JP3109464B2 - Particle trajectory simulation method and sputtering simulation method using the same - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a particle orbital simulation method for analyzing the motion state of a system including a plurality of particles colliding with each other and calculating the orbit of each particle, and more particularly to a sputtered particle which is a molecule composed of a plurality of atoms. The present invention relates to a method for simulating the trajectory of a sputtered particle using a molecular dynamics calculation method of a collision between a particle and a background gas particle which is a molecule composed of a plurality of atoms.

[0002]

2. Description of the Related Art With the increase in the number of elements mounted on a semiconductor integrated circuit device, there is an urgent need to develop a technique for embedding a conductive film in a miniaturized contact hole. A sputter device or the like is used as one of the techniques for embedding a conductive film. However, since development of the device requires a lot of time and cost, it is necessary to efficiently develop the device by simulation.

As such a simulation technique, in particular, in the simulation of a compound-based sputtering apparatus such as TiN, for example, as described in Japanese Patent Application No. 9-48686, sputtered particles of molecules consisting of a plurality of atoms are used. Calculates the collision of a molecule consisting of atoms with background gas particles by molecular dynamics method, etc., sets the scattering angle and velocity after the collision in a table, and emits sputtered particles from the target using Monte Carlo method Then, when calculating the collision between the sputtered particles and the background gas in the sputtering apparatus, the scattering angle and velocity after the collision are read from the table described above, and the trajectory of the sputtered particles is calculated. Calculate the buried shape of the sputtered metal in the contact hole of the semiconductor device using the string model There is a method. According to this method, the efficiency of the development of the sputtering apparatus can be increased by predicting in advance the shape of the sputtering apparatus or the embedded shape when parameters such as apparatus pressure and temperature are changed.

Here, as the prior art, only collision calculation by molecular dynamics between a sputtered particle of a molecule composed of a plurality of atoms and a background gas particle of a molecule composed of a plurality of atoms related to the present invention is shown in FIG. FIG. 3 shows a schematic flowchart.

In the prior art, for example, collisions between sputtered TiN particles, which are diatomic molecules, and N ₂ background gas particles, which are diatomic molecules, are calculated by a molecular dynamics method using, for example, the Velret method.

First, a two-body potential U is calculated with respect to the interatomic distance r for each type of atom pair to be handled, and a gradient dU / dr with respect to r with respect to the interatomic distance of the potential.
Tabulate the values of

At time t = 0, initial values are set for Δt and the particle position, velocity, and acceleration vectors x, v, and a for all atoms involved in the collision, and time t is set to t + Δt
Proceed to

Next, the position, velocity, and time at time t = t−Δt for all atoms involved in collision using the Velet method.
Acceleration vector x (t−Δt), v (t−Δt), a
From (t−Δt), x (t) = x (t−Δt) + Δt · v (t−Δt) +1
/ 2 △ {t ² a (t- △ t) v (t- ／ · △ t) = v (t- △ t) 十 1 / 2１ ／
Find t · a (t−Δt).

Next, from the position vector x (t) for all atoms involved in the collision and the position vector x _i (t) of the i-th atom other than x (t), the direction vector r _i (t) =
x _i (t) −x (t) is obtained, and the slope of two potentials (dU) corresponding to the interatomic distance | r _i (t) |
/ Dr) reads _i , acceleration vector a at time t
(T) is calculated by: a (t) = (1 / m) · Σ (dU / dr) · r (t) /
| R (t) | Here, m is the mass of the atom.

Further, for all atoms involved in collision, v (t− ／ · ／ t) and a (t), v (t) = v (t−−１ · △ t) + ／ · △ t ・ a
(T) is obtained.

Next, a very large value such as 1 × 10 ³² is set for the time step Δt at the next time. Then, for a moving distance for determination 判定 x having a predetermined value that is predetermined as a calculation parameter, Δx = ｘt _tmp × (| v (t) | ² + a (t) · v (t )
Using the Newton method, the value of the temporary time step Δt _tmp at the next time is calculated so that Δt _tmp + ／ · | a (t) | ² · Δt _tmp ² ) ^1/2 is satisfied. Ask. Further, Δt _tmp thus obtained is compared with Δt, and
If t _tmp <Δt, then Δt = Δt _tmp .

