CN111800932B - Plasma discharge process simulation method and system - Google Patents

Plasma discharge process simulation method and system Download PDF

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CN111800932B
CN111800932B CN202010787014.8A CN202010787014A CN111800932B CN 111800932 B CN111800932 B CN 111800932B CN 202010787014 A CN202010787014 A CN 202010787014A CN 111800932 B CN111800932 B CN 111800932B
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CN111800932A (en
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张钰如
高飞
王友年
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Dalian University of Technology
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    • H—ELECTRICITY
    • H05—ELECTRIC TECHNIQUES NOT OTHERWISE PROVIDED FOR
    • H05H—PLASMA TECHNIQUE; PRODUCTION OF ACCELERATED ELECTRICALLY-CHARGED PARTICLES OR OF NEUTRONS; PRODUCTION OR ACCELERATION OF NEUTRAL MOLECULAR OR ATOMIC BEAMS
    • H05H1/00—Generating plasma; Handling plasma
    • H05H1/24—Generating plasma
    • H05H1/46—Generating plasma using applied electromagnetic fields, e.g. high frequency or microwave energy
    • H—ELECTRICITY
    • H05—ELECTRIC TECHNIQUES NOT OTHERWISE PROVIDED FOR
    • H05H—PLASMA TECHNIQUE; PRODUCTION OF ACCELERATED ELECTRICALLY-CHARGED PARTICLES OR OF NEUTRONS; PRODUCTION OR ACCELERATION OF NEUTRAL MOLECULAR OR ATOMIC BEAMS
    • H05H1/00—Generating plasma; Handling plasma
    • H05H1/24—Generating plasma
    • H05H1/46—Generating plasma using applied electromagnetic fields, e.g. high frequency or microwave energy
    • H05H1/4645—Radiofrequency discharges
    • H—ELECTRICITY
    • H05—ELECTRIC TECHNIQUES NOT OTHERWISE PROVIDED FOR
    • H05H—PLASMA TECHNIQUE; PRODUCTION OF ACCELERATED ELECTRICALLY-CHARGED PARTICLES OR OF NEUTRONS; PRODUCTION OR ACCELERATION OF NEUTRAL MOLECULAR OR ATOMIC BEAMS
    • H05H1/00—Generating plasma; Handling plasma
    • H05H1/24—Generating plasma
    • H05H1/46—Generating plasma using applied electromagnetic fields, e.g. high frequency or microwave energy
    • H05H1/4645—Radiofrequency discharges
    • H05H1/466—Radiofrequency discharges using capacitive coupling means, e.g. electrodes
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    • H05—ELECTRIC TECHNIQUES NOT OTHERWISE PROVIDED FOR
    • H05H—PLASMA TECHNIQUE; PRODUCTION OF ACCELERATED ELECTRICALLY-CHARGED PARTICLES OR OF NEUTRONS; PRODUCTION OR ACCELERATION OF NEUTRAL MOLECULAR OR ATOMIC BEAMS
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Abstract

The invention discloses a plasma discharge process simulation method and a plasma discharge process simulation system, and relates to the technical field of plasma sources. The method comprises the following steps: calculating a bipolar electric field in the main plasma region using a bipolar diffusion approximation and a drift diffusion approximation; calculating the electron temperature of the current radio frequency period by using a bipolar electric field and an electron energy equation; in the sheath region, determining the density and the speed of positive ions by using the electron temperature, a continuity equation and a momentum equation; determining an instantaneous electric field by using a Poisson equation and a current balance equation; and if the difference between the densities of the same charged particles in two adjacent radio frequency periods is smaller than a preset density threshold, determining the energy distribution and the angle distribution of the ions by using the positive ion density, the positive ion speed and the instantaneous electric field of the sheath region by using an ion Monte Carlo method. According to the invention, the main plasma region and the sheath layer are processed in a partitioning manner, and solving of a Poisson equation is avoided by adopting bipolar diffusion approximation in the main plasma region, so that the operation efficiency of the simulation method is increased, and the simulation speed is increased.

Description

Plasma discharge process simulation method and system
Technical Field
The invention relates to the technical field of plasma sources, in particular to a plasma discharge process simulation method and system.
Background
The chip is widely applied to the fields of computers, communication terminals, consumer electronics, automotive electronics, medical instruments, aerospace, national defense and the like. The chip industry is taken as an industrial foundation, is directly concerned with economic development, information safety and national defense construction, and is one of important marks for measuring the comprehensive strength of a country. In the fabrication of semiconductor chips, there are thousands of processes from front-end processing to back-end packaging, one third of which involves plasma techniques such as etching, deposition, ashing, implantation, cleaning, etc., where the plasma used is what is known as industrial plasma. As semiconductor chip processing technology is developing towards etched line widths of several nanometers (nm) and 450 millimeters (mm) of large wafer processing, the design and development of industrial plasma sources are facing a number of critical scientific issues that need to be addressed. In order to solve these key scientific problems, intensive research into the generation, transport and interaction of the process plasma with the surface of the material has to be carried out. In the research process of the industrial plasma source, only an experimental method is adopted, so that the cost is high and the period is long. Therefore, it is very important to provide a method capable of completely, rapidly and effectively simulating the industrial plasma discharge process. However, the stability of the poisson equation directly restricts the operation of the whole existing industrial plasma discharge process simulation method, and the operation of the poisson equation in the simulation method consumes a large amount of time, so that the existing industrial plasma discharge process simulation method has the problem of low simulation speed.
Disclosure of Invention
The invention aims to provide a plasma discharge process simulation method and a plasma discharge process simulation system, which solve the problem of low simulation speed of the existing industrial plasma discharge process simulation method.
In order to achieve the purpose, the invention provides the following scheme:
a plasma discharge process simulation method is applied to a plasma chamber, and the plasma chamber comprises: the device comprises a radio frequency source, a coil, a dielectric window, a chamber side wall, a lower polar plate, a capacitor and a bias power supply;
the radio frequency source is connected with the coil; the coil is placed on the upper surface of the dielectric window; the side surface of the dielectric window is connected with the side wall of the chamber, and the side wall of the chamber is grounded; the lower polar plate is positioned below the dielectric window, and the lower surface of the dielectric window is opposite to the upper surface of the lower polar plate; a plasma region is formed in the space among the dielectric window, the side wall of the cavity and the lower polar plate; the first end of the capacitor is connected with the lower surface of the lower polar plate, the second end of the capacitor is connected with one end of the bias power supply, and the other end of the bias power supply is grounded;
the plasma discharge process simulation method comprises the following steps:
acquiring the sheath thickness calculated in the previous radio frequency period, and dividing the plasma region in the current radio frequency period into a main plasma region and a sheath region according to the sheath thickness;
acquiring the electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region and the positive ion velocity of the sheath region which are calculated in the last radio frequency period;
solving a continuity equation of ions and a quasi-neutral condition of the plasma by using the electron temperature of the previous radio frequency period, the positive ion flux of the main plasma region of the previous radio frequency period and the negative ion flux of the main plasma region of the previous radio frequency period to obtain the charged particle density of the main plasma region of the current radio frequency period; the charged particle density of the main plasma region comprises a positive ion density of the main plasma region, a negative ion density of the main plasma region and an electron density of the main plasma region;
solving a Helmholtz equation of the main plasma region by using the electron density of the main plasma region to obtain the deposition power of the radio-frequency electromagnetic field in the main plasma region in the current radio-frequency period;
solving drift diffusion approximation and bipolar diffusion approximation of the charged particle flux of the main plasma region by using the charged particle density of the main plasma region to obtain a bipolar electric field in the main plasma region of the current radio frequency period; the charged particle flux of the main plasma region comprises a positive ion flux of the main plasma region, a negative ion flux of the main plasma region and an electron flux of the main plasma region;
solving an electron energy equation by utilizing the bipolar electric field, the deposition power of the radio-frequency electromagnetic field and the bias source deposition power of the sheath region of the previous radio-frequency period to obtain the electron temperature of the main plasma region of the current radio-frequency period;
solving a continuity equation of positive ions and a momentum equation of the positive ions by using the electron temperature of the current radio frequency period and the positive ion velocity of the sheath region of the previous radio frequency period to obtain the positive ion density of the sheath region and the positive ion velocity of the sheath region of the current radio frequency period;
solving a Poisson equation and a current balance equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current radio frequency period; the charged particle density of the sheath region comprises an electron density of the sheath region, a negative ion density of the sheath region, and a positive ion density of the sheath region;
judging whether the difference value between the densities of the same charged particles in two adjacent radio frequency periods is smaller than a corresponding density preset threshold value or not to obtain a first judgment result;
if the first judgment result is yes, determining the energy distribution and the angle distribution of ions incident to the lower polar plate by using the positive ion density of the sheath region, the positive ion speed of the sheath region and the instantaneous electric field by adopting an ion Monte Carlo method;
if the first judgment result is negative, calculating the positive ion flux of the main plasma region and the negative ion flux of the main plasma region in the current radio frequency period by using the charged particle density of the main plasma region and the bipolar electric field, calculating the sheath thickness of the current radio frequency period by using the charged particle density of the sheath region, calculating the bias source deposition power of the sheath region in the current radio frequency period by using the sheath thickness of the current radio frequency period and the sheath potential drop, updating the next radio frequency period to the current radio frequency period, returning to the step of obtaining the sheath thickness calculated in the previous radio frequency period, and dividing the plasma region in the current radio frequency period into the main plasma region and the sheath region according to the sheath thickness.
Optionally, the obtaining of the charged particle density of the main plasma region in the current radio frequency cycle by solving an ion continuity equation and a quasi-neutral condition of the plasma by using the electron temperature of the previous radio frequency cycle, the positive ion flux of the main plasma region in the previous radio frequency cycle, and the negative ion flux of the main plasma region in the previous radio frequency cycle specifically includes:
determining the generation and loss source terms of j-th ions caused by collision in the main plasma region by using the electron temperature and the type of the reaction gas in the plasma chamber;
solving the continuity equation of the ions by using the positive ion flux of the main plasma region, the negative ion flux of the main plasma region and the generation and loss source terms of the j-th ions caused by collision in the main plasma region
Figure BDA0002622369160000031
Determining the positive ion density and the negative ion density of the main plasma region of the current radio frequency period;
solving the quasi-neutral condition of the plasma by using the positive ion density of the main plasma region and the negative ion density of the main plasma region
Figure BDA0002622369160000041
Determining an electron density of the main plasma region of the current radio frequency cycle;
in the formula, njDenotes the density of positive ions or negative ions in the main plasma region, and when j is + nj＝n+Represents the positive ion density of the main plasma region, and n isj＝n_Represents a negative ion density of the main plasma region; t represents time;
Figure BDA0002622369160000042
represents the divergence of the positive ion flux or the divergence of the negative ion flux of the main plasma region; sjA source term representing the generation and loss of the jth ion caused by collisions in the main plasma region; e represents a elementary charge; n iseRepresents an electron density of the main plasma region; z+Represents the charge amount charged by positive ions of the main plasma region; z-Representing the charge amount charged by the negative ions of the main plasma region.
