete-time distribution refers to a calculation result that a physical quantity can output over a plurality of calculation cycles. Discrete spatial distribution refers to a calculation result in which a physical quantity can be output on a plurality of meshes.

Example 2

Fig. 2 is a block diagram of a simulation system of a plasma source according to an embodiment of the present invention. As shown in fig. 2, the simulation system of the plasma source in the present embodiment includes:

a mesh dividing module 201, configured to divide the target area into a plurality of orthogonal meshes connected to each other; the target region includes a discharge chamber and a dielectric region of a capacitively coupled plasma source to be simulated.

And a current-time flow state parameter determining module 202, configured to determine a flow state parameter of the capacitively coupled plasma source to be simulated at the current time.

Specifically, the module 202 for determining the current flow state parameter specifically includes:

the current-time flow state parameter determining submodule 2021 is configured to determine a flow state parameter of electrons and a flow state parameter of heavy particles of the capacitively coupled plasma source to be simulated at the current time; the flow state parameters of the electrons comprise density and temperature, and the flow state parameters of the heavy particles comprise density, flow velocity and temperature; the heavy particles include: ions and neutral particles.

And the system of equations and discrete form electrostatic field equation determining module 203 is used for determining the system of equations and discrete form electrostatic field equations at the current moment according to the flow state parameters at the current moment.

Specifically, the equation set and discrete form electrostatic field equation determining module 203 specifically includes:

the electronic fluid equation set determining submodule 2031, configured to determine the electronic fluid equation set at the current time according to the flow state parameter of the electrons at the current time; the system of electron fluid equations includes: an electron continuity equation and an electron energy equation.

The heavy particle fluid equation set determining sub-module 2032 is configured to determine the heavy particle fluid equation set at the current time according to the flow state parameter of the heavy particles at the current time; the set of heavy particle fluid equations includes: a heavy particle continuity equation, a heavy particle momentum equation, and a heavy particle energy equation.

The electrostatic field equation determining module 2033 is configured to determine the electrostatic field equation at the current time according to the flow state parameter of the electrons at the current time and the flow state parameter of the ions at the current time.

An ultra-relaxation acceleration parameter determining module 2034, configured to obtain an ultra-relaxation acceleration parameter at the current time according to the flow state parameter of the electrons at the current time, the flow state parameter of the electrons at the first set time period, the flow state parameter of the heavy particles at the current time, and the flow state parameter of the heavy particles at the first set time period; the first set time period is a time when the starting time is a first time, the ending time of the first set time period is a second time, the first time is a time when the step of dividing the target area into a plurality of mutually connected orthogonal grids begins, and the second time is a previous time.

The source item and transport coefficient determining module 2035 is configured to, in combination with the ultra-relaxation acceleration method, obtain the source item of the electron at the current time, the source item of the heavy particle at the current time, the transport coefficient of the electron at the current time, and the transport coefficient of the heavy particle at the current time according to the flow state parameter of the electron at the current time, the flow state parameter of the heavy particle at the current time, and the ultra-relaxation acceleration parameter at the current time.

A first discretization module 2036, configured to discretize the electrostatic field equation at the current time by using a finite volume method and a semi-implicit format, to obtain a discrete-form electrostatic field equation at the current time on the orthogonal grid; the semi-implicit format is a semi-implicit format in which electromigration diffusion is corrected.

And an electric potential and electric field strength determining module 204 for determining the electric potential and the electric field strength on the orthogonal grid at the current moment.

Specifically, the electric potential and electric field strength determining module 204 specifically includes:

and the potential calculating module 2041 is configured to substitute the current-time flow state parameter of electrons, the current-time flow state parameter of ions, and the current-time transport coefficient of electrons into the current-time discrete electrostatic field equation and solve the result to obtain the potential of the current time on the orthogonal grid.

The electric field strength calculating submodule 2042 is configured to calculate the electric field strength on the orthogonal grid at the current time according to the electric potential on the orthogonal grid at the current time.

The next-time flowing state parameter determining module 205 is configured to determine a flowing state parameter at a next time according to the equation set at the current time.

Specifically, the flow state parameter determining module 205 at the next time specifically includes:

the second discretization module 2051 is configured to discretize the system of electronic fluid equations at the current time by using a finite volume method, a windward format, and a multi-step explicit euler method to obtain a discrete-form system of electronic fluid equations at the current time on the orthogonal grid.

And the electron flow state parameter calculation module 2052 is configured to substitute the flow state parameter of the electrons at the current time, the source item of the electrons at the current time, the transport coefficient of the electrons at the current time, the potential at the current time, and the electric field strength at the current time into the discrete-form electron fluid equation set at the current time to obtain the flow state parameter of the electrons at the next time.

A third discretization module 2053 is configured to discretize the set of heavy particle fluid equations at the current time by using a finite volume method and a single-step explicit euler method to obtain a discrete-form set of heavy particle fluid equations at the current time on the orthogonal grid.

And the heavy particle flow state parameter calculation module 2054 is configured to substitute the flow state parameter of the heavy particles at the current time, the source item of the heavy particles at the current time, the transport coefficient of the heavy particles at the current time, the electric potential at the current time, and the electric field strength at the current time into the discrete form heavy particle fluid equation set at the current time to obtain the flow state parameter of the heavy particles at the next time.

The determining module 206 is configured to determine whether the current time meets a preset termination condition.

If the current time meets the preset termination condition, a first execution module 207 is executed, and the first execution module 207 is used for determining the flow state parameters, the electric potential and the electric field intensity of the second set time period as the simulation result.

Specifically, the first executing module 207 is specifically configured to: determining the flow state parameters of electrons, the flow state parameters of heavy particles, the electric potential and the electric field intensity in a second set time period as the simulation result of the capacitively coupled plasma source to be simulated; the starting time of the second set time period is the first time, the ending time of the second set time period is the third time, and the third time is the time meeting the preset termination condition.

If the current time does not satisfy the preset termination condition, the second execution module 208 is executed, and the system of equations and the discrete-form electrostatic field equation determination module (the electronic fluid equation system determination module 2031) are returned. The second execution module 208 is configured to replace the flow state parameter at the current time with the flow state parameter at the next time.

Specifically, the second executing module 208 specifically includes:

the first execution sub-module 2081 is configured to replace the flow state parameter of the electrons at the current time with the flow state parameter of the electrons at the next time.

The second execution sub-module 2082 is configured to replace the flow state parameter of the heavy particle at the current time with the flow state parameter of the heavy particle at the next time.

As an alternative embodiment, the electronic fluid equation set determining module 2031 specifically includes:

and the electronic continuity equation determining unit is used for determining the electronic continuity equation at the current moment according to the density of the electrons at the current moment.

And the electron energy equation determining unit is used for determining the electron energy equation at the current moment according to the temperature of the electrons at the current moment.

And the electronic fluid equation set determining unit is used for determining the electronic continuity equation at the current moment and the electronic energy equation at the current moment as the electronic fluid equation set at the current moment.

As an alternative embodiment, the module 2032 for determining the heavy particle fluid equation set specifically includes:

and the heavy particle continuity equation determining unit is used for determining the heavy particle continuity equation of the current moment according to the density of the heavy particles of the current moment.

And the heavy particle momentum equation determining unit is used for determining the heavy particle momentum equation at the current moment according to the flow velocity of the heavy particles at the current moment.

And the heavy particle energy equation determining unit is used for determining the heavy particle energy equation at the current moment according to the temperature of the heavy particles at the current moment.

And the heavy particle fluid equation set determining unit is used for determining the heavy particle continuity equation at the current moment, the heavy particle momentum equation at the current moment and the heavy particle energy equation at the current moment as the heavy particle fluid equation set at the current moment.

As an optional implementation manner, the discrete algebraic equation determining module 2033 specifically includes:

and the electrostatic field equation determining unit is used for determining the electrostatic field equation of the orthogonal grid at the current moment.

And the correcting unit is used for correcting the charge density item in the electrostatic field equation to obtain the corrected electrostatic field equation of the orthogonal grid at the current moment.

And the discrete algebraic equation determining unit is used for converting the corrected electrostatic field equation at the current moment into a discrete algebraic equation on the orthogonal grid by using a finite volume method.

Example 3

The present invention also provides an electronic device, comprising:

one or more processors;

a storage device having one or more programs stored thereon;

the one or more programs, when executed by the one or more processors, cause the one or more processors to implement the method of embodiment 1.

Example 4

The present invention also provides a storage medium having a computer program stored thereon, wherein the computer program implements the method in embodiment 1 when executed by a processor.

The method has the advantages that:

(1) The simulation speed of the method is extremely fast, and can even reach more than 100 times of that of some common simulation methods, because the method pertinently solves a plurality of common performance limiting factors in the capacitive coupling plasma simulation: the problem of overlarge calculation amount of the Poisson equation caused by coupling of an electric field and electrons is solved through semi-implicit correction of the Poisson equation; the problems of poor stability of the electronic fluid equation set and large single-step calculation amount of a hidden format are solved through the explicit windward format of the electronic fluid equation set; the problem that the total calculated amount of the electronic fluid equation set is overlarge due to the fact that the stability of a few grids is too poor when high voltage or an electrode tip exists is solved through a multi-time step method of the electronic fluid equation set; by scaling the time step of the neutral particle flow equation and the coefficients or source terms of some important collision reactions, the problem of excessive total calculation due to slow convergence in some cases is solved. The combined use of the multiple targeted solutions enables the whole set of simulation method to have extremely high simulation efficiency.

(2) The high calculation efficiency of the simulation method is completely derived from the design of a logarithmic value method, and the simulation method does not depend on a specific calculation case or a software and hardware platform, or a large-scale parallel algorithm or a high-performance linear (nonlinear) solver. This makes the high performance of the simulation method simple to implement, easy to reproduce, and stable and controllable.

The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other. For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.

The principles and embodiments of the present invention have been described herein using specific examples, which are set forth only to help understand the apparatus and its core concepts of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, this summary should not be construed as limiting the invention.

Claims

1. A method of simulating a plasma source, the method comprising:

dividing a target area into a plurality of orthogonal grids connected with each other; the target region comprises a discharge chamber and a dielectric region of a capacitively coupled plasma source to be simulated;

determining an electronic fluid equation set at the current moment according to the flow state parameters of the electrons at the current moment; the system of electron-fluid equations includes: an electron continuity equation and an electron energy equation;

combining a super-relaxation acceleration method, and obtaining a source item of electrons at the current moment, a source item of heavy particles at the current moment, a transport coefficient of electrons at the current moment and a transport coefficient of heavy particles at the current moment according to the flow state parameters of electrons at the current moment, the flow state parameters of heavy particles at the current moment and the super-relaxation acceleration parameters at the current moment;

judging whether the current moment meets a preset termination condition;

if so, determining the flow state parameter of the electrons, the flow state parameter of the heavy particles, the electric potential and the electric field intensity in a second set time period as a simulation result of the capacitively coupled plasma source to be simulated; the starting time of the second set time period is the first moment, the ending time of the second set time period is a third moment, and the third moment is a moment meeting the preset termination condition;

2. A simulation system for a plasma source, the system comprising:

the current-time flow state parameter determining module is used for determining the flow state parameters of electrons and the flow state parameters of heavy particles of the to-be-simulated capacitively coupled plasma source at the current time; the flow state parameters of the electrons comprise density and temperature, and the flow state parameters of the heavy particles comprise density, flow velocity and temperature; the heavy particles include: ions and neutral particles;

the electrostatic field equation determining module is used for determining the electrostatic field equation at the current moment according to the flow state parameters of the electrons and the ions at the current moment;

the ultra-relaxation acceleration parameter determination module is used for obtaining the ultra-relaxation acceleration parameter at the current moment according to the flow state parameter of the electrons at the current moment, the flow state parameter of the electrons in the first set time period, the flow state parameter of the heavy particles at the current moment and the flow state parameter of the heavy particles in the first set time period; the first set time period is the time when the starting time is the first time, the deadline of the first set time period is the second time, the first time is the time when the step of dividing the target area into a plurality of mutually connected orthogonal grids begins, and the second time is the last time;

a third discrete module, configured to utilize a finite volume method and a single-step explicit euler method to discretize the heavy particle fluid equation set at the current time to obtain a discrete form heavy particle fluid equation set at the current time on the orthogonal grid;

the first execution module is used for determining the flow state parameter of the electrons, the flow state parameter of the heavy particles, the electric potential and the electric field strength in a second set time period as a simulation result of the to-be-simulated capacitively coupled plasma source if the current moment meets a preset termination condition; the starting time of the second set time period is the first moment, the ending time of the second set time period is a third moment, and the third moment is a moment meeting the preset termination condition;

3. An electronic device, comprising:

one or more processors;

a storage device having one or more programs stored thereon;

the one or more programs, when executed by the one or more processors, cause the one or more processors to implement the method of claim 1.

4. A storage medium having stored thereon a computer program, wherein the computer program, when executed by a processor, implements the method of claim 1.