As described above, Δt _tmp is calculated for all the atoms involved in the collision, compared with Δt, and the time step Δt at the next time obtained last is used to calculate the time t = t
+ △ t, and this operation is repeated until the time t reaches the time when the collision ends.

[0013]

In the prior art, when a movement is calculated at a large potential gradient, a change in the potential value before and after the movement is large, and the position and velocity values after the movement follow the change in the potential value. The value of the position and the velocity calculated in this way is significantly different from that of the velocity value of the sputtered particle after the collision.

[0014] It is an object of the present invention to improve the accuracy of particle orbit calculation by the molecular dynamics method at a place where the potential gradient is large.

[0015]

According to the present invention for solving the above-mentioned problems, a particle orbit simulation method for analyzing a motion state of a system including a plurality of particles colliding with each other and calculating an orbit of each particle is provided. A particle trajectory simulation method is provided, wherein the trajectory of each particle is calculated by sequentially repeating the following steps (A) to (C) until a predetermined time is reached. (A) Step of obtaining acceleration, velocity, and position at time t for one or two or more two-particle systems involved in the collision. (B) The amount of change in the two-body potential of the two-particle system becomes equal to a preset value. Step of calculating time step Δt _tmp (C) Δ calculated for each two-particle system involved in collision
and Delta] t Select the smallest Delta] t _tmp of t _tmp, when calculating the movement value of the t at large step potential gradient replacing the value of t + Delta] t, is calculated accuracy because the change in potential is smaller than a predetermined value improves. The steps (A) to (C) are performed for one or two or more two-particle systems involved in collision, but are preferably performed for all two-particle systems substantially involved in collision.

According to the present invention, there is provided a particle trajectory simulation method for analyzing a motion state of a system including a plurality of particles colliding with each other and calculating a trajectory of each particle. A particle trajectory simulation method is provided, wherein the trajectory of each particle is calculated by sequentially repeating steps until a predetermined time is reached. (A) Step of obtaining acceleration, velocity, and position at time t for one or two or more two-particle systems involved in collision. (B) The amount of change in the two-body potential of the two-particle system is substantially equal to a preset value. Calculating the time step Δt _tmp so that the moving distance of each particle constituting the two-particle system becomes smaller than a preset reference value. (C) For each two-particle system involved in collision Calculated Δ
and Delta] t Select the smallest Delta] t _tmp of t _tmp, in step the particle trajectory simulation method to replace the value of the t to the value of t + Delta] t, the reference value for the moving distance by providing a step (B) A comparison is made. As a result, the calculation accuracy can be further improved. This is because, if the moving distance is too large, the error may increase because the influence of other particles existing between the moving distances cannot be ignored. The steps (A) to (C) are performed for one or two or more two-particle systems involved in collision, but are preferably performed for all two-particle systems substantially involved in collision.

Further, according to the present invention, a plurality of collisions with each other can be achieved.
Analyze the motion state of a system containing a number of particles and calculate the trajectory of each particle.
In the calculated particle orbit simulation method,
Steps (A) to (D) in order until a predetermined time is reached.
Calculating the trajectory of each particle by repeating the following
Particle orbit simulation method characterized by
You. (A) One or more two-particle systems involved in collisions
To obtain the distance r (t) between particles at time t,
Two-body potential of the two-particle system corresponding to the distance r (t)
Particles whose change amount is equal to a preset value
Distance movement amount Δr ^*(B) calculating the (t) (B) the two-body pote corresponding to the inter-particle distance r (t)
The differential value dU / dR of the differential is obtained, and the differential value dU / dR is obtained.
From R, acceleration a of each particle constituting the two-particle system
(T) Step of calculating velocity v (t) (C) Acceleration a (t), velocity v (t) and time step
Distance between particles expressed as a function of
r = Δr^*Set the time step value so that
(D) Δ calculated for each two-particle system involved in collision
t ′, the smallest Δt ′ is selected as Δt,
Step of replacing the value of t with the value of t + Δt Δr^* Is expressed as a function of r, and given r, Δ
r^*Is determined. On the other hand, given r, dU / dR is found.
That is, the acceleration a is obtained. Then, the speed v
I get it. So given any time step
Then, this time step, a, v
The change amount Δr is calculated. This Δr and Δr ^*Becomes equal
Thus, the value of the time step is determined and set to Δt ′. This
By doing so, the time step is
So that the amount of change in the initial is approximately equal to the preset value
Is set to The steps (A) to (D) above
Describes one or more two-particle systems involved in collisions
But for all two-particle systems that are substantially involved in the collision.
It is preferable to carry out.

Further, according to the present invention, a plurality of collisions with each other are provided.
Analyze the motion state of a system containing a number of particles and calculate the trajectory of each particle.
In the calculated particle orbit simulation method,
Steps (A) to (F) in order until a predetermined time is reached.
Calculating the trajectory of each particle by repeating the following
Particle orbit simulation method characterized by
You. (A) One or more two-particle systems involved in collisions
To obtain the distance r (t) between particles at time t,
Two-body potential of the two-particle system corresponding to the distance r (t)
Particles whose change amount is equal to a preset value
Distance movement amount Δr ^*(B) calculating the (t) (B) the two-body pote corresponding to the inter-particle distance r (t)
The differential value dU / dR of the differential is obtained, and the differential value dU / dR is obtained.
From R, acceleration a of each particle constituting the two-particle system
(T) Step of calculating velocity v (t) (C) Acceleration a (t), velocity v (t) and time step
Distance between particles expressed as a function of
r = Δr^*Set the time step value so that
(D) the moving distance of the first particles constituting the two-particle system is
Time step and second particle with a preset value
Time step when the travel distance of the
Step of setting the smaller one to Δt ″ (E) If Δt ′ ≦ Δt ″, Δt ″_tmp= Δt ', Δ
Δt if t ′> Δt ″_tmp= Δt ″
(F) Δ calculated for each two-particle system involved in collision
t_tmpThe smallest Δt_tmpIs selected as Δt, and
Step of replacing the value of t with the value of t + Δt
The steps (A) to (F) involve one or more steps involved in the collision.
Is performed on two or more two-particle systems, but is substantially
It is preferred to carry out for all two-particle systems provided.

Further, according to the present invention, a plurality of collisions with each other are provided.
Analyze the motion state of a system containing a number of particles and calculate the trajectory of each particle.
In the calculated particle orbit simulation method,
Steps (A) to (G) in order until a predetermined time is reached.
Calculating the trajectory of each particle by repeating the following
Particle orbit simulation method characterized by
You. (A) One or more two-particle systems involved in collisions
To obtain the distance r (t) between particles at time t,
Two-body potential of the two-particle system corresponding to the distance r (t)
Particles whose change amount is equal to a preset value
Distance movement amount Δr ^*(B) calculating the (t) (B) the two-body pote corresponding to the inter-particle distance r (t)
The differential value dU / dR of the differential is obtained, and the differential value dU / dR is obtained.
From R, acceleration a of each particle constituting the two-particle system
(T) Step of calculating velocity v (t) (C) Acceleration a (t), velocity v (t) and time step
Distance between particles expressed as a function of
r = Δr^*Set the time step value so that
(D) Δ calculated for each two-particle system involved in collision
t ′, the smallest Δt ′ is selected, and Δt_tmpTo be
Step (E): The moving distance of the first particles constituting the two-particle system is
Time step and second particle with a preset value
Time step when the travel distance of the
Step of setting smaller one to Δt ″ (F) Δt_tmpIf <Δt ″, Δt = Δt_tmp, Δt
_tmpIf> Δt ″, the step of setting Δt = Δt ″ (G) The step of replacing the value of t with the value of t + Δt The steps (A) to (G) involve the collision
One or two or more two-particle systems
It is preferred to do this for all qualitatively involved two-particle systems.
New

Further, according to the present invention, the motion state of a molecule composed of atoms emitted from a sputter target and a molecule contained in a background gas is analyzed by using any one of the above-described particle orbital simulation methods. Is provided.

[0021]

DESCRIPTION OF THE PREFERRED EMBODIMENTