JP2783174B2 - Simulation method of ion implantation impurity distribution - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method of simulating the distribution of ions implanted into a substrate by ion implantation in the depth direction, and more particularly to the method of simulating the distribution of ions implanted into a semiconductor substrate in a semiconductor manufacturing process. A method for performing a simulation.

[0002]

2. Description of the Related Art Ion implantation technology is widely used in processes such as the manufacture of semiconductor devices. When ions are implanted into a substrate, even if the energy of the implanted ions is constant, the distribution of ions in the substrate in the depth direction has a certain spread, and the ion concentration varies depending on the location. Therefore, if ion implantation is performed to form an impurity region of a semiconductor device, it is necessary to examine the ion distribution in the substrate after the implantation in advance, and obtain the simulation of the distribution of the implanted ions. Things are going on.

As one method of performing ion implantation simulation, an impurity profile after ion implantation is determined by an analytical expression using moments R _p , ΔR _p , β, and γ defined by equations (1) to (4). There is a way to ask. Where "A
nalysis and Simulation of semiconductor Devices ",
Based on the description of Springer-Verlag and Wien New York, the definition formula of each moment is shown.

[0004]

(Equation 1) Here, R is the depth from the substrate surface, and f (R) is

[0005]

(Equation 2) Is an impurity distribution in the depth direction (impurity in this specification refers to ions implanted into a substrate). R _p is a first-order moment amount, and represents an average range of the ion in the depth direction. ΔR _p is a second-order moment, which represents the standard deviation of the distribution in the depth direction, and γ is a third-order moment, which is an amount called skewness, which represents the distortion of the distribution. β is a fourth-order moment amount, which is called kurtosis or excess, which indicates the sharpness of the distribution near the maximum value.

According to this book, Gaussian distribution,
There are a joint Gaussian distribution and a Pearson distribution, and the Gaussian distribution is expressed as follows using only R _p and △ R _p .

[0007]

(Equation 3) The joint Gaussian distribution is expressed as follows using R _p , △ R _p and γ.

[0008]

(Equation 4) Here, σ ₁ , σ ₂ , and R _m are parameters that satisfy the following equation.

[0009]

(Equation 5) In addition, Pearson type distribution, using the R _p and △ R _p and γ and β, differential equations

[0010]

(Equation 6) Given by Where y, a, b ₀ , b ₂ , A

[0011]

(Equation 7) To be satisfied.

[0012] Among the above-mentioned moment, the method for determining the R _p and the standard deviation △ R _p projected range, and J. Lindhard and M. Scharff
HE Schiott, Mat. Fys. Medd. Dan. Vid. Selsk.
Vol. 33, No. 14, (1963), which is called LSS theory. Here, the basic concept of the LSS theory will be described.

It is assumed that ions incident on an amorphous substrate with energy E collide with atomic nuclei and electrons in the substrate and advance while losing energy, and stop at a distance R from the substrate surface. The probability is represented by P (R, E ), P (R, E) is

[0014]

(Equation 8) According to Here, N is the atomic density of the substrate, T is the energy lost by the ions colliding with the atoms in the substrate, and dσ (T) is the differential cross section.

The differential cross section dσ (T) is defined as the energy of ions before collision, the atomic weight of incident ions, the atomic weight of substrate atoms,
The mass of the nucleus of the incident ion, the mass of the nucleus of the substrate atom,
Since the incident ion is a function of the energy T given to the nuclei of the substrate atoms, P (R, E) is a function that depends on the material of the substrate, the ion species of the implanted ions, and the energy of the implanted ions. Further, when the energy of the incident ion is E, the above-described impurity distribution f (R) is given by P (R,
It can be considered to represent the equivalent of E).

By modifying equation (17), R _{p of} equation (1)
And △ R _p in equation (2) is

[0017]

(Equation 9) Given by Where S (E) is the stopping power of substrate atoms by nuclei and electrons.

[0018]

(Equation 10) And Ω ² (E) is

[0019]

[Equation 11] It is.

It is considered that the stopping power by the nucleus and the stopping power by the electron are independent of each other,

[0021]

(Equation 12) And Here, S _n (E) is the stopping power (nuclear stopping power) by the nucleus,

[0022]

(Equation 13)It is expressed as Where T_nMeans that the ions are
Energy to lose, dσ_nIs the scattering of ions and nuclei
Differential cross section at T_nIs a function of Also S _e(E)
Is the stopping power by electrons (electron stopping power).
It is determined as a quantity proportional to speed.

After all, in the LSS theory, R _p and ΔR _p are obtained by sequentially executing the following procedures 1 to 4. Procedure 1: Using the scattering cross section σ _n (T _n ) when ions are scattered to the nucleus, calculate the nuclear stopping power S _n (E) according to equation (23), and further stop the electron stopping power S _e (E ) Is determined as a quantity proportional to the speed. Step 2: Stopping power is the sum of nuclear stopping power and electron stopping power (22)
Calculate by formula. Step 3: For equation (21), the scattering cross section at the nucleus gives

[0024]

[Equation 14] And give. Step 4: By substituting equation (22) into equation (18) and substituting equations (22) and (24) into equation (19), R _p and △ R _p are determined.

JB Sanders describes "Ranges of projectile
s in amorphous materials ", Can.J. Phys., 46 , 445 (1
968) developed the LSS theory and showed a method of generally calculating higher order moments in addition to R _p and △ R _p .
Gibbons et al., JE Gibbons, WS Johnson and S.
W. Mylroie, "Projected Range Statistics (Semiconduc
tors and Related Materials, second edition) ", pp.
In 7-27, the method of JBSanders, is calculating γ the equation (1) R _p and (2) of △ R _p and (3).

When performing an ion implantation simulation based on the LSS theory, first, a moment is determined in advance by the LSS theory as a function of the ion species of the implanted ions, the implantation energy, and the material of the substrate, and the actual simulation is performed. With ion species and implantation energy,
When the material of the substrate is designated, a moment corresponding to the designation is given to equations (1) to (4) to obtain a profile after ion implantation.

[0027]

Conventional theories for calculating moments assume that the substrate is amorphous, as represented by the LSS theory. For this reason, the result of a simulation performed using the moment according to the conventional theory does not very well represent the impurity profile when ion implantation is performed on the crystal. The cause is considered to be a phenomenon called channeling that occurs when ions are implanted into the crystal. Channeling is a phenomenon in which atoms incident on a crystal at a specific angle penetrate deep into a substrate without being greatly scattered by nuclei because atoms are arranged in a regular structure in the crystal. On the other hand, such a channeling phenomenon does not occur in an amorphous substrate having no regular arrangement order of atoms.

Assuming that phosphorus (P) ions having an energy of 1000 keV are implanted into a silicon crystal,
Comparing the value of the moment calculated by Gibbons according to the LSS theory with the value of the moment obtained by actual ion implantation and measurement, according to the LSS theory, R _p = 1.
1231, △ R _p = 0.1827, γ = −1.007, whereas in actual measurement, R _p = 1.10446, △
R _p = 0.194635, γ = −0.3980 are obtained,
The value of the third-order moment γ is more than doubled.

In order to accurately obtain the moment of the impurity profile due to ion implantation into the crystal, it is a good method to actually measure the impurity profile, but the actual measurement is very expensive. When one measurement is performed by specifying the ion species, energy, and substrate material, typically about 1
It costs 10,000 yen. For example, 20 levels of energy for one ion species from 100 KeV to 2000 KeV,
In addition, the dose amount is from 1 × 10 ¹³ cm ⁻² to 1 × 10 ¹⁶ cm ^−2.
If you measure up to 4 levels, 100,000 yen x 20 levels x 4
It costs 8 million yen in total.

As a method for obtaining the distribution of the implanted ions without depending on the actual measurement, there is a method based on Monte Carlo simulation. According to the Monte Carlo simulation, it is possible to perform an ion implantation simulation in consideration of a crystal structure, to obtain an impurity distribution (profile) close to an actually measured value, and to obtain a good simulation result. However, the Monte Carlo simulation has a drawback that it takes a considerable time for one execution. For example, when a simulation of implanting phosphorus ions at an energy of 1000 KeV with 10,000 particles is performed, a workstation requires more than 10 hours. It takes calculation time. Therefore, performing the Monte Carlo simulation every time the impurity distribution is to be obtained requires an enormous amount of calculation time.

On the other hand, the time required for the ion implantation simulation based on the analytical formula using the moment is about several minutes to several tens of minutes when the same workstation is used. However, it is significant that the ion implantation simulation can be performed with high accuracy by the analytical formula using the moment.

An object of the present invention is to provide a method capable of accurately obtaining a moment of ion implantation into a crystal without actually measuring, and to provide a method capable of performing an accurate ion implantation simulation in a short time. It is in.

[0033]

Means for Solving the Problems] simulation method of ion implanted impurities of the first invention is a simulation method for determining the distribution in the depth direction of the ion implanted impurity ions into the substrate, Monte Carlo ions considering crystal structure A first step of performing an implantation simulation to calculate a profile of the ions, a second step of normalizing the profile with respect to a dose, and calculating a moment of distribution from the normalized profile; A third step of executing a simulation by an analytical expression for a profile after ion implantation using the moment amount.

According to a second aspect of the present invention, there is provided a method of simulating an ion-implanted impurity in a simulation method for obtaining a distribution of impurity ions implanted in a substrate in a depth direction. A first step of executing a simulation and calculating the profile of the impurity ion is compared with a profile obtained by the first step in which the profile of the impurity ion by the analytical formula is obtained. A second step of extracting a moment amount that minimizes an error between the two, and a third step of executing a simulation on the profile after ion implantation using the above-mentioned analytical expression using the moment amount.

In each of the above inventions, the third step can be repeatedly executed after the execution of the second step. In other words, the memory corresponding to multiple ion implantation levels
The amount of the weight is determined using the first step and the second step.
And then set the required ion implantation level
The third step can be performed using
You. Further, as the moment amount, one or more types selected from average range, standard deviation of distribution, skewness, and kurtosis can be used. An expression representing one or more distributions selected from a distribution and a Pearson-type distribution can be used.

[0036]

According to the present invention, the moment is obtained not from the LSS theory but from the result of Monte Carlo ion implantation simulation in consideration of the crystal structure.
Since the simulation is performed by applying the obtained moment amount to the analytical formula, the simulation is performed using the moment parameters close to the actual measurement, and the calculation time as a whole is greatly reduced when multiple simulations are performed. it can.

[0037]

Next, embodiments of the present invention will be described with reference to the drawings.

Embodiment 1 FIG. 1 is a flowchart showing a rough flow of an ion implantation simulation in Embodiment 1.

First, a Monte Carlo ion implantation simulation taking into account the crystal structure is performed to calculate the ion distribution (impurity distribution) after ion implantation (step 10).
1). Then, the obtained impurity distribution f (R) is

[0040]

(Equation 15) (Step 102). Next, the respective moments R _p , △ R _p , γ, and β are obtained by numerically integrating the above defined equations (1) to (4) using the standardized impurity distribution (step 103). ), Using the obtained moments R _p , △ R _p , γ, and β, and substituting these moments into an analytical expression to obtain an impurity distribution under desired conditions (step 104). As an analytical expression, for example,
An expression representing a Gaussian distribution, a joint Gaussian distribution, or a Pearson distribution is used. Then, it is determined whether to end or repeat the simulation (step 105).
If it is to be continued, the process returns to step 103 to obtain the ion distribution again, and if not, the process is terminated.

In this embodiment, if steps 101 to 103 are performed once, the moments R _p , △ R _p ,
Since γ and β are obtained, if the values of these moments are stored, then it is possible to execute the simulation any number of times only by executing step 104 only.

Next, a Monte Carlo simulation considering the crystal structure will be described. The method of Monte Carlo ion implantation simulation considering the crystal structure is described in M.Ha
ne et al., Masami Hane and Masao Fukuma, "Ion Imp
lantation Model Considering Crystal structure Effe
cts ", IEEE Transaction on Electron Devices", Vol.
37, No. 9, pp. 1959-1963, Septembwer 1990.

FIG. 2 is a diagram schematically showing a place where accelerated ions j are incident on a place (crystal) where atoms a to i are regularly arranged. Atoms are thermally oscillating, and in order to simulate this, a deviation from the equilibrium position due to the thermal oscillations is given by random numbers so that the existence probability of the atoms becomes a Gaussian distribution. The incident ion j is
It approaches the atom d first and is scattered by the atom d in order to exert both forces. The scattered ions j are
Changing the angle by θ _L and losing energy T,
The scattering angle θ _L and the energy T are uniquely determined if the distance p shown in FIG. 3 and the energy E of the incident ion 11 are determined if the ion species of the incident ion 11 and the type of the nucleus 12 are constant. Further, ions lose energy due to electrons, and the energy lost also depends on p and E in FIG.

An ion whose energy and direction have been changed by scattering loses energy by being scattered by another atom, but this process is repeated to determine the position of the ion when the ion energy finally becomes zero. Ask.
Further, this calculation is performed for ions of a specified number of particles (several thousands to tens of thousands), and finally, an impurity distribution in the crystal can be obtained.

FIG. 4 shows a processing procedure of a Monte Carlo ion implantation simulation in consideration of a crystal structure.
First, kinetic energy, direction, and position are set for one of the implanted ion particles (step 11).
1). Then, the position of the atom approaching the ion is calculated, and the position of the atom that scatters the ion is searched (step 112). Subsequently, the scattering angle of the ions due to the scattering of the atoms and the energy lost by the ions are calculated (step 113), and the energy, position, and direction of the scattered ion particles are re-established based on the scattering angles and the lost energy values. It is defined (step 114). Then, it is determined whether the ion energy is greater than 0 or 0 at that time (step 115). If not, the process returns to step 112 to consider the next scattering. If the energy is 0 in step 115, this ion particle has stopped, so the processing for this ion is finished, and it is determined whether or not the processing has been completed for ions of the specified number of particles (step 116). If there is an unprocessed ion, the process returns to step 111 to perform the same process for the next ion. If the process has been completed for all the particles, all the processes are completed. In this way, a Monte Carlo simulation taking into account the crystal structure is performed.

Next, an example of a simple method for obtaining a moment by numerical integration will be described.

It is assumed that a numerical sequence as shown in Table 1 has been obtained by Monte Carlo ion implantation simulation in consideration of the crystal structure. In this table, R represents the depth and f represents the impurity concentration at the depth R. For simplicity of description, the depth R is represented by five levels (R1 to R5).

[0048]

[Table 1] By applying this numerical sequence to equation (1) and performing numerical integration, R _p is obtained. In order to perform numerical integration, R × f is obtained as shown in Table 2 using R and f obtained in Table 1.

[0049]

[Table 2] FIG. 5 shows R and R × f in a graph.
Integrating equation (1) and the polygon ABCDEF of FIG.
Since finding the area of GHIJ is the same, use the quadratic quadrature formula,

[0050]

(Equation 16) By calculating, R _p can be calculated.
The other moments ΔR _p , γ, and β can be calculated in the same manner.

Although the first embodiment has been described above, when the impurity profile is obtained by an analytical expression using the moment extracted by the method shown in the first embodiment, the phosphorus ion is 5%.
In the high energy region of 00 keV or more, the profile given by the Gaussian distribution or the joint Gaussian distribution of the analytical expression agrees well with the profile obtained by Monte Carlo simulation. However, at a low energy of about 20 KeV, it does not match well. The reason is that the profile obtained by ion implantation at high energy is smooth as shown in FIG. 6 (a), and is similar to the Gaussian distribution and the joint Gaussian distribution of the analytical expression, and is easily expressed by these analytical expressions. On the other hand, it is considered that the profile in the case of low energy has a shape cut off on the surface as shown in FIG. 6 (b), and this shape cannot be expressed by an analytical expression. Example 2 shows a method for obtaining a moment that gives a profile that matches well with the profile obtained by Monte Carlo ion implantation simulation even in the case of low energy.

Embodiment 2 FIG. 7 is a flowchart showing a process in Embodiment 2. First, as in the case of the first embodiment, a Monte Carlo ion implantation simulation is performed in consideration of the crystal structure to obtain an ion profile after ion implantation (step 121). Then, using the obtained impurity profile, curve fitting is performed so that the distribution overlaps each of the Gaussian distribution, the joint Gaussian distribution, and the Pearson distribution, and a moment for each distribution is extracted by the least squares method. (Step 122). Then, an impurity distribution is determined by substituting the obtained moment (step 123).
Thereafter, it is determined whether or not to continue the simulation (step 124).
Is repeated, and if not, the entire process ends.

Here, the curve fitting and the least square method for the Gaussian distribution will be described. According to this method, the profile obtained by the Monte Carlo ion implantation simulation has a representative point a as shown in FIG.
₁ , a ₂ , a ₃ , a ₄ , a ₅ , and the moment R _p ,
The points on the Gaussian distribution when ΔR _p is given are b ₁ , b ₂ ,
Suppose that given by b ₃ , b ₄ , b ₅

[0054]

[Equation 17] Is a method of obtaining moments R _p and △ R _p such that the minimum value is obtained. In this method, if fitting is performed on a Gaussian distribution, R _p , △ R _p can be obtained. If fitting is performed on a joint Gaussian distribution, R _p , △ can be obtained.
R _p , γ can be obtained, and R _p , に 対 し て R _p , γ, β can be obtained by fitting a Pearson type distribution.

As described above, when the moment is determined by fitting, even if the profile to be fitted has a shape cut off on the surface, the profile obtained by substituting the obtained moment into the analytical expression is as follows: FIG.
As shown in the figure, in the region inside the substrate, the profile matches well with the Monte Carlo ion simulation profile.