Optionally, the solving a helmholtz equation of the main plasma region by using the electron density of the main plasma region to obtain the deposition power of the radio frequency electromagnetic field in the main plasma region in the current radio frequency cycle specifically includes:
converting the Maxwell equation set of the main plasma region into a Helmholtz equation;
solving the Helmholtz equation by using the electron density of the main plasma region to obtain a radio-frequency electric field in the main plasma region of the current radio-frequency period;
using the radio frequency electric field and formula
Figure BDA0002622369160000043
Determining the deposition power of the radio frequency electromagnetic field in the main plasma region of the current radio frequency period;
in the formula, PindRepresenting a deposition power of the radio frequency electromagnetic field; sigmapIndicating the plasma conductivity; erRepresenting a component of the radio frequency electric field in a radial direction of the plasma region; ezRepresenting a component of the radio frequency electric field in an axial direction of the plasma region; eθRepresents the component of the radio frequency electric field in the theta direction of the plasma region, and theta represents the angle around the symmetry axis of the plasma region.
Optionally, the solving of the drift diffusion approximation and the bipolar diffusion approximation of the charged particle flux of the main plasma region by using the charged particle density of the main plasma region to obtain the bipolar electric field in the main plasma region of the current radio frequency cycle specifically includes:
solving drift diffusion and bipolar diffusion approximations of the charged particle flux of the main plasma region using the charged particle density of the main plasma region
Figure BDA0002622369160000051
Obtaining a bipolar electric field of the main plasma region of the current radio frequency cycle;
in the above formula,. mu.+Represents the mobility of positive ions of the main plasma region; n is+Represents a positive ion density of the main plasma region; esRepresenting a bipolar electric field; d+Represents a diffusion coefficient of positive ions of the main plasma region;
Figure BDA0002622369160000052
a positive ion density gradient representing the main plasma region; mu.s-Represents the mobility of negative ions in the main plasma region; n is-Represents a negative ion density of the main plasma region; d-Represents a diffusion coefficient of negative ions of the main plasma region;
Figure BDA0002622369160000053
a negative ion density gradient representing the main plasma region; mu.seRepresents the mobility of electrons of the main plasma region; n iseRepresents an electron density of the main plasma region; deA diffusion coefficient representing electrons of the main plasma region;
Figure BDA0002622369160000054
representing an electron density gradient of the main plasma region.
Optionally, the obtaining the electron temperature of the main plasma region in the current radio frequency cycle by solving an electron energy equation using the bipolar electric field, the deposition power of the radio frequency electromagnetic field, and the deposition power of the bias source of the sheath region in the previous radio frequency cycle specifically includes:
deposition power and deposition rate using the bipolar electric field, the radio frequency electromagnetic fieldSolving an electron energy equation by the bias source deposition power of the sheath region of the radio frequency period
Figure BDA0002622369160000055
Obtaining the electron temperature of the main plasma region in the current radio frequency period;
in the formula, neRepresents an electron density of the main plasma region; t iseRepresents an electron temperature of the main plasma region; t represents time;
Figure BDA0002622369160000056
a divergence representing electron fluence; e represents a elementary charge; esRepresenting the bipolar electric field; gamma-shapedeRepresents the electron flux of the main plasma region; pindRepresents the deposition power, P, of the radio-frequency electromagnetic fieldbiasRepresenting the bias source deposition power; weIndicating the energy loss caused by the collision.
Optionally, the solving a continuity equation of positive ions and a momentum equation of positive ions by using the electron temperature of the current radio frequency cycle and the positive ion velocity of the sheath region of the previous radio frequency cycle to obtain the positive ion density of the sheath region and the positive ion velocity of the sheath region of the current radio frequency cycle specifically includes:
determining the positive ion flux of the sheath region of the current rf cycle using the positive ion velocity of the sheath region of the previous rf cycle;
determining a source term for generation and loss of positive ions caused by collisions in the sheath region of the current radio frequency cycle using the electron temperature of the current radio frequency cycle and the type of reactant gas in the plasma chamber;
solving a positive ion continuity equation using the positive ion flux of the sheath region of the current radio frequency cycle and the collisional-induced positive ion generation and loss source terms in the sheath region
Figure BDA0002622369160000061
Obtaining said sheath region for said current RF cycleA positive ion density;
solving a momentum equation of positive ions using the positive ion density of the sheath region for the current RF cycle
Figure BDA0002622369160000062
Obtaining the positive ion velocity of the sheath region of the current radio frequency cycle;
n 'in the formula'+Representing the positive ion density of the sheath region; t represents time;
Figure BDA0002622369160000063
represents divergence of positive ion flux of the sheath region; s'+Representing the source terms of the generation and loss of positive ions caused by collisions in the sheath region; u'+Represents the velocity of the positive ions;
Figure BDA0002622369160000064
is represented by (n'+u'+u'+) Divergence of (d);
Figure BDA0002622369160000065
a positive ion gas pressure gradient indicative of the sheath region; p'+A positive ion gas pressure indicative of the sheath region; m'+Mass of positive ions representing the sheath region; z'+Representing the amount of charge charged by positive ions of the sheath region; e represents a elementary charge; eshA transient electric field indicative of the sheath region; m+Indicating the momentum transfer due to the collision.
Optionally, the solving a poisson equation and a current balance equation by using the charged particle density of the sheath region in the current radio frequency cycle to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current radio frequency cycle specifically includes:
solving a Poisson equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain an instantaneous electric field of the sheath region in the current radio frequency period;
by solving the current balance equation Ii+Ie+Id＝Ibiassin(ωbiast), obtaining the sheath potential drop of the current radio frequency period;
in the formula,
Figure BDA0002622369160000066
Iirepresents the ion flow; i iseRepresenting the electron flow; i isdRepresents a displacement current; i isbiasRepresenting a current magnitude of the bias power supply; omegabiasRepresenting a frequency of the bias power supply; t represents time; csRepresenting an equivalent capacitance of the sheath region; vsRepresenting the sheath potential drop; epsilon0Represents a dielectric constant in a vacuum; a represents the area of the lower polar plate; dsThe sheath thickness is indicated.
Optionally, the calculating, by using the charged particle density of the main plasma region and the bipolar electric field, a positive ion flux of the main plasma region and a negative ion flux of the main plasma region in the current radio frequency cycle, calculating a sheath thickness in the current radio frequency cycle by using the charged particle density of the sheath region, and calculating a bias source deposition power of the sheath region in the current radio frequency cycle by using the sheath thickness in the current radio frequency cycle and the sheath potential drop specifically includes:
solving a formula using the charged particle density of the main plasma region and the dipole electric field
Figure BDA0002622369160000071
And formula
Figure BDA0002622369160000072
Obtaining the positive ion flux of the main plasma region and the negative ion flux of the main plasma region in the current radio frequency period;
in the formula, gamma+Represents a positive ion flux of the main plasma region; mu.s+Represents the mobility of positive ions of the main plasma region; n is+Represents a positive ion density of the main plasma region; esRepresenting a bipolar electric field; d+Representing the main plasmaDiffusion coefficient of positive ions of the sub-region;
Figure BDA0002622369160000073
a positive ion density gradient representing the main plasma region; gamma-shaped-Represents a negative ion flux of the main plasma region; mu.s_Represents the mobility of negative ions in the main plasma region; n is_Represents a negative ion density of the main plasma region; d_Represents a diffusion coefficient of negative ions of the main plasma region;
Figure BDA0002622369160000074
a negative ion density gradient representing the main plasma region;
calculating the sheath thickness of the current radio frequency cycle by using the charged particle density of the sheath region;
solving a formula by using the sheath thickness of the current radio frequency period and the sheath potential drop
Figure BDA0002622369160000075
And
Figure BDA0002622369160000076
obtaining the bias source deposition power of the sheath region in the current radio frequency period;
in the formula, PbiasRepresenting a bias source deposition power; t represents a radio frequency period; vsRepresenting the sheath potential drop; i isiRepresents the ion flow; i iseRepresenting the electron flow; i isdRepresents a displacement current; t represents time; csRepresenting the equivalent capacitance of the sheath; epsilon0Represents a dielectric constant in a vacuum; a represents the area of the lower polar plate; dsThe sheath thickness is indicated.
A plasma discharge process simulation system, comprising:
the dividing module is used for acquiring the sheath thickness calculated in the previous radio frequency period and dividing the plasma region in the current radio frequency period into a main plasma region and a sheath region according to the sheath thickness;
the acquisition module is used for acquiring the electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region and the positive ion speed of the sheath region which are calculated in the last radio frequency period;
a charged particle density determination module, configured to solve an ion continuity equation and a quasi-neutral condition of the plasma by using the electron temperature of the previous radio frequency period, the positive ion flux of the main plasma region of the previous radio frequency period, and the negative ion flux of the main plasma region of the previous radio frequency period, so as to obtain a charged particle density of the main plasma region of the current radio frequency period; the charged particle density of the main plasma region comprises a positive ion density of the main plasma region, a negative ion density of the main plasma region and an electron density of the main plasma region;
the deposition power determining module is used for solving a Helmholtz equation of the main plasma region by using the electron density of the main plasma region to obtain the deposition power of the radio-frequency electromagnetic field in the main plasma region in the current radio-frequency period;
a bipolar electric field determination module, configured to solve drift diffusion approximation and bipolar diffusion approximation of the charged particle flux of the main plasma region by using the charged particle density of the main plasma region, so as to obtain a bipolar electric field in the main plasma region in the current radio frequency cycle; the charged particle flux of the main plasma region comprises a positive ion flux of the main plasma region, a negative ion flux of the main plasma region and an electron flux of the main plasma region;
the electronic temperature determination module is used for solving an electronic energy equation by utilizing the bipolar electric field, the deposition power of the radio-frequency electromagnetic field and the bias source deposition power of the sheath region of the previous radio-frequency period to obtain the electronic temperature of the main plasma region of the current radio-frequency period;
the positive ion density and velocity determination module is used for solving a continuity equation of positive ions and a momentum equation of the positive ions by utilizing the electron temperature of the current radio frequency period and the positive ion velocity of the sheath region of the previous radio frequency period to obtain the positive ion density of the sheath region of the current radio frequency period and the positive ion velocity of the sheath region;
the sheath parameter determining module is used for solving a Poisson equation and a current balance equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current radio frequency period; the charged particle density of the sheath region comprises an electron density of the sheath region, a negative ion density of the sheath region, and a positive ion density of the sheath region;
the first judgment module is used for judging whether the difference value between the densities of the same charged particles in two adjacent radio frequency periods is smaller than a corresponding density preset threshold value or not to obtain a first judgment result;
the ion energy angle distribution determining module is used for determining the ion energy distribution and the angle distribution of the incident ions to the lower polar plate by using the positive ion density of the sheath region, the positive ion velocity of the sheath region and the instantaneous electric field and adopting an ion Monte Carlo method when the first judgment result is yes;
and a returning module, configured to, when the first determination result is negative, calculate a positive ion flux of the main plasma region and a negative ion flux of the main plasma region in the current radio frequency cycle by using the charged particle density of the main plasma region and the bipolar electric field, calculate a sheath thickness in the current radio frequency cycle by using the charged particle density of the sheath region, calculate a bias source deposition power of the sheath region in the current radio frequency cycle by using the sheath thickness in the current radio frequency cycle and the sheath potential drop, update a next radio frequency cycle to the current radio frequency cycle, and execute the dividing module.
