JP2933076B2 - Simulation method of sputtering equipment - 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 for simulating a sputtering apparatus, and more particularly to a method for simulating a sputtering apparatus for calculating the trajectory of sputtered particles of diatomic molecules.

[0002]

2. Description of the Related Art With the increase in the number of mounted elements in a semiconductor integrated circuit device, there is an urgent need to develop a technique for embedding a conductive film in a miniaturized contact hole.

[0003] In order to solve this problem, it is necessary to develop a sputtering apparatus. Simulation techniques have been developed to increase the efficiency of these developments. this is,
For example, H. IEDM by Yamada et al. (1994)
553-556 "Practical Monte Carlo Sputter Depo
sition Simulation with Quasi-Axis-Symmetrical (QAS)
As disclosed in “Approximation”, this is a method of calculating the trajectory in the device and calculating the shape using a string model. However, the collision calculation used here is
It uses collisions between atoms. Recently, TiN sputtering has become an important technique for barrier film growth. In order to support TiN, it is necessary to simulate a sputtering apparatus for diatomic molecules. A corresponding prior art of a sputtering apparatus simulation will be described with reference to a schematic flowchart shown in FIG.

[0004] First, TiN particles are generated from the surface of a TiN target at a probability proportional to the erosion distribution.
Uniform random numbers e1, e returning numbers in the range of 0 to 1 with the same probability
2, the horizontal angle Φ (Φ = 2π · e1) and the vertical angle θ {θ = acos ((e2) ^1/2 )} are obtained, and the emission energy Ee and the velocity Vs of the TiN molecule Vs = (2Ee / m) ^1/2 is determined by a rejection method using uniform random numbers e3 and e4 so as to follow the Thompson distribution. still,
Here, the uniform random number is a random number that returns a number in the range of 0 to 1 with the same probability.

Further, the velocity vg of the N ₂ molecule according to the Maxwell distribution of the gas temperature of the apparatus is obtained by the Box-Muller method using the uniform random numbers e5 and e6 as shown in equations 1 and 2 to obtain the TiN molecule and the N ₂ molecule. Relative velocity vrel = vs
-Vg is obtained by vector calculation.

[0006]

Vg = (− 2 · ln (e5) · (kT / Mg))
^1/2 cos (2π · e6)

[0007]

Vg ′ = (− 2 · ln (e5) · (kT / M
g)) ^1/2 sin (2π · e6) Here, vg ′ is a value that is simultaneously determined due to the properties of the Box-Muller method, and is used in the next calculation of the velocity of the N ₂ molecule.

Next, assuming a two-body potential between Ti-N and between N-N, for example, a velocity level
By the et method, the Newton's equation is solved by classical mechanics to perform molecular dynamics calculations, and the position, velocity,
Calculate acceleration.

The initial values of the molecular dynamics calculation at this time are, as shown in FIG. 5, an angular velocity ω1, a frequency vl, and a frequency vl of TiN.
The angular velocity ω2 of N ₂ and the frequency v2 are calculated by Ma at the background gas temperature.
xwell distribution and uniform random numbers e7, e8, e9, e1
Box- using O, e11, e12, e13, e14
According to the Muller method, it is obtained as shown in Expressions 3 to 10.

[0010]

Ω1 = (− 2 · ln (e7) · (kT / Is))
^1/2 cos (2π · e8)

[0011]

Ω1 ′ = (− 2 · ln (e7) · (kT / I
s)) ^1/2 sin (2π · e8)

[0012]

V1 = (− 2 · ln (e9) · (kT / Ks))
^1/2 cos (2π ・ e1O)

[0013]

V1 ′ = (− 2 · ln (e9) · (kT / K
s)) ^1/2 sin (2π · e1O)

[0014]

Ω2 = (− 2 · ln (e11) · (kT / I)
g)) ^1/2 cos (2π · e12)

[0015]

Ω2 ′ = (− 2 · ln (e11) · (kT / I)
g)) ^1/2 sin (2π · e12)

[0016]

## EQU9 ## v2 = (− 2 · ln (e13) · (kT / K
g)) ^1/2 cos (2π · e14)

[0017]

V2 ′ = (− 2 · ln (e13) · (kT / K
g)) ^1/2 sin (2π · e14) where ω1 ′, v1 ′, ω2 ′, and v2 ′ are Box−
Since it is a value obtained simultaneously due to the properties of the Muller method, it is used as an initial value for calculating the molecular orbital of the next orbital.