Next, the results of comparing the moment extracted according to the present invention with parameters obtained from the conventional LSS theory when phosphorus is ion-implanted into a silicon crystal substrate at an energy of 1000 keV will be described.
According to the method of the present invention, as the moment, R _p = 1.1
185, ΔR _p = 0.169076, γ = −0.3405
87, and in the LSS theory, R _p = 1.1231, △
R _p = 0.12727 and γ = -1.007 were obtained. In the actual measurement, R _p = 1.1446, △ Rp = 0.19
4635, γ = −0.3980 was obtained. According to the method of the present invention, the value of the third moment γ is much closer to the actual measurement than the LSS theory. Regarding R _p , up to the second digit, both of the LSS theory and those obtained by the present invention agree with the actually measured values, but the values of the present invention are slightly closer to the actually measured values than those of the LSS theory. I have. Regarding ΔR _p , up to the first digit, both the LSS theory and the value obtained by the present invention agree with the actually measured value, but the value based on the LSS theory is slightly closer to the actually measured value than the present invention. In the LSS theory, since the value of the third-order moment γ is about three times different from the actually measured value, the method of the present invention generally extracts a moment parameter closer to the actually measured value than the LSS theory. It can be said that it can be done.

In the present invention, in order to obtain the moment, it is only necessary to start up one workstation for several hours to several tens of hours.
The subsequent ion-implantation simulation using the analytical formulas takes only a few minutes to several tens of minutes per calculation.
No additional costs are required other than the cost of these calculations.

[0058]

As described above, according to the present invention, instead of obtaining the moment from the LSS theory, the moment is obtained from the result of Monte Carlo ion implantation simulation in consideration of the crystal structure, and the obtained moment is converted into an analytical expression. Since the simulation is performed by applying the simulation, there is an effect that it is possible to perform the simulation with the same accuracy as the Monte Carlo ion implantation simulation in a shorter time.

[Brief description of the drawings]

FIG. 1 is a flowchart illustrating a processing procedure according to a first embodiment.

FIG. 2 is a diagram schematically showing a state in which ions are incident on a crystal.

FIG. 3 is a diagram illustrating a relationship between a collision coefficient p and a scattering angle θ _L.

FIG. 4 is a flowchart showing a processing procedure of a Monte Carlo ion implantation simulation in consideration of a crystal structure.

FIG. 5 is a graph showing the relationship between R and R × f.

6A is a diagram illustrating a depth distribution of an impurity by ion implantation at a high energy, and FIG. 6B is a diagram illustrating a depth distribution of an impurity by ion implantation at a low energy.

FIG. 7 is a flowchart illustrating a processing procedure according to a second embodiment.

FIG. 8 is a graph showing data and functions used in the least squares method.

FIG. 9 is a diagram showing a distribution obtained by Monte Carlo ion implantation simulation and a distribution obtained by an analytical expression.

[Explanation of symbols]

101-105, 111-116, 121-124
Steps

Claims (5)

(57) [Claims] 1. A simulation method for obtaining a distribution of impurity ions implanted into a substrate in a depth direction, wherein a Monte Carlo ion implantation simulation is performed in consideration of a crystal structure to calculate a profile of the impurity ions. Step, a second step of normalizing the profile with respect to the dose amount, and calculating a moment amount of the distribution from the standardized profile, and using the moment amount, for the profile after ion implantation, using an analytical expression. And a third step of performing a simulation. 2. A method for calculating a depth direction distribution of impurity ions implanted into a substrate, comprising: performing a Monte Carlo ion implantation simulation in consideration of a crystal structure to calculate a profile of the impurity ions. And a step of comparing the profile of the impurity ion obtained by the analytical expression with the profile obtained in the first step, and extracting a moment amount that minimizes an error between both profiles by a least square method. And a third step of performing a simulation on the profile after the ion implantation using the amount of moment by the analytical formula. 3. The mode corresponding to a plurality of ion implantation levels.
The first step and the second step.
Ion implantation, and then ion implantation
Simulation method of ion implantation impurity according to claim 1 or 2 run the third step using the moment of levels. 4. The ion according to claim 1, wherein the amount of moment is at least one selected from an average range, a standard deviation of distribution, a skewness, and a kurtosis. Simulation method for implanted impurities. 5. The ion according to claim 1, wherein the analytic expression is an expression representing one or more distributions selected from a Gaussian distribution, a joint Gaussian distribution, and a Pearson-type distribution. Simulation method for implanted impurities.