Optionally, the bipolar electric field determination module specifically includes:
a bipolar electric field determination unit for solving a drift diffusion approximation and a bipolar diffusion approximation of the charged particle flux of the main plasma region using the charged particle density of the main plasma region
Figure BDA0002622369160000091
Obtaining a bipolar electric field of the main plasma region of the current radio frequency cycle;
in the above formula,. mu.+Represents the mobility of positive ions of the main plasma region; n is+Represents a positive ion density of the main plasma region; esRepresenting a bipolar electric field; d+Represents a diffusion coefficient of positive ions of the main plasma region;
Figure BDA0002622369160000092
a positive ion density gradient representing the main plasma region; mu.s-Represents the mobility of negative ions in the main plasma region; n is-Represents a negative ion density of the main plasma region; d-Represents a diffusion coefficient of negative ions of the main plasma region;
Figure BDA0002622369160000093
a negative ion density gradient representing the main plasma region; mu.seRepresents the mobility of electrons of the main plasma region; n iseRepresents an electron density of the main plasma region; deA diffusion coefficient representing electrons of the main plasma region;
Figure BDA0002622369160000094
representing an electron density gradient of the main plasma region.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects:
the invention provides a plasma discharge process simulation method and system. The method comprises the following steps: acquiring the sheath thickness calculated in the previous radio frequency period, and dividing the plasma region of the current radio frequency period into a main plasma region and a sheath region according to the sheath thickness; acquiring the electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region and the positive ion velocity of the sheath region which are calculated in the last radio frequency period; solving a continuity equation of ions and a quasi-neutral condition of the plasma by using the electron temperature of the previous radio frequency period, the positive ion flux of the main plasma region of the previous radio frequency period and the negative ion flux of the main plasma region of the previous radio frequency period to obtain the charged particle density of the main plasma region of the current radio frequency period; the charged particle density of the main plasma region comprises the positive ion density of the main plasma region, the negative ion density of the main plasma region and the electron density of the main plasma region; solving a Helmholtz equation of the main plasma region by using the electron density of the main plasma region to obtain the deposition power of the radio-frequency electromagnetic field in the main plasma region of the current radio-frequency period; solving drift diffusion approximation and bipolar diffusion approximation of the charged particle flux of the main plasma region by using the charged particle density of the main plasma region to obtain a bipolar electric field in the main plasma region of the current radio frequency period; the charged particle flux of the main plasma region comprises positive ion flux of the main plasma region, negative ion flux of the main plasma region and electron flux of the main plasma region; solving an electron energy equation by utilizing the bipolar electric field, the deposition power of the radio-frequency electromagnetic field and the bias source deposition power of the sheath region in the previous radio-frequency period to obtain the electron temperature of the main plasma region in the current radio-frequency period; solving a continuity equation of positive ions and a momentum equation of the positive ions by using the electron temperature of the current radio frequency period and the positive ion velocity of the sheath region of the previous radio frequency period to obtain the positive ion density of the sheath region of the current radio frequency period and the positive ion velocity of the sheath region; solving a Poisson equation and a current balance equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current radio frequency period; the charged particle density of the sheath region comprises the electron density of the sheath region, the negative ion density of the sheath region and the positive ion density of the sheath region; judging whether the difference value between the densities of the same charged particles in two adjacent radio frequency periods is smaller than a corresponding density preset threshold value or not to obtain a first judgment result; if the first judgment result is yes, determining the energy distribution and the angle distribution of ions incident to the lower polar plate by using the positive ion density of the sheath region, the positive ion speed of the sheath region and the instantaneous electric field and by adopting an ion Monte Carlo method; if the first judgment result is negative, calculating the positive ion flux of the main plasma region and the negative ion flux of the main plasma region in the current radio frequency period by using the charged particle density of the main plasma region and the bipolar electric field, calculating the sheath thickness in the current radio frequency period by using the charged particle density of the sheath region, calculating the bias source deposition power of the sheath region in the current radio frequency period by using the sheath thickness and the sheath potential drop in the current radio frequency period, updating the next radio frequency period to the current radio frequency period, returning to the step of obtaining the sheath thickness calculated in the previous radio frequency period, and dividing the plasma region in the current radio frequency period into the main plasma region and the sheath region according to the sheath thickness. According to the plasma discharge process simulation method, the main plasma area and the sheath layer are processed in a partitioning mode, and bipolar diffusion approximation is adopted in the main plasma area, so that a Poisson equation is avoided being solved, the operation efficiency of the simulation method is improved, and the simulation speed is increased; and (3) simulating the dynamic behavior of the sheath by adopting a complete sheath model, and self-exactly including the influence of the sheath on the main plasma. The sheath model includes: a positive ion continuity equation, a positive ion momentum equation, a poisson equation, and a current balance equation.
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 method for simulating a plasma discharge process according to an embodiment of the present invention;
FIG. 2 is a block diagram of a plasma chamber provided by an embodiment of the present invention;
fig. 3 is a structural diagram of a plasma discharge process simulation system according to an embodiment of the present invention.
Description of the symbols: 1. a radio frequency source; 2. a coil; 3. a dielectric window; 4. a chamber sidewall; 5. a lower polar plate; 6. a capacitor; 7. a bias power supply.
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 plasma discharge process simulation method and a plasma discharge process simulation system, which solve the problem of low simulation speed of the existing industrial plasma discharge process simulation method.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
Fig. 2 is a structural diagram of a plasma chamber according to an embodiment of the present invention. Referring to fig. 2, the plasma chamber includes: an RF source 1, a coil 2, a dielectric window 3, a chamber side wall 4, a lower polar plate 5, a capacitor 6 and a bias power supply 7. The plasma chamber adopts an inductive coupling discharge chamber. In this embodiment, a comprehensive, fast and effective plasma discharge process simulation method is established according to a plasma source commonly used in the industry, such as an inductively coupled plasma source, the plasma chamber of this embodiment is a simplified simulated chamber structure, and in practical experiments, the plasma chamber may further include other devices.
The radio frequency source 1 is connected with the coil 2; the coil 2 is arranged on the upper surface of the dielectric window 3; the side surface of the dielectric window 3 is connected with the chamber side wall 4, and the chamber side wall 4 is grounded; the lower polar plate 5 is positioned below the dielectric window 3, and the lower surface of the dielectric window 3 is opposite to the upper surface of the lower polar plate 5; a plasma region is formed in the space among the dielectric window 3, the chamber side wall 4 and the lower polar plate 5, and comprises a main plasma region and a sheath region; the first end of the capacitor 6 is connected with the lower surface of the lower pole plate 5, the second end of the capacitor 6 is connected with one end of the bias power supply 7, and the other end of the bias power supply 7 is grounded.
Fig. 1 is a flowchart of a plasma discharge process simulation method according to an embodiment of the present invention. Referring to fig. 1, the plasma discharge process simulation method includes:
step 101, obtaining the sheath thickness calculated in the previous radio frequency cycle, and dividing the plasma region of the current radio frequency cycle into a main plasma region and a sheath region according to the sheath thickness.
In general, a simulated plasma process chamber has an axisymmetric structure, so that a two-dimensional coordinate system (r, z) is established in an inductively coupled discharge chamber with the center of the lower surface of the lower electrode plate as an origin, the direction perpendicular to the lower electrode plate as an axial direction, and the direction parallel to the lower electrode plate as a radial direction, wherein r represents a radial coordinate, and z represents an axial coordinate.
And in the first radio frequency period, acquiring the initialized sheath layer thickness, and dividing the plasma region of the first radio frequency period into a main plasma region and a sheath layer region according to the initialized sheath layer thickness. And when iteration is carried out from the second radio frequency period, dividing the plasma region according to the sheath layer thickness calculated in the previous radio frequency period.
And 102, acquiring the electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region and the positive ion velocity of the sheath region, which are calculated in the last radio frequency period. And in the first radio frequency period, acquiring the initialized electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region and the positive ion velocity of the sheath region. The plasma discharge process simulation method of this embodiment calculates the parameter of the current rf cycle using the parameter calculated in the previous rf cycle, and when the current rf cycle is the first rf cycle, the parameter used is the initialized parameter. The parameters include: sheath thickness, electron temperature, positive ion flux in the main plasma region, negative ion flux in the main plasma region, bias source deposition power in the sheath region, positive ion velocity in the sheath region, positive ion density in the sheath region, potential in the sheath region, and instantaneous electric field in the sheath region.
103, solving a continuity equation of ions and a quasi-neutral condition of the plasma by using the electron temperature of the previous radio frequency period, the positive ion flux of the main plasma region of the previous radio frequency period and the negative ion flux of the main plasma region of the previous radio frequency period to obtain the charged particle density of the main plasma region of the current radio frequency period; the charged particle density of the main plasma region includes a positive ion density of the main plasma region, a negative ion density of the main plasma region, and an electron density of the main plasma region.
Step 103 specifically comprises: and determining the generation and loss source terms of j-th ions caused by collision in the main plasma area of the current radio frequency period by using the electron temperature of the last radio frequency period and the type of the reaction gas in the plasma chamber. During the first RF cycle, S is determined by the initialized electron temperature and the type of reactant gas in the plasma chamberj。
Solving the continuity equation of the ions by using the positive ion flux of the main plasma region in the last radio frequency period, the negative ion flux of the main plasma region in the last radio frequency period and the generation and loss source terms of the j-th ions caused by collision in the main plasma region in the current radio frequency period
Figure BDA0002622369160000131
And determining the positive ion density and the negative ion density of the main plasma region of the current radio frequency period. The first rf cycle uses the positive ion flux of the initialized main plasma region and the negative ion flux of the initialized main plasma region.
Solving the quasi-neutral condition of the plasma by using the positive ion density of the main plasma region of the current radio frequency period and the negative ion density of the main plasma region of the current radio frequency period
Figure BDA0002622369160000132
The electron density of the main plasma region of the current rf cycle is determined.
In the formula, njDenotes the density of positive or negative ions in the main plasma region, and when j is + nj＝n+Indicating the main plasma regionWhen j is-then nj＝n-Represents the negative ion density of the main plasma region; t represents time; gamma-shapedjRepresents the positive ion flux or the negative ion flux of the main plasma region;
Figure BDA0002622369160000133
represents the divergence of the positive ion flux or the divergence of the negative ion flux of the main plasma region; sjA source term representing the generation and loss of the jth ion caused by collisions in the main plasma region; e represents a elementary charge; n iseRepresents the electron density of the main plasma region; z+Represents the charge amount charged by the positive ions in the main plasma region; z-Indicating the amount of charge carried by the negative ions in the main plasma region.
And 104, solving a Helmholtz equation of the main plasma region by using the electron density of the main plasma region to obtain the deposition power of the radio-frequency electromagnetic field in the main plasma region in the current radio-frequency period.
Step 104 specifically includes: the inductively coupled plasma source mainly depends on the circular electric field excited by the circular current in the coil to drive the discharge. The electromagnetic field distribution in the plasma can be obtained by solving maxwell's equations. Maxwell's system of equations is:
Figure BDA0002622369160000141
wherein,
Figure BDA0002622369160000142
x represents rotation of vector, E represents radio frequency electric field, B represents magnetic field, t represents time, mu0Is the magnetic permeability in vacuum, J denotes the plasma current, ε0Which represents the dielectric constant in vacuum. Equation (1) is two equations in a standard maxwell system of equations. According to ohm's law, J ═ σpE, where σpIs the plasma conductivity. E. The three physical quantities B and J are functions of time variable and space variable, i.e. the three physical quantities vary both in time and in spaceAnd (4) transforming.