The distance L between the TiN molecule and the N ₂ molecule is L
Is set to, for example, 30 A, which is sufficiently large with respect to the cut-off distance of the two-body potential, to fix the N ₂ molecule and give the TiN molecule a relative velocity Vrel. Also, the collision radius bma
The value of x is larger than the cut-off distance of the two-body potential, for example, about 15 A, and is obtained from the collision parameter b = bmax · e15 using the uniform random number e15.

Using these initial values, the position, velocity, and acceleration of the TiN molecule at each time are recorded at time intervals of about 2 fsec by the molecular dynamics method, and the traveling directions before and after the collision as viewed from the center of gravity are recorded. Calculate the scattering angle χ and calculate the Ti
From the translation energy Epb of the N molecule and the translation energy Epa of the TiN molecule after collision, the energy loss κ = Epa
/ Epb is calculated, and the direction and velocity of the TiN molecule after collision are calculated. Further, the mean free path λ0 is calculated from the collision radius bmax, a Poisson distribution is assumed, and a uniform random number e1 is calculated.
6, the distance dL = λ0 · vrel · | ln (e16) | until the TiN molecule causes the next collision is calculated.

By these calculations, the orbit of the TiN molecule after collision is calculated. In this manner, the trajectory of the particle is repeatedly calculated until the particle reaches the side wall of the apparatus or the wafer. Extract the orbit of TiN molecule in a certain area on the wafer,
The result is used to calculate the shape using a string model or the like.

[0021]

In the prior art, the scattering angle and the energy loss at the collision of diatomic molecules are calculated by the molecular dynamics method for each collision. To calculate the trajectory of 10,000 particles, 1
Even an EWS (engineering workstation) with a performance of 90 MIPS takes about three months and is not practical.

The object of the present invention is to carry out a scattering calculation of sputtered particles of diatomic molecules in consideration of the rotation of the molecules, etc.
It is another object of the present invention to reduce the calculation time required for calculating the trajectory of the sputtered particles in the apparatus.

[0023]

In order to achieve the above object, the present invention provides the following: (1) First, the scattering angle and energy loss of sputtered particles of diatomic molecules are considered by molecular dynamics method in consideration of the rotation of molecules and the like. And calculate. This calculation is performed with a plurality of relative energies and a sufficient number of times for each of a plurality of collision parameters to obtain a distribution, a collision radius is calculated from a collision parameter having a scattering angle of 0, and the relationship between the relative energy and the collision radius is calculated. Create the first table shown. Next, for each relative energy, the scattering angle and the maximum value of the energy loss are divided, the probability density is calculated from the number of scattering angles and the data of the energy loss entering each divided region, and the scattering angle is calculated for each collision parameter. And a second table showing the probability density of energy loss. (2) In the collision calculation between the sputtered particles and the surrounding gas molecules using the Monte Carlo method, first, the temperature Ma in the apparatus is measured.
The velocity of the gas particles is determined from the xwell distribution, and the relative velocity and relative energy are determined. Read the first table calculated by the molecular dynamics method to find the collision radius for the relative energy, read the probability density from the second table calculated by the molecular dynamics method, and use the rejection method to calculate the collision parameters, scattering angle, and energy loss. Decide. Further, the collision position of the sputtered particles is calculated from the Poisson distribution, and the trajectory of the sputtered particles after the collision is calculated. In this way, the trajectory of the particle is repeatedly calculated until the particle reaches the side wall of the apparatus or the wafer, extracted in a specific region on the wafer, and the deposition shape is calculated using a string model or the like.

[Action] Since molecular dynamics calculations are performed in consideration of the rotation of molecules and the probability densities of the scattering angle and the energy loss are tabulated, when calculating the orbits of the particles by the Monte Carlo method, only the table calculations are required. Is calculated using
The trajectory calculation of sputtered particles of atomic molecules can be performed in a short calculation time.

[0025]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS A sputtering apparatus simulation method according to an embodiment of the present simulation method will be described with reference to the drawings.

[Embodiment 1] FIG. 1 is a schematic flowchart of a sputtering apparatus simulation method according to the present invention.

In this embodiment, for example, 2
Simulate an atomic and molecular sputtering system.

First, assuming that the initial value of the collision parameter is 0, a two-body potential is assumed between Ti-N and N-N. For example, Newton's equation is solved classically by the velocity velret method, and molecular dynamics is solved. Perform calculations and calculate the position, velocity, and acceleration of atoms at each time.

[0029] At this time, the initial value of the molecular dynamics calculations, as shown in FIG. 2, the initial value L of the distance between TiN-N _2, 2
For example, 30A is set to be sufficiently large with respect to the cut-off distance of the body potential, N ₂ molecules are fixed, and TiN molecules are
When the relative energy to the N ₂ molecule is E, TiN
An initial velocity of the molecule V = (2E / m) ^1/2 was given. TiN
The initial angular velocity ω1 and initial frequency v1 of the molecule, and the initial angular velocity ω2 and initial frequency v2 of the N ₂ molecule are uniform random numbers e1 that return numbers in the range of 0 to 1 with the same probability from the Maxwell distribution at the background gas temperature. , E2, e3, e4, e5,
The values calculated by the Box-Muller method using e6, e7, and e8 as in Equations 11 to 18 were used.

[0030]

Ω1 = (− 2 · ln (e1) · (kT / I)
s)) ^1/2 cos (2π · e2)

[0031]

Ω1 ′ = (− 2 · ln (e1) · (kT / I)
s)) ^1/2 sin (2π · e2)

[0032]

## EQU13 ## v1 = (− 2 · ln (e3) · (kT / K
s)) ^1/2 cos (2π · e4)

[0033]

V1 ′ = (− 2 · ln (e3) · (kT / K
s)) ^1/2 sin (2π · e4)

[0034]

Ω2 = (− 2 · ln (e5) · (kT / I)
g)) ^1/2 cos (2π · e6)

[0035]

Ω2 ′ = (− 2 · ln (e5) · (kT / I)
g)) ^1/2 sin (2π · e6)

[0036]

## EQU17 ## v2 = (− 2 · ln (e7) · (kT / K
g)) ^1/2 cos (2π · e8)

[0037]

V2 ′ = (− 2 · ln (e7) · (kT / K
g)) ^1/2 sin (2π · e8) where ω1 ′, v1 ′, ω2 ′, and v2 ′ are Box−
Since it is a value obtained simultaneously due to the properties of the Muller method, it is used as an initial value for calculating the molecular orbital of the next orbital. Here, the uniform random number is a random number that returns a number in the range of 0 to 1 with the same probability.

With such initial values, the position, velocity and acceleration of the TiN molecule at each time are recorded at a time interval of about 2 fsec by the molecular dynamics method, and the scattering formed in the traveling direction before and after the collision as viewed from the center of gravity is obtained. Calculate the angle χ, TiN before collision
From the translational energy Epb of the molecule and the translational energy Epa of the TiN molecule after collision, the energy loss κ = Epa / E
Calculate pb and record each.

The calculation of the scattering angle χ and the energy loss κ is performed, for example, about 300 times for one collision parameter b as a number sufficient for obtaining the distribution. The same calculation is performed by increasing the collision parameter b by db until all the scattering angles χ become 0, and the collision parameter b that makes the scattering angle ０ 0 becomes the collision radius bmax. The above-described molecular dynamics calculation is performed by setting the relative energy E to, for example, 1, 4, 16, 64,
Each is calculated for the case of 256 eV, and a first table showing the relationship between the relative energy and the collision radius is created.

Next, for each relative energy E, the maximum value π of the scattering angle and the maximum value κmax of the energy loss are, for example, 2
It is assumed that the number of data of the scattering angle χ and the energy loss κ which are divided into 0 and enter each divided area are Nχ and Nκ. Furthermore, Nχ,
Normalized using the maximum values Nχmax and Nκmax of Nκ, the probability density R (χ) = Nχ / Nχmax, R (κ) =
Calculate Nκ / Nκmax and create a second table showing the scattering angle and the probability density of energy loss for each collision parameter.

Next, the trajectory of the sputtered particles is calculated by the following procedure using the Monte Carlo method.

First, TiN particles are generated from the surface of the TiN target at a probability proportional to the erosion distribution, and the horizontal angle Φ (Φ = Φ) is generated using uniform random numbers e9 and e1O.
2π · e9), vertical angle θ ｛(θ = acos
((E1O) ^1/2 )}, and then the emission energy E
velocity of e and TiN molecules Vs = (2Ee / Ms) ^1/2
Is determined using the rejection method using e11 and e12 so as to follow the Thompson distribution.

Further, assuming that the temperature in the apparatus follows the Maxwell distribution, the velocity vg of the N ₂ molecule is set to a uniform random number e13,
Using the Box-Muller method using e14,
9. The relative speed vrel = vs−vg between the TiN molecule and the N ₂ molecule is obtained by vector calculation as shown in Expression 20.

[0044]

Vg = (− 2 · ln (e13) · (kT / M
g)) ^1/2 cos (2π · e14)

[0045]

Vg ′ = (− 2 · ln (e13) · (kT / M
g)) ^1/2 sin (2π · e14) Here, vg ′ is a value that is simultaneously determined due to the properties of the Box-Muller method, and is therefore used for calculating the velocity of the N ₂ molecule in the molecular orbital calculation of the next orbital.

Next, referring to the first table calculated by the molecular dynamics method, the relative energy E calculated from the relative velocity Vrel of the N ₂ molecule and the TiN sputtered molecule is calculated by the energy Eprev and Eafter before and after the relative energy E. From the values of the collision radii bmax_prev and bmax_after, the collision radius bmax = bmax_prev + (bma
x_after-bmax_prev) · (E-Epr
ev) / (Eafter-Eprev).

Next, as shown in FIG. 3, the collision parameter b is calculated as b = int (bmax · e
15 / db) · db, and a uniform random number e1
6, the scattering angle 散乱 = π · e16 and the energy loss κ = κmax · e17 are obtained using e17, and the probability density R (χ),
R (κ) is read from the second table calculated by molecular dynamics.
Here, int (x) is a function that returns an integer value of x.
Furthermore, uniform random numbers e18, e19 and R (χ), R (κ)
And when R (χ)> e18 and R (κ)> e19, b, χ, κ are adopted, and R (χ) <e18 or R
If (κ) <e19, reject it and create a new uniform random number e1
Using 5 ′, e16 ′, and e17 ′, the collision parameter is b = int (bmax · e15 ′ / db) · db, the scattering angle is χ = π · e16 ′, and the energy loss is κ = 1 · e1.
7 'occurs and the decision of acceptance or rejection is made again. This operation is performed until b, χ, and κ are adopted.

Assuming a Poisson distribution, a uniform random number e2
Using 0, the distance dL = λ0 · vrel · | ln (e20) | until the TiN molecule causes the next collision is calculated.

Further, the trajectory of the sputtered particles after collision is calculated from the scattering angle χ and the energy loss κ. In this manner, the trajectory of the particle is repeatedly calculated until the particle reaches the side wall of the apparatus or the wafer. The trajectory of the sputter particles calculated in this manner is extracted in a certain area on the wafer, and the deposition shape is calculated using the result by using a string model or the like.

At this time, the trajectory calculation time of the sputtered particles by the Monte Carlo method is calculated using the table divided into about 400, and is about 3 hours in the EWS of 190 MIPS, which is a practically reduced practically compared with the prior art. Calculation time.