The electromagnetic field and plasma current of the main plasma region of the current rf cycle are approximated harmonically, i.e., assuming that the three physical quantities E, B and J are simple harmonically related over time and proportional to exp (-iwt), E, B and J can be written as follows:
Figure BDA0002622369160000143
wherein,
Figure BDA0002622369160000144
representing the spatial distribution of the radio frequency electric field, only in relation to spatial variables;
Figure BDA0002622369160000145
represents the spatial distribution of the magnetic field, only with respect to spatial variables;
Figure BDA0002622369160000146
represents the spatial distribution of the plasma current, and is only related to spatial variables; exp represents an exponential function; i represents an imaginary symbol; w represents the radio source frequency; t represents time.
Under the harmonic approximation described above, the maxwell's system of equations for the main plasma region can be converted to helmholtz equations. The helmholtz equation is:
Figure BDA0002622369160000147
wherein,
Figure BDA0002622369160000148
denotes the Laplace operator, k0Is wave number, epsilonpIs the plasma dielectric constant.
And solving a Helmholtz equation by using the electron density of the main plasma region to obtain the radio-frequency electric field in the main plasma region of the current radio-frequency period. Determination of the plasma dielectric constant epsilon using the electron density of the main plasma regionpThe dielectric constant ε of the plasmapSubstituting the formula (3) to solve the Helmholtz equation. Since the characteristic time scale (nanosecond) of the time variation of the radio frequency electromagnetic field in the main plasma region is much smaller than that of the plasma transport (microsecond), the density of the plasma can be approximately considered to be constant, i.e. the local density is approximate, within tens or even hundreds of radio frequency cycles, so that the dielectric constant epsilon of the plasma is approximatepConstant over tens or even hundreds of radio frequency cycles. By solving the Helmholtz equation and the formula (2), a radio frequency electric field E can be obtained, and then the radio frequency electric field E is substituted into the Maxwell equation set to obtain a magnetic field B.
And (4) determining the deposition power of the radio-frequency electromagnetic field in the main plasma region of the current radio-frequency period by using the radio-frequency electric field and the formula (4). And (4) determining the deposition power of the radio frequency electromagnetic field in the plasma in the main plasma region of the current radio frequency period by using the components of the radio frequency electric field E in the r direction, the z direction and the theta direction and the formula (4).
Figure BDA0002622369160000151
In the formula, PindRepresenting the deposition power of the radio frequency electromagnetic field; re represents the real part of the corresponding complex number; erRepresents the component of the radio frequency electric field in the radial direction (r direction) of the plasma region; ezRepresents the component of the radio frequency electric field in the axial direction (z direction) of the plasma region; eθRepresents the component of the rf electric field in the direction of theta in the plasma region, theta representing the angle around the axis of symmetry of the plasma region.
The invention adopts harmonic wave approximation and local density approximation, and the strong coupling between the electromagnetic field and the plasma is weak coupling, thereby accelerating the simulation speed.
105, solving drift diffusion approximation and bipolar diffusion approximation of the charged particle flux of the main plasma region by using the charged particle density of the main plasma region to obtain a bipolar electric field in the main plasma region in the current radio frequency period; the charged particle flux of the main plasma region includes a positive ion flux of the main plasma region, a negative ion flux of the main plasma region, and an electron flux of the main plasma region.
Step 105 specifically includes: solving drift diffusion approximation and bipolar diffusion approximation of charged particle flux in main plasma region using charged particle density in main plasma region
Figure BDA0002622369160000152
The bipolar electric field of the main plasma region of the current radio frequency cycle is obtained. In the main plasma region, a bipolar diffusion approximation is used, i.e. assuming that the sum of the fluxes of all positive ions is equal to the sum of the fluxes of all negatively charged particles (electrons and negative ions):
Figure BDA0002622369160000153
wherein, gamma is+、Γ-、ΓePositive ion flux, negative ion flux and electron flux, respectively, "+" and "-" below the summation symbol indicate the summation of positive and negative ions, respectively. The electron and ion fluxes are approximated by drift diffusion, as follows:
Figure BDA0002622369160000161
substituting the electron density, the positive ion density and the negative ion density of the main plasma region of the current radio frequency period into the formula (6), the bipolar electric field E of the main plasma region of the current radio frequency period can be calculatedsThereby avoiding solving the poisson equation to accelerate the simulation speed.
In the above formula,. mu.+Represents the mobility of positive ions in the main plasma region; n is+Represents the positive ion density of the main plasma region; esRepresenting a bipolar electric field; d+Represents the diffusion coefficient of positive ions in the main plasma region;
Figure BDA0002622369160000162
a positive ion density gradient representing the main plasma region; mu.s-Negative ions representing the main plasma region(ii) mobility of the seed; n is-Represents the negative ion density of the main plasma region; d-Represents the diffusion coefficient of negative ions in the main plasma region;
Figure BDA0002622369160000163
a negative ion density gradient representing the main plasma region; mu.seRepresents the mobility of electrons in the main plasma region; n iseRepresents the electron density of the main plasma region; deRepresents the diffusion coefficient of electrons in the main plasma region;
Figure BDA0002622369160000164
representing the electron density gradient of the main plasma region.
And 106, solving an electron energy equation by using the bipolar electric field, the deposition power of the radio frequency electromagnetic field and the bias source deposition power of the sheath region in the previous radio frequency period to obtain the electron temperature of the main plasma region in the current radio frequency period.
Step 106 specifically includes: and solving an electron energy equation by using the bipolar electric field in the current radio frequency period, the deposition power of the radio frequency electromagnetic field in the current radio frequency period and the deposition power of the bias source in the sheath region in the previous radio frequency period to obtain the electron temperature of the main plasma region in the current radio frequency period. The electron energy equation is:
Figure BDA0002622369160000165
in the formula, neRepresents the electron density of the main plasma region; t iseRepresents the electron temperature of the main plasma region; t represents time; q. q.seElectron fluence;
Figure BDA0002622369160000166
a divergence representing electron fluence; e represents a elementary charge; esRepresenting a bipolar electric field; gamma-shapedeRepresents the electron flux of the main plasma region; pindDenotes the deposition power, P, of the radio-frequency electromagnetic fieldbiasRepresenting a bias source deposition power; weTo representEnergy loss due to collision. During the first RF cycle, power is deposited using an initialized bias source.
And 107, solving a continuity equation of the positive ions and a momentum equation of the positive ions by using the electron temperature of the current radio frequency period and the positive ion velocity of the sheath region of the previous radio frequency period to obtain the positive ion density of the sheath region of the current radio frequency period and the positive ion velocity of the sheath region. In the sheath region, the quasi-neutral condition is not met, so that a sheath model is required to describe the instantaneous oscillation behavior of the sheath region, and the electron density, the negative ion density, the positive ion velocity, the electric potential of the plasma sheath region and the electron temperature of the main plasma region of the current radio frequency cycle are taken as boundary conditions of the sheath model.
Step 107 specifically includes: and determining the positive ion flux of the sheath region of the current radio frequency period by using the positive ion velocity of the sheath region of the previous radio frequency period. And multiplying the positive ion velocity of the sheath region of the previous radio frequency period by the positive ion density to obtain the positive ion flux of the sheath region of the current radio frequency period. And in the first radio frequency period, the initialized positive ion speed is adopted.
The electron temperature of the current radio frequency cycle and the type of the reaction gas in the plasma chamber are used to determine the generation and loss source terms of the positive ions caused by collision in the sheath region of the current radio frequency cycle.
Solving a continuity equation of positive ions by using the flux of positive ions in the sheath region of the current radio frequency cycle and the source terms of generation and loss of positive ions caused by collisions in the sheath region
Figure BDA0002622369160000171
And obtaining the positive ion density of the sheath region of the current radio frequency period.
And solving the momentum equation of the positive ions by using the positive ion density of the sheath region in the current radio frequency period to obtain the positive ion velocity of the sheath region in the current radio frequency period. Solving momentum equation of positive ions by using positive ion density of sheath region of current radio frequency period and instantaneous electric field of sheath region of last radio frequency period
Figure BDA0002622369160000172
And obtaining the positive ion velocity of the sheath region of the current radio frequency period.
N 'in the formula'+Represents the positive ion density of the sheath region; t represents time; Γ 'of'+Represents the positive ion flux of the sheath region;
Figure BDA0002622369160000173
represents the divergence of the positive ion flux of the sheath region; s'+Showing the source terms of the generation and loss of positive ions caused by collisions in the sheath region; u'+Represents the velocity of the positive ions;
Figure BDA0002622369160000174
is represented by (n'+u'+u'+) Divergence of (d); p'+Positive ion gas pressure indicating sheath region;
Figure BDA0002622369160000175
a positive ion gas pressure gradient indicative of a sheath region; m'+Mass of positive ions representing sheath region; z'+The amount of charge carried by the positive ions representing the sheath region; e represents a elementary charge; eshThe instantaneous electric field representing the sheath region of the previous radio frequency cycle; m+Indicating the momentum transfer due to the collision. In the first radio frequency period, the instantaneous electric field E of the initialized sheath region is adoptedshAnd acquiring the instantaneous electric field of the sheath region of the previous radio frequency cycle when iteration is carried out from the second radio frequency cycle.
Step 108, solving a Poisson equation and a current balance equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current radio frequency period; the charged particle density of the sheath region includes an electron density of the sheath region, a negative ion density of the sheath region, and a positive ion density of the sheath region.
Step 108 specifically includes: and solving the Boltzmann distribution by using the electron temperature of the current radio frequency period and the electric potential of the sheath region of the previous radio frequency period to determine the electron density of the sheath region of the current radio frequency period. And in the first radio frequency period, acquiring the potential of the sheath region in the previous radio frequency period by adopting the potential of the initialized sheath region from the beginning of the second radio frequency period.
And solving the Boltzmann distribution by using the preset negative ion temperature and the electric potential of the sheath region of the previous radio frequency period to determine the negative ion density of the sheath region of the current radio frequency period. The negative ion temperature is generally set to room temperature.
Solving a Poisson equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain the instantaneous electric field of the sheath region in the current radio frequency period. Substituting the charged particle density of the sheath region in the current radio frequency period into a Poisson equation, solving to obtain the potential of the sheath region in the current radio frequency period, and substituting the potential of the sheath region in the current radio frequency period into a formula
Figure BDA0002622369160000181
Solving to obtain the instantaneous electric field E of the sheath region of the current radio frequency periodsh；
Figure BDA0002622369160000182
Representing the gradient of the potential. The poisson equation is:
Figure BDA0002622369160000183
wherein φ represents an electric potential, e represents a elementary charge, ε0Denotes the dielectric constant in vacuum, Z+Represents the charge amount, n, charged to the positive ion+Denotes the positive ion density, Z-Represents the amount of charge charged in the negative ion, n-Denotes the negative ion density, neRepresents the electron density.
By solving the current balance equation Ii+Ie+Id＝Ibiassin(ωbiast) and obtaining the sheath potential drop of the current radio frequency period.
In the formula,
Figure BDA0002622369160000184
Iirepresents the ion flow; i iseIndicating electricitySub-streams; i isdRepresents a displacement current; i isbiasRepresenting the magnitude of the current of the bias supply; omegabiasRepresenting the frequency of the bias power supply; t represents time; csRepresenting the equivalent capacitance of the sheath region; vsRepresents the sheath potential drop; a represents the area of the lower polar plate; dsRepresenting the sheath thickness of the previous rf cycle.
The instantaneous electric field of the sheath area, the potential drop of the sheath, the thickness of the sheath, the density of positive ions, the speed of the positive ions and the deposition power of the bias source can be determined through a sheath model. The sheath model includes: a positive ion continuity equation, a positive ion momentum equation, a poisson equation, a current balance equation, and equation (9).
Step 109, judging whether the difference value between the densities of the same charged particles in two adjacent radio frequency periods is smaller than a corresponding density preset threshold value, and obtaining a first judgment result. Judging whether the difference value between the positive ion densities of two adjacent radio frequency periods is smaller than a preset positive ion density threshold value, judging whether the difference value between the negative ion densities of two adjacent radio frequency periods is smaller than a preset negative ion density threshold value, and judging whether the difference value between the electron densities of two adjacent radio frequency periods is smaller than a preset electron density threshold value to obtain a first judgment result.
And 110, if the first judgment result is yes, determining the energy distribution and the angle distribution of the ions incident to the lower polar plate by using the positive ion density of the sheath region, the positive ion speed of the sheath region and the instantaneous electric field and by using an ion Monte Carlo method.
Step 110 specifically includes: and uniformly scattering a large amount of positive ions on the boundary of the sheath layer according to the preset proportion of the number of the scattered particles and the density of the positive ions by utilizing the density of the positive ions in the sheath layer of the current radio frequency period. And updating the position and the speed of the positive ions by utilizing the instantaneous electric field of the sheath region in the current radio frequency period, the Newton's equation of motion of the particles and the collision process of the positive ions in the sheath layer. When all positive ions reach the lower electrode, counting the ratio of ions bombarded to the surface of the lower polar plate in each energy interval to obtain energy distribution; the angular distribution is similar to the energy distribution.
And step 111, if the first judgment result is negative, calculating the positive ion flux of the main plasma region and the negative ion flux of the main plasma region in the current radio frequency period by using the charged particle density of the main plasma region and the bipolar electric field, calculating the sheath thickness in the current radio frequency period by using the charged particle density of the sheath region, calculating the bias source deposition power of the sheath region in the current radio frequency period by using the sheath thickness and the sheath potential drop in the current radio frequency period, updating the next radio frequency period to the current radio frequency period, returning to the step 101 to obtain the sheath thickness calculated in the previous radio frequency period, and dividing the plasma region in the current radio frequency period into the main plasma region and the sheath region according to the sheath thickness.
Step 111 specifically includes:
solving drift diffusion approximation expression of flux using charged particle density and dipole electric field of main plasma region of current radio frequency cycle
Figure BDA0002622369160000191
And
Figure BDA0002622369160000192
and obtaining the positive ion flux of the main plasma region and the negative ion flux of the main plasma region of the current radio frequency period.
In the formula, gamma+Represents the positive ion flux of the main plasma region; mu.s+Represents the mobility of positive ions in the main plasma region; n is+Represents the positive ion density of the main plasma region; esRepresenting a bipolar electric field; d+Represents the diffusion coefficient of positive ions in the main plasma region;
Figure BDA0002622369160000201
a positive ion density gradient representing the main plasma region; gamma-shaped-Represents the negative ion flux of the main plasma region; mu.s_Represents the mobility of negative ions in the main plasma region; n is_Represents the negative ion density of the main plasma region; d_Represents the diffusion coefficient of negative ions in the main plasma region;
Figure BDA0002622369160000202
representing the negative ion density gradient of the main plasma region. The drift diffusion approximation expression further includes:
Figure BDA0002622369160000203
and calculating the thickness of the sheath layer in the current radio frequency period by using the charged particle density of the sheath layer in the current radio frequency period. The ratio of the electron density to the positive ion density is 1, the position closest to the lower polar plate is the boundary of the sheath layer, and the distance between the position and the lower polar plate is the thickness of the sheath layer.
Solving formula by using sheath thickness and sheath potential drop of current radio frequency period
Figure BDA0002622369160000204
And
Figure BDA0002622369160000205
and obtaining the deposition power of the bias source of the sheath region in the current radio frequency period. Specifically, the equivalent capacitance of the sheath is calculated by utilizing the sheath thickness of the current radio frequency period
Figure BDA0002622369160000206
In the formula, epsilon0Represents a dielectric constant in a vacuum; a represents the area of the lower polar plate; dsThe sheath thickness is indicated.
Equivalent capacitance C using sheathsAnd sheath potential drop V of the current radio frequency cyclesCalculating the displacement current
Figure BDA0002622369160000207
Will shift the current IdSubstituting the formula (9) to obtain the deposition power of the bias source of the sheath region of the current radio frequency period.
Figure BDA0002622369160000208
In the formula, PbiasRepresenting a bias source deposition power; t represents a radio frequency period; i isiRepresents the ion flow; i iseRepresenting the electron flow; t represents time.
The plasma discharge process simulation method of this embodiment may further divide the plasma region into grids through a two-dimensional coordinate system (r, z), and obtain the spatial distribution of the plasma according to the calculation in steps 103 and 108, where the spatial distribution of the plasma includes: the electromagnetic field at each grid point, the density of the charged particles, the dipole electric field, the flux of the charged particles, the electron temperature and the deposition power of the radio frequency electromagnetic field in the plasma, and the instantaneous electric field, the positive ion density and the positive ion velocity at each grid point in the sheath region.
Fig. 3 is a structural diagram of the plasma discharge process simulation system according to the embodiment of the present invention. Referring to fig. 3, the plasma discharge process simulation system includes:
the dividing module 201 is configured to obtain a sheath thickness calculated in a previous radio frequency cycle, and divide a plasma region in a current radio frequency cycle into a main plasma region and a sheath region according to the sheath thickness.
The obtaining module 202 is configured to obtain the electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region, and the positive ion velocity of the sheath region, which are calculated in the previous radio frequency cycle. And in the first radio frequency period, acquiring the initialized electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region and the positive ion velocity of the sheath region.
The charged particle density determining module 203 is configured to solve a continuity equation of ions and a quasi-neutral condition of the plasma by using the electron temperature of the previous radio frequency period, the positive ion flux of the main plasma region of the previous radio frequency period, and the negative ion flux of the main plasma region of the previous radio frequency period, so as to obtain a charged particle density of the main plasma region of the current radio frequency period; the charged particle density of the main plasma region includes a positive ion density of the main plasma region, a negative ion density of the main plasma region, and an electron density of the main plasma region.
Charged particle density determination moduleBlock 203 specifically includes: and the generation and loss source term determining unit is used for determining the generation and loss source terms of j-th ions caused by collision in the main plasma region of the current radio frequency period by using the electron temperature of the last radio frequency period and the type of the reaction gas in the plasma chamber. During the first RF cycle, S is determined by the initialized electron temperature and the type of reactant gas in the plasma chamberj。
A positive and negative ion density determination unit for solving the continuity equation of the ions by using the positive ion flux of the main plasma region in the last radio frequency period, the negative ion flux of the main plasma region in the last radio frequency period and the generation and loss source terms of the j-th ions caused by collision in the main plasma region in the current radio frequency period
Figure BDA0002622369160000211
And determining the positive ion density and the negative ion density of the main plasma region of the current radio frequency period. The first rf cycle uses the positive ion flux of the initialized main plasma region and the negative ion flux of the initialized main plasma region.
An electron density determination unit for solving the quasi-neutral condition of the plasma by using the positive ion density of the main plasma region of the current radio frequency cycle and the negative ion density of the main plasma region of the current radio frequency cycle
Figure BDA0002622369160000221
The electron density of the main plasma region of the current rf cycle is determined.
In the formula, njDenotes the density of positive or negative ions in the main plasma region, and when j is + nj＝n+Denotes the positive ion density of the main plasma region, j ═ then, nj＝n-Represents the negative ion density of the main plasma region; t represents time; gamma-shapedjRepresents the positive ion flux or the negative ion flux of the main plasma region;
Figure BDA0002622369160000222
divergence or negative ion of the positive ion flux representing the main plasma regionDivergence of the sub-flux; sjA source term representing the generation and loss of the jth ion caused by collisions in the main plasma region; e represents a elementary charge; n iseRepresents the electron density of the main plasma region; z+Represents the charge amount charged by the positive ions in the main plasma region; z-Indicating the amount of charge carried by the negative ions in the main plasma region.
And the deposition power determining module 204 is configured to solve a helmholtz equation of the main plasma region by using the electron density of the main plasma region, so as to obtain the deposition power of the rf electromagnetic field in the main plasma region of the current rf cycle.
The deposition power determining module 204 specifically includes:
and the conversion unit is used for converting the Maxwell equation set of the main plasma region into Helmholtz equation.
And the radio frequency electric field determining unit is used for solving a Helmholtz equation by using the electron density of the main plasma region to obtain the radio frequency electric field in the main plasma region of the current radio frequency period.
A deposition power determining unit for determining deposition power by using RF electric field and formula
Figure BDA0002622369160000223
Determining the deposition power of the radio frequency electromagnetic field in the main plasma region of the current radio frequency period.
In the formula, PindRepresenting the deposition power of the radio frequency electromagnetic field; re represents the real part of the corresponding complex number; erRepresents the component of the radio frequency electric field in the radial direction (r direction) of the plasma region; ezRepresents the component of the radio frequency electric field in the axial direction (z direction) of the plasma region; eθRepresents the component of the rf electric field in the direction of theta in the plasma region, theta representing the angle around the axis of symmetry of the plasma region.
A bipolar electric field determining module 205, configured to solve drift diffusion approximation and bipolar diffusion approximation of the charged particle flux in the main plasma region by using the charged particle density in the main plasma region, so as to obtain a bipolar electric field in the main plasma region in the current radio frequency period; the charged particle flux of the main plasma region includes a positive ion flux of the main plasma region, a negative ion flux of the main plasma region, and an electron flux of the main plasma region.
The bipolar electric field determination module 205 specifically includes: a bipolar electric field determination unit for solving drift diffusion approximation and bipolar diffusion approximation of the charged particle flux of the main plasma region using the charged particle density of the main plasma region
Figure BDA0002622369160000231
The bipolar electric field of the main plasma region of the current radio frequency cycle is obtained.
In the above formula,. mu.+Represents the mobility of positive ions in the main plasma region; n is+Represents the positive ion density of the main plasma region; esRepresenting a bipolar electric field; d+Represents the diffusion coefficient of positive ions in the main plasma region;
Figure BDA0002622369160000232
a positive ion density gradient representing the main plasma region; mu.s-Represents the mobility of negative ions in the main plasma region; n is-Represents the negative ion density of the main plasma region; d-Represents the diffusion coefficient of negative ions in the main plasma region;
Figure BDA0002622369160000233
a negative ion density gradient representing the main plasma region; mu.seRepresents the mobility of electrons in the main plasma region; n iseRepresents the electron density of the main plasma region; deRepresents the diffusion coefficient of electrons in the main plasma region;
Figure BDA0002622369160000234
representing the electron density gradient of the main plasma region.
And the electronic temperature determination module 206 is configured to solve an electronic energy equation by using the bipolar electric field, the deposition power of the radio-frequency electromagnetic field, and the deposition power of the bias source of the sheath region in the previous radio-frequency cycle, so as to obtain the electronic temperature of the main plasma region in the current radio-frequency cycle.
Electronic temperature determination module 206The body includes: an electronic temperature determination unit for solving an electronic energy equation by using the bipolar electric field of the current radio frequency cycle, the deposition power of the radio frequency electromagnetic field of the current radio frequency cycle and the deposition power of the bias source of the sheath region of the previous radio frequency cycle
Figure BDA0002622369160000235
And obtaining the electron temperature of the main plasma region of the current radio frequency period.
In the formula, neRepresents the electron density of the main plasma region; t iseRepresents the electron temperature of the main plasma region; t represents time; q. q.seElectron fluence;
Figure BDA0002622369160000236
a divergence representing electron fluence; e represents a elementary charge; esRepresenting a bipolar electric field; gamma-shapedeRepresents the electron flux of the main plasma region; pindDenotes the deposition power, P, of the radio-frequency electromagnetic fieldbiasRepresenting a bias source deposition power; weIndicating the energy loss caused by the collision. During the first RF cycle, power is deposited using an initialized bias source.
And the positive ion density and velocity determining module 207 is used for solving a continuity equation of positive ions and a momentum equation of the positive ions by using the electron temperature of the current radio frequency period and the positive ion velocity of the sheath region of the previous radio frequency period to obtain the positive ion density of the sheath region of the current radio frequency period and the positive ion velocity of the sheath region.
The positive ion density velocity determination module 207 specifically includes: and the positive ion flux determining unit is used for determining the positive ion flux of the sheath region of the current radio frequency period by using the positive ion speed of the sheath region of the previous radio frequency period. And multiplying the positive ion velocity of the sheath region of the previous radio frequency period by the positive ion density to obtain the positive ion flux of the sheath region of the current radio frequency period. And in the first radio frequency period, the initialized positive ion speed is adopted.
And the loss source item determining unit is used for determining the generation and loss source items of the positive ions caused by collision in the sheath region of the current radio frequency period by using the electron temperature of the current radio frequency period and the type of the reaction gas in the plasma chamber.
A positive ion density determination unit for solving a continuity equation of the positive ions by using the positive ion flux of the sheath region of the current radio frequency cycle and the source terms of generation and loss of the positive ions caused by collision in the sheath region
Figure BDA0002622369160000241
And obtaining the positive ion density of the sheath region of the current radio frequency period.
And the positive ion velocity determining unit is used for solving a momentum equation of the positive ions by using the positive ion density of the sheath region in the current radio frequency period to obtain the positive ion velocity of the sheath region in the current radio frequency period. Solving momentum equation of positive ions by using positive ion density of sheath region of current radio frequency period and instantaneous electric field of sheath region of last radio frequency period
Figure BDA0002622369160000242
And obtaining the positive ion velocity of the sheath region of the current radio frequency period.
N 'in the formula'+Represents the positive ion density of the sheath region; t represents time; Γ 'of'+Represents the positive ion flux of the sheath region;
Figure BDA0002622369160000243
represents the divergence of the positive ion flux of the sheath region; s'+Showing the source terms of the generation and loss of positive ions caused by collisions in the sheath region; u'+Represents the velocity of the positive ions;
Figure BDA0002622369160000244
is represented by (n'+u'+u'+) Divergence of (d); p'+Positive ion gas pressure indicating sheath region;
Figure BDA0002622369160000245
a positive ion gas pressure gradient indicative of a sheath region; m'+Mass of positive ions representing sheath region; z'+The amount of charge carried by the positive ions representing the sheath region; e represents a elementary charge; eshIs shown onA transient electric field of a sheath region of a radio frequency cycle; m+Indicating the momentum transfer due to the collision. In the first radio frequency period, the instantaneous electric field E of the initialized sheath region is adoptedshAnd acquiring the instantaneous electric field of the sheath region of the previous radio frequency cycle when iteration is carried out from the second radio frequency cycle.
The sheath parameter determining module 208 is configured to solve a poisson equation and a current balance equation by using the charged particle density of the sheath region in the current radio frequency cycle to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current radio frequency cycle; the charged particle density of the sheath region includes an electron density of the sheath region, a negative ion density of the sheath region, and a positive ion density of the sheath region.
The sheath parameter determining module 208 specifically includes: and the electron density determining unit is used for solving the Boltzmann distribution by using the electron temperature of the current radio frequency period and the electric potential of the sheath region of the previous radio frequency period to determine the electron density of the sheath region of the current radio frequency period. And in the first radio frequency period, acquiring the potential of the sheath region in the previous radio frequency period by adopting the potential of the initialized sheath region from the beginning of the second radio frequency period.
And the negative ion density determining unit is used for solving the Boltzmann distribution by utilizing the preset negative ion temperature and the electric potential of the sheath region of the previous radio frequency period to determine the negative ion density of the sheath region of the current radio frequency period. The negative ion temperature is generally set to room temperature.
And the instantaneous electric field determining unit is used for solving a Poisson equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain the instantaneous electric field of the sheath region in the current radio frequency period. The method is specifically used for substituting the charged particle density of the sheath region in the current radio frequency period into the Poisson equation, solving to obtain the potential of the sheath region in the current radio frequency period, and substituting the potential of the sheath region in the current radio frequency period into a formula
Figure BDA0002622369160000251
Solving to obtain the instantaneous electric field E of the sheath region of the current radio frequency periodsh；
Figure BDA0002622369160000252
Representing the gradient of the potential.
A sheath potential drop determination unit for determining a potential drop of the sheath by solving a current balance equation Ii+Ie+Id＝Ibiassin(ωbiast) and obtaining the sheath potential drop of the current radio frequency period.
In the formula,
Figure BDA0002622369160000253
Iirepresents the ion flow; i iseRepresenting the electron flow; i isdRepresents a displacement current; i isbiasRepresenting the magnitude of the current of the bias supply; omegabiasRepresenting the frequency of the bias power supply; t represents time; csRepresenting the equivalent capacitance of the sheath region; vsRepresents the sheath potential drop; a represents the area of the lower polar plate; dsRepresenting the sheath thickness of the previous rf cycle.
The first determining module 209 is configured to determine whether a difference between densities of the same charged particle in two adjacent radio frequency periods is smaller than a corresponding density preset threshold, so as to obtain a first determination result. Judging whether the difference value between the positive ion densities of two adjacent radio frequency periods is smaller than a preset positive ion density threshold value, judging whether the difference value between the negative ion densities of two adjacent radio frequency periods is smaller than a preset negative ion density threshold value, and judging whether the difference value between the electron densities of two adjacent radio frequency periods is smaller than a preset electron density threshold value to obtain a first judgment result.
And the ion energy angle distribution determining module 210 is configured to determine, by using the positive ion density of the sheath region, the positive ion velocity of the sheath region, and the instantaneous electric field, the ion energy distribution and the angle distribution of the ions incident to the lower plate by using an ion monte carlo method, when the first determination result is yes. The method is particularly used for uniformly scattering a large number of positive ions on the boundary of the sheath layer according to the preset proportion of the number of the particles to be scattered and the positive ion density by utilizing the positive ion density of the sheath layer region in the current radio frequency period. And updating the position and the speed of the positive ions by utilizing the instantaneous electric field of the sheath region in the current radio frequency period, the Newton's equation of motion of the particles and the collision process of the positive ions in the sheath layer. When all positive ions reach the lower electrode, counting the ratio of ions bombarded to the surface of the lower polar plate in each energy interval to obtain energy distribution; the angular distribution is similar to the energy distribution.
And a returning module 211, configured to, if the first determination result is negative, calculate a positive ion flux of the main plasma region and a negative ion flux of the main plasma region in the current radio frequency cycle by using the charged particle density of the main plasma region and the bipolar electric field, calculate a sheath thickness in the current radio frequency cycle by using the charged particle density of the sheath region, calculate a bias source deposition power of the sheath region in the current radio frequency cycle by using the sheath thickness and the sheath potential drop in the current radio frequency cycle, update the next radio frequency cycle as the current radio frequency cycle, and execute the dividing module.
The returning module 211 specifically includes: an ion flux determination unit for solving a drift-diffusion approximation expression of the flux using the charged particle density and the bipolar electric field of the main plasma region of the current radio frequency cycle
Figure BDA0002622369160000261
And
Figure BDA0002622369160000262
and obtaining the positive ion flux of the main plasma region and the negative ion flux of the main plasma region of the current radio frequency period.
In the formula, gamma+Represents the positive ion flux of the main plasma region; mu.s+Represents the mobility of positive ions in the main plasma region; n is+Represents the positive ion density of the main plasma region; esRepresenting a bipolar electric field; d+Represents the diffusion coefficient of positive ions in the main plasma region;
Figure BDA0002622369160000263
a positive ion density gradient representing the main plasma region; gamma-shaped-Represents the negative ion flux of the main plasma region; mu.s-Represents the mobility of negative ions in the main plasma region; n is-Represents the negative ion density of the main plasma region; d-Representing negative ions of the main plasma zoneA diffusion coefficient;
Figure BDA0002622369160000264
representing the negative ion density gradient of the main plasma region.
And the sheath thickness determining unit is used for calculating the sheath thickness of the current radio frequency period by using the charged particle density of the sheath region of the current radio frequency period. The ratio of the electron density to the positive ion density is 1, the position closest to the lower polar plate is the boundary of the sheath layer, and the distance between the position and the lower polar plate is the thickness of the sheath layer.
A bias source deposition power determining unit for solving the formula by using the sheath thickness and the sheath potential drop of the current RF cycle
Figure BDA0002622369160000265
And
Figure BDA0002622369160000266
and obtaining the deposition power of the bias source of the sheath region in the current radio frequency period. Specifically, the equivalent capacitance of the sheath is calculated by utilizing the sheath thickness of the current radio frequency period
Figure BDA0002622369160000271
In the formula, epsilon0Represents a dielectric constant in a vacuum; a represents the area of the lower polar plate; dsThe sheath thickness is indicated. Equivalent capacitance C using sheathsAnd sheath potential drop V of the current radio frequency cyclesCalculating the displacement current
Figure BDA0002622369160000272
Will shift the current IdSubstitution formula
Figure BDA0002622369160000273
And obtaining the deposition power of the bias source of the sheath region in the current radio frequency period.
In the formula, PbiasRepresenting a bias source deposition power; t represents a radio frequency period; i isiRepresents the ion flow; i iseRepresenting the electron flow; t represents time.
At present, the existing simulation method almost adopts a 'strong coupling' technology, so that the numerical simulation process is slower. The electric field E and the magnetic field B depend on the plasma current J, as known from maxwell's system of equations, for example. In the fluid mechanics simulation method, the plasma current J is determined by the plasma dynamics equation, which in turn depends on the electromagnetic field. That is, the electromagnetic field is closely related to the state of the plasma. Therefore, in the existing simulation methods, the electromagnetic field and the plasma state are all related in a 'strong coupling' manner, that is, the electron density is required to be called when the electromagnetic field is solved at each time step, and the electromagnetic field is required to be called when the electron density is solved at each time step, thereby consuming a great deal of calculation time. The invention adopts harmonic wave approximation and local density approximation, and the strong coupling between the electromagnetic field and the plasma is weak coupling, thus accelerating the simulation speed; the stability of the poisson equation directly restricts the operation of the whole simulation method, so that the operation of the poisson equation in the simulation method consumes a large amount of time. The invention divides the main plasma area and the sheath layer into blocks for processing, and adopts bipolar diffusion approximation in the main plasma area, thereby avoiding solving the Poisson equation, accelerating the operation efficiency of the simulation method and the system, and improving the simulation speed. In addition, the existing simulation methods are almost all studied in a block mode, that is, only a certain part of the plasma source is concerned. For example, some simulation methods assume a fixed power deposition profile without self-righting the electromagnetic field distribution; some simulation methods only consider the influence of a coil power supply, but ignore the influence of a bias power supply commonly adopted in the actual process; some simulation methods only study the characteristics of the sheath layer near the lower polar plate, but do not consider the characteristics of the main plasma region; some simulation methods only study the characteristics of the main plasma region, neglect the characteristics of the plasma on the surface of the lower electrode plate, and actually the characteristics of the plasma on the surface of the upper electrode plate and the lower electrode plate directly influence the process. The invention has comprehensive functions, and the electromagnetic field, the main plasma region and the sheath region are coupled together through the step 101-.
The invention establishes a complete multi-field coupling model facing to the actual industrial plasma source, comprehensively considers the influence of different external factors such as chamber structure, bias power supply frequency, power, discharge pressure, discharge gas and the like on the actual discharge process, provides a new coupling technology according to the time characteristic scale and the space characteristic scale of different physical quantity changes, changes 'strong coupling' into 'weak coupling', accelerates the numerical simulation speed, and deeply discloses the control behavior of the external factors on the discharge process, plasma parameters and space uniformity so as to adapt to the numerical simulation requirement of the industrial plasma source. In addition, through a multi-field coupling sheath model, the space-time distribution of various physical quantities on the surface of the lower polar plate can be obtained, and the influence of the instantaneous oscillation of the sheath region on the characteristics of the main plasma region is automatically and properly included, namely, the deposition power of a bias source is obtained through calculation of the sheath model, the deposition power of the bias source is used for calculating the electron temperature of the main plasma region, and the electron temperature can pass through SjThe density of various charged particles is influenced, and further the influence of the sheath layer on the main plasma area is reflected. The multi-field coupling model comprises an electromagnetic field, a temperature field, a density field and the like.
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 provided only to help understand the method and the core concept 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, the present disclosure should not be construed as limiting the invention.

Claims (8)

1. A plasma discharge process simulation method is applied to a plasma chamber, and the plasma chamber comprises: the device comprises a radio frequency source, a coil, a dielectric window, a chamber side wall, a lower polar plate, a capacitor and a bias power supply;
the radio frequency source is connected with the coil; the coil is placed on the upper surface of the dielectric window; the side surface of the dielectric window is connected with the side wall of the chamber, and the side wall of the chamber is grounded; the lower polar plate is positioned below the dielectric window, and the lower surface of the dielectric window is opposite to the upper surface of the lower polar plate; a plasma region is formed in the space among the dielectric window, the side wall of the cavity and the lower polar plate; the first end of the capacitor is connected with the lower surface of the lower polar plate, the second end of the capacitor is connected with one end of the bias power supply, and the other end of the bias power supply is grounded;
the plasma discharge process simulation method comprises the following steps:
acquiring the sheath thickness calculated in the previous radio frequency period, and dividing the plasma region in the current radio frequency period into a main plasma region and a sheath region according to the sheath thickness;
acquiring the electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region and the positive ion velocity of the sheath region which are calculated in the last radio frequency period;
solving a continuity equation of ions and a quasi-neutral condition of the plasma by using the electron temperature of the previous radio frequency period, the positive ion flux of the main plasma region of the previous radio frequency period and the negative ion flux of the main plasma region of the previous radio frequency period to obtain the charged particle density of the main plasma region of the current radio frequency period; the charged particle density of the main plasma region comprises a positive ion density of the main plasma region, a negative ion density of the main plasma region and an electron density of the main plasma region; the method specifically comprises the following steps: determining the generation and loss source terms of j-th ions caused by collision in the main plasma region by using the electron temperature and the type of the reaction gas in the plasma chamber; utilizing positive ion flux of the main plasma region, the mainSolving the continuity equation of ions by using the negative ion flux of the plasma region and the generation and loss source terms of the jth ion caused by collision in the main plasma region
Figure FDA0003080331300000021
Determining the positive ion density and the negative ion density of the main plasma region of the current radio frequency period; solving the quasi-neutral condition of the plasma by using the positive ion density of the main plasma region and the negative ion density of the main plasma region
Figure FDA0003080331300000022
Determining an electron density of the main plasma region of the current radio frequency cycle; in the formula, njDenotes the density of positive ions or negative ions in the main plasma region, and when j is + nj＝n+Represents the positive ion density of the main plasma region, and n isj＝n-Represents a negative ion density of the main plasma region; t represents time;
Figure FDA0003080331300000023
represents the divergence of the positive ion flux or the divergence of the negative ion flux of the main plasma region; sjA source term representing the generation and loss of the jth ion caused by collisions in the main plasma region; e represents a elementary charge; n iseRepresents an electron density of the main plasma region; z+Represents the charge amount charged by positive ions of the main plasma region; z-Represents the charge amount charged by the negative ions in the main plasma region;
solving a Helmholtz equation of the main plasma region by using the electron density of the main plasma region to obtain the deposition power of the radio-frequency electromagnetic field in the main plasma region in the current radio-frequency period, which specifically comprises the following steps: converting the Maxwell equation set of the main plasma region into a Helmholtz equation; solving the Helmholtz equation by using the electron density of the main plasma region to obtain the radio frequency in the main plasma region of the current radio frequency periodAn electric field; using the radio frequency electric field and formula
Figure FDA0003080331300000024
Determining the deposition power of the radio frequency electromagnetic field in the main plasma region of the current radio frequency period; in the formula, PindRepresenting a deposition power of the radio frequency electromagnetic field; sigmapIndicating the plasma conductivity; erRepresenting a component of the radio frequency electric field in a radial direction of the plasma region; ezRepresenting a component of the radio frequency electric field in an axial direction of the plasma region; eθRepresenting the component of the radio frequency electric field in the theta direction of the plasma region, theta represents the angle around the symmetry axis of the plasma region;
solving drift diffusion approximation and bipolar diffusion approximation of the charged particle flux of the main plasma region by using the charged particle density of the main plasma region to obtain a bipolar electric field in the main plasma region of the current radio frequency period; the charged particle flux of the main plasma region comprises a positive ion flux of the main plasma region, a negative ion flux of the main plasma region and an electron flux of the main plasma region;
solving an electron energy equation by utilizing the bipolar electric field, the deposition power of the radio-frequency electromagnetic field and the bias source deposition power of the sheath region of the previous radio-frequency period to obtain the electron temperature of the main plasma region of the current radio-frequency period;
solving a continuity equation of positive ions and a momentum equation of the positive ions by using the electron temperature of the current radio frequency period and the positive ion velocity of the sheath region of the previous radio frequency period to obtain the positive ion density of the sheath region and the positive ion velocity of the sheath region of the current radio frequency period;
solving a Poisson equation and a current balance equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current radio frequency period; the charged particle density of the sheath region comprises an electron density of the sheath region, a negative ion density of the sheath region, and a positive ion density of the sheath region;
judging whether the difference value between the densities of the same charged particles in two adjacent radio frequency periods is smaller than a corresponding density preset threshold value or not to obtain a first judgment result;
if the first judgment result is yes, determining the energy distribution and the angle distribution of ions incident to the lower polar plate by using the positive ion density of the sheath region, the positive ion speed of the sheath region and the instantaneous electric field by adopting an ion Monte Carlo method;
if the first judgment result is negative, calculating the positive ion flux of the main plasma region and the negative ion flux of the main plasma region in the current radio frequency period by using the charged particle density of the main plasma region and the bipolar electric field, calculating the sheath thickness of the current radio frequency period by using the charged particle density of the sheath region, calculating the bias source deposition power of the sheath region in the current radio frequency period by using the sheath thickness of the current radio frequency period and the sheath potential drop, updating the next radio frequency period to the current radio frequency period, returning to the step of obtaining the sheath thickness calculated in the previous radio frequency period, and dividing the plasma region in the current radio frequency period into the main plasma region and the sheath region according to the sheath thickness.
2. The method according to claim 1, wherein the solving drift diffusion approximation and bipolar diffusion approximation of the charged particle flux of the main plasma region by using the charged particle density of the main plasma region to obtain the bipolar electric field in the main plasma region of the current rf cycle comprises:
solving drift diffusion and bipolar diffusion approximations of the charged particle flux of the main plasma region using the charged particle density of the main plasma region
Figure FDA0003080331300000041
Obtaining a bipolar electric field E of the main plasma region of the current RF cycles；
In the above formula,. mu.+Represents the mobility of positive ions of the main plasma region; n is+Represents a positive ion density of the main plasma region; d+Represents a diffusion coefficient of positive ions of the main plasma region;
Figure FDA0003080331300000042
a positive ion density gradient representing the main plasma region; mu.s-Represents the mobility of negative ions in the main plasma region; n is-Represents a negative ion density of the main plasma region; d-Represents a diffusion coefficient of negative ions of the main plasma region;
Figure FDA0003080331300000043
a negative ion density gradient representing the main plasma region; mu.seRepresents the mobility of electrons of the main plasma region; n iseRepresents an electron density of the main plasma region; deA diffusion coefficient representing electrons of the main plasma region;
Figure FDA0003080331300000051
representing an electron density gradient of the main plasma region.
3. The method for simulating a plasma discharge process according to claim 1, wherein the solving an electron energy equation using the bipolar electric field, the deposition power of the rf electromagnetic field, and the deposition power of the bias source in the sheath region of the previous rf cycle to obtain the electron temperature of the main plasma region of the current rf cycle includes:
solving an electron energy equation by using the bipolar electric field, the deposition power of the radio frequency electromagnetic field and the deposition power of the bias source of the sheath region of the previous radio frequency period
Figure FDA0003080331300000052
Obtaining the main plasma of the current RF cycleElectron temperature of the zone;
in the formula, neRepresents an electron density of the main plasma region; t iseRepresents an electron temperature of the main plasma region; t represents time;
Figure FDA0003080331300000053
a divergence representing electron fluence; e represents a elementary charge; esRepresenting the bipolar electric field; gamma-shapedeRepresents the electron flux of the main plasma region; pindRepresents the deposition power, P, of the radio-frequency electromagnetic fieldbiasRepresenting the bias source deposition power; weIndicating the energy loss caused by the collision.
4. The method for simulating a plasma discharge process according to claim 1, wherein the solving a continuity equation of positive ions and a momentum equation of positive ions by using the electron temperature of the current rf cycle and the positive ion velocity of the sheath region of the previous rf cycle to obtain the positive ion density of the sheath region and the positive ion velocity of the sheath region of the current rf cycle includes:
determining the positive ion flux of the sheath region of the current rf cycle using the positive ion velocity of the sheath region of the previous rf cycle;
determining a source term for generation and loss of positive ions caused by collisions in the sheath region of the current radio frequency cycle using the electron temperature of the current radio frequency cycle and the type of reactant gas in the plasma chamber;
solving a positive ion continuity equation using the positive ion flux of the sheath region of the current radio frequency cycle and the collisional-induced positive ion generation and loss source terms in the sheath region
Figure FDA0003080331300000061
Obtaining the positive ion density of the sheath region of the current radio frequency cycle;
solving for positive ion density using the sheath region of the current RF cycleEquation of momentum of ions
Figure FDA0003080331300000062
Obtaining the positive ion velocity of the sheath region of the current radio frequency cycle;
n 'in the formula'+Representing the positive ion density of the sheath region; t represents time;
Figure FDA0003080331300000063
represents divergence of positive ion flux of the sheath region; s'+Representing the source terms of the generation and loss of positive ions caused by collisions in the sheath region; u'+Represents the velocity of the positive ions;
Figure FDA0003080331300000064
is represented by (n'+u'+u'+) Divergence of (d);
Figure FDA0003080331300000065
a positive ion gas pressure gradient indicative of the sheath region; p'+A positive ion gas pressure indicative of the sheath region; m'+Mass of positive ions representing the sheath region; z'+Representing the amount of charge charged by positive ions of the sheath region; e represents a elementary charge; eshA transient electric field indicative of the sheath region; m+Indicating the momentum transfer due to the collision.
5. The method for simulating a plasma discharge process according to claim 1, wherein the solving a poisson equation and a current balance equation by using the charged particle density of the sheath region in the current rf cycle to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current rf cycle includes:
solving a Poisson equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain an instantaneous electric field of the sheath region in the current radio frequency period;
by solving the current balance equation Ii+Ie+Id＝Ibias sin(ωbiast), obtaining the sheath potential drop of the current radio frequency period;
in the formula,
Figure FDA0003080331300000071
Iirepresents the ion flow; i iseRepresenting the electron flow; i isdRepresents a displacement current; i isbiasRepresenting a current magnitude of the bias power supply; omegabiasRepresenting a frequency of the bias power supply; t represents time; csRepresenting an equivalent capacitance of the sheath region; vsRepresenting the sheath potential drop; epsilon0Represents a dielectric constant in a vacuum; a represents the area of the lower polar plate; dsThe sheath thickness is indicated.
6. The method for simulating a plasma discharge process according to claim 1, wherein the calculating a positive ion flux of the main plasma region and a negative ion flux of the main plasma region of the current rf cycle using the charged particle density of the main plasma region and the bipolar electric field, calculating a sheath thickness of the current rf cycle using the charged particle density of the sheath region, and calculating a bias source deposition power of the sheath region of the current rf cycle using the sheath thickness of the current rf cycle and the sheath potential drop comprises:
solving a formula using the charged particle density of the main plasma region and the dipole electric field
Figure FDA0003080331300000072
And formula
Figure FDA0003080331300000073
Obtaining the positive ion flux of the main plasma region and the negative ion flux of the main plasma region in the current radio frequency period;
in the formula, gamma+Represents a positive ion flux of the main plasma region; mu.s+Represents the mobility of positive ions of the main plasma region; n is+Represents a positive ion density of the main plasma region; esRepresenting a bipolar electric field; d+Represents a diffusion coefficient of positive ions of the main plasma region;
Figure FDA0003080331300000074
a positive ion density gradient representing the main plasma region; gamma-shaped-Represents a negative ion flux of the main plasma region; mu.s-Represents the mobility of negative ions in the main plasma region; n is-Represents a negative ion density of the main plasma region; d-Represents a diffusion coefficient of negative ions of the main plasma region;
Figure FDA0003080331300000081
a negative ion density gradient representing the main plasma region;
calculating the sheath thickness of the current radio frequency cycle by using the charged particle density of the sheath region;
solving a formula by using the sheath thickness of the current radio frequency period and the sheath potential drop
Figure FDA0003080331300000082
And
Figure FDA0003080331300000083
obtaining the bias source deposition power of the sheath region in the current radio frequency period;
in the formula, PbiasRepresenting a bias source deposition power; t represents a radio frequency period; vsRepresenting the sheath potential drop; i isiRepresents the ion flow; i iseRepresenting the electron flow; i isdRepresents a displacement current; t represents time; csRepresenting the equivalent capacitance of the sheath; epsilon0Represents a dielectric constant in a vacuum; a represents the area of the lower polar plate; dsThe sheath thickness is indicated.
7. A plasma discharge process simulation system, applied to a plasma chamber, comprising: the device comprises a radio frequency source, a coil, a dielectric window, a chamber side wall, a lower polar plate, a capacitor and a bias power supply;
the radio frequency source is connected with the coil; the coil is placed on the upper surface of the dielectric window; the side surface of the dielectric window is connected with the side wall of the chamber, and the side wall of the chamber is grounded; the lower polar plate is positioned below the dielectric window, and the lower surface of the dielectric window is opposite to the upper surface of the lower polar plate; a plasma region is formed in the space among the dielectric window, the side wall of the cavity and the lower polar plate; the first end of the capacitor is connected with the lower surface of the lower polar plate, the second end of the capacitor is connected with one end of the bias power supply, and the other end of the bias power supply is grounded;
the plasma discharge process simulation system includes:
the dividing module is used for acquiring the sheath thickness calculated in the previous radio frequency period and dividing the plasma region of the current radio frequency period into a main plasma region and a sheath region according to the sheath thickness;
the acquisition module is used for acquiring the electron temperature, the positive ion flux of the main plasma region, the negative ion flux of the main plasma region, the bias source deposition power of the sheath region and the positive ion speed of the sheath region which are calculated in the last radio frequency period;
the charged particle density determining module is used for solving a continuity equation of ions and a quasi-neutral condition of the plasma by using the electron temperature of the previous radio frequency period, the positive ion flux of the main plasma region of the previous radio frequency period and the negative ion flux of the main plasma region of the previous radio frequency period to obtain the charged particle density of the main plasma region of the current radio frequency period; the charged particle density of the main plasma region includes a positive ion density of the main plasma region, a negative ion density of the main plasma region, and an electron density of the main plasma region, and specifically includes: determining the j (th) species caused by collisions in the main plasma region using the electron temperature and the type of reactant gas in the plasma chamberThe generation and loss source terms of ions; solving the continuity equation of the ions by using the positive ion flux of the main plasma region, the negative ion flux of the main plasma region and the generation and loss source terms of the j-th ions caused by collision in the main plasma region
Figure FDA0003080331300000091
Determining the positive ion density and the negative ion density of the main plasma region of the current radio frequency period; solving the quasi-neutral condition of the plasma by using the positive ion density of the main plasma region and the negative ion density of the main plasma region
Figure FDA0003080331300000092
Determining an electron density of the main plasma region of the current radio frequency cycle; in the formula, njDenotes the density of positive ions or negative ions in the main plasma region, and when j is + nj＝n+Represents the positive ion density of the main plasma region, and n isj＝n-Represents a negative ion density of the main plasma region; t represents time;
Figure FDA0003080331300000093
represents the divergence of the positive ion flux or the divergence of the negative ion flux of the main plasma region; sjA source term representing the generation and loss of the jth ion caused by collisions in the main plasma region; e represents a elementary charge; n iseRepresents an electron density of the main plasma region; z+Represents the charge amount charged by positive ions of the main plasma region; z-Represents the charge amount charged by the negative ions in the main plasma region;
the deposition power determining module is configured to solve a helmholtz equation of the main plasma region by using the electron density of the main plasma region to obtain the deposition power of the radio-frequency electromagnetic field in the main plasma region in the current radio-frequency cycle, and specifically includes: converting the Maxwell equation set of the main plasma region into a Helmholtz equation; using said master, etcSolving the Helmholtz equation by the electron density of the ion region to obtain a radio frequency electric field in the main plasma region of the current radio frequency period; using the radio frequency electric field and formula
Figure FDA0003080331300000101
Determining the deposition power of the radio frequency electromagnetic field in the main plasma region of the current radio frequency period; in the formula, PindRepresenting a deposition power of the radio frequency electromagnetic field; sigmapIndicating the plasma conductivity; erRepresenting a component of the radio frequency electric field in a radial direction of the plasma region; ezRepresenting a component of the radio frequency electric field in an axial direction of the plasma region; eθRepresenting the component of the radio frequency electric field in the theta direction of the plasma region, theta represents the angle around the symmetry axis of the plasma region;
a bipolar electric field determination module, configured to solve drift diffusion approximation and bipolar diffusion approximation of the charged particle flux of the main plasma region by using the charged particle density of the main plasma region, so as to obtain a bipolar electric field in the main plasma region in the current radio frequency cycle; the charged particle flux of the main plasma region comprises a positive ion flux of the main plasma region, a negative ion flux of the main plasma region and an electron flux of the main plasma region;
the electronic temperature determination module is used for solving an electronic energy equation by utilizing the bipolar electric field, the deposition power of the radio-frequency electromagnetic field and the bias source deposition power of the sheath region of the previous radio-frequency period to obtain the electronic temperature of the main plasma region of the current radio-frequency period;
the positive ion density and velocity determination module is used for solving a continuity equation of positive ions and a momentum equation of the positive ions by utilizing the electron temperature of the current radio frequency period and the positive ion velocity of the sheath region of the previous radio frequency period to obtain the positive ion density of the sheath region of the current radio frequency period and the positive ion velocity of the sheath region;
the sheath parameter determining module is used for solving a Poisson equation and a current balance equation by utilizing the charged particle density of the sheath region in the current radio frequency period to obtain an instantaneous electric field and a sheath potential drop of the sheath region in the current radio frequency period; the charged particle density of the sheath region comprises an electron density of the sheath region, a negative ion density of the sheath region, and a positive ion density of the sheath region;
the first judgment module is used for judging whether the difference value between the densities of the same charged particles in two adjacent radio frequency periods is smaller than a corresponding density preset threshold value or not to obtain a first judgment result;
the ion energy angle distribution determining module is used for determining the ion energy distribution and the angle distribution of the incident ions to the lower polar plate by using the positive ion density of the sheath region, the positive ion velocity of the sheath region and the instantaneous electric field and adopting an ion Monte Carlo method when the first judgment result is yes;
and a returning module, configured to, when the first determination result is negative, calculate a positive ion flux of the main plasma region and a negative ion flux of the main plasma region in the current radio frequency cycle by using the charged particle density of the main plasma region and the bipolar electric field, calculate a sheath thickness in the current radio frequency cycle by using the charged particle density of the sheath region, calculate a bias source deposition power of the sheath region in the current radio frequency cycle by using the sheath thickness in the current radio frequency cycle and the sheath potential drop, update a next radio frequency cycle to the current radio frequency cycle, and execute the dividing module.
8. The plasma discharge process simulation system of claim 7, wherein the bipolar electric field determination module specifically comprises:
a bipolar electric field determination unit for solving a drift diffusion approximation and a bipolar diffusion approximation of the charged particle flux of the main plasma region using the charged particle density of the main plasma region
Figure FDA0003080331300000121
Obtaining a bipolar charge of the main plasma region of the current RF cycleField Es；
In the above formula,. mu.+Represents the mobility of positive ions of the main plasma region; n is+Represents a positive ion density of the main plasma region; d+Represents a diffusion coefficient of positive ions of the main plasma region;
Figure FDA0003080331300000122
a positive ion density gradient representing the main plasma region; mu.s-Represents the mobility of negative ions in the main plasma region; n is-Represents a negative ion density of the main plasma region; d-Represents a diffusion coefficient of negative ions of the main plasma region;
Figure FDA0003080331300000123
a negative ion density gradient representing the main plasma region; mu.seRepresents the mobility of electrons of the main plasma region; n iseRepresents an electron density of the main plasma region; deA diffusion coefficient representing electrons of the main plasma region;
Figure FDA0003080331300000124
representing an electron density gradient of the main plasma region.
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