[Embodiment 2] In Embodiment 2, a procedure for creating a third table from Embodiment 1 and the calculation results by the molecular dynamics method, and a third procedure for calculating the trajectory of particles by the Monte Carlo method, are described. Is different in the procedure of reading the table of the above and calculating the scattering angle and the energy loss. Therefore, only these two points will be described.

In the second embodiment of the present invention, similar to the first embodiment,
After the first table showing the relationship between the relative energy and the collision radius is created using the result of calculating the collision of diatomic molecules by the molecular dynamics method, as shown in FIG. A graph in which the horizontal axis is the parameter b and the vertical axis is the scattering angle 作成 is created, and the maximum value π of the scattering angle and the maximum value bmax of the collision parameter b are divided into 20, and the scattering angle に 入 る entering each divided region is calculated. The number N of data is counted. At this time, as shown in FIG. 6A, the i-th b divided region b
i is normalized using the maximum value Nχmax of Nχ, and the probability density R (χ) = Nχ / Nχmax of the scattering angle with χ as a variable
Is calculated. Further, the peak value χpeak of the probability density distribution is obtained from the scattering angle χ when Nχmax. next,
As shown in FIG. 6B, a graph is created with the horizontal axis representing the scattering angle χ and the vertical axis representing the scattering angle probability density R (χ). ,
The respective standard deviations σli and σri are obtained.

In the same procedure as above, for the energy loss κ, the probability density of the energy loss using κ as a variable in each divided region of each b is obtained from a graph in which the horizontal axis represents the collision parameter b and the vertical axis represents the energy loss κ. R (κ) and the peak value κpeak of the distribution are obtained. Further, a graph is made with the horizontal axis representing the energy loss κ and the vertical axis representing the probability density R (κ) of the energy loss. By approximation, standard deviations σkli and σkri are obtained.

From the above results, the peak value χpe of the probability density distribution of the scattering angle for the i-th collision parameter bi
aki, when the probability density distribution of the scattering angle is approximated to a normal distribution, the standard deviation σli, σri on the left and right when the probability density distribution of the scattering angle is approximated to the normal distribution, and the peak value κpeak of the probability density distribution of the energy loss is approximated to the normal distribution. A third table is created by associating the left and right standard deviations σkli and σkri of the peak value of.

Next, when calculating the trajectory of the particles by the Monte Carlo method, refer to the first table based on the relative energy E calculated from the relative velocity Vrel of the N ₂ molecule and the TiN sputtered molecule as in the first embodiment. After obtaining the collision radius bmax, the collision parameter bi is determined using a uniform random number.

Next, the standard deviations σli, σri of the probability density of the scattering angle corresponding to bi, and the peak value χ of the probability density
peaki is read from the third table, and for a uniform random number e1, e
If 1 <σli / (σli + σri), use σli,
The scattering angle is calculated by the Box-Muller method from the uniform random numbers e2 and e3, and χ = χpeaki- (2 · ln (e2) · (σli))
^1/2 · cos (2π · e3), and if e1> σli / (σli + σri), σr
Using i, the scattering angle χ is given by: χ = χpeak + (2 · ln (e2) · (σri))
^1/2 · cos (2π · e3)

Similarly, the standard deviations σkli and σkri of the probability density of energy loss corresponding to bi and the peak value κpeak of the probability density are read from the third table,
For a uniform random number e4, e4> σkli / (σkli + σ
kri), the energy loss κ is calculated as κ = κpeak− (2 · ln (e5) · (σkl
i)) ^1/2 · sin (2π · e6) and e4 <σkli / (σkli + σkr)
If i), then κ = κpeak + (2 · ln (e5) · (σkr
i)) It is obtained as ^1/2 · sin (2π · e6).

The subsequent method of calculating the trajectory by the Monte Carlo method using χ and κ is the same as in the first embodiment.

Further, by using the embodiment of the present invention, the trajectory calculation of the sputtered particles by the Monte Carlo method is as follows.
Calculated by the Box-Muller method using the standard deviation and the peak value when the probability density distribution of the scattering angle and the probability density distribution of the energy loss are approximated by a normal distribution, so that the EWS with a performance of 190 MIPS is about 1 hour. Compared with the first embodiment, there is an advantage that it can be reduced to about one third.

[0060]

According to the present invention, molecular dynamics calculations are performed in consideration of the rotation of molecules, etc., and the scattering angles and the probability densities of energy loss are tabulated. Therefore, when calculating the orbits of particles by the Monte Carlo method, only the table calculations are required. Trajectory calculation of the sputtered particles of the diatomic molecule system is performed using the conventional branching technique.
It can be performed in a calculation time of about 1/0.

[Brief description of the drawings]

FIG. 1 is a schematic flowchart of a simulation method for a sputtering apparatus according to the present invention.

FIG. 2 is a schematic diagram showing a molecular dynamics calculation of the present invention.

FIG. 3 is a schematic view showing calculation of a scattering angle of sputtered particles of the present invention.

FIG. 4 is a schematic flowchart of a simulation method of a conventional sputtering apparatus.

FIG. 5 is a schematic diagram showing a prior art molecular dynamics calculation.

FIGS. 6A and 6B are schematic views showing calculation of a scattering angle in Embodiment 2 of the present invention.

[Explanation of symbols]

1 Distribution of scattering angle に 対 す る for collision parameter b based on molecular dynamics calculation 2 Region defining probability density R (χ) of scattering angle に 対 す る for collision parameter b based on molecular dynamics calculation

Claims (4)

(57) [Claims] 1. A method for simulating a sputtering apparatus for diatomic molecules forming a deposited film shape in a contact hole, wherein collision calculation between sputter particles and a background gas by molecular dynamics is repeated using relative energy and collision parameters as parameters. Calculating a first table showing the relationship between the relative energy and the collision radius; and, using the relative energy as a parameter, a second angle indicating a scattering angle and a probability density of energy loss with respect to the collision parameter. And a step of creating a table. 2. The step of obtaining the position, angle, and energy of sputtered particles emitted from a target surface by a Monte Carlo method, and determining a collision radius between the sputtered particles and the background gas using the first table. Obtaining a collision parameter from the collision radius, obtaining a scattering angle and energy loss from the second table by a rejection method, and obtaining a collision position from a Poisson distribution using the half collision. Calculating the trajectory of the sputtered particles after the collision from the scattering angle and the energy loss; and extracting a trajectory group of the specific sputtered particles in a specific region on the wafer,
2. The method according to claim 1, further comprising: calculating a shape of a deposited film by using the orbital group of the specific sputtered particles. Calculating a probability density of the scattering angle and a probability density of the energy loss with respect to the collision parameter; and determining a peak of the probability density distribution of the scattering angle from a maximum value of the respective probability densities. Calculating the peak value of the probability density distribution of the energy loss and the energy loss, and calculating each standard deviation by approximating each of the probability densities with a normal distribution on each of the left and right sides of the peak value of the probability density distribution. With respect to the value of the collision parameter, the standard deviation of the probability density distribution of the scattering angle, the peak value of the probability density distribution of the scattering angle, the standard deviation of the probability density distribution of the energy loss, and the probability density distribution of the energy loss Creating a third table showing the correspondence between the peak values of the sputtering apparatus. Method. 4. The standard deviation of the probability density distribution of the scattering angle with respect to the collision parameter, the peak value of the probability density distribution of the scattering angle, and the standard deviation of the probability density distribution of the energy loss from the third table. Reading the peak value of the probability density distribution of energy loss, the standard deviation of the probability density distribution of the scattering angle, the peak value of the probability density distribution of the scattering angle, the standard deviation of the probability density distribution of the energy loss, and the energy loss 4. The method for simulating a sputtering apparatus according to claim 3, further comprising a step of obtaining the scattering angle and the energy loss by a Box-Muller method using a peak value of the probability density distribution of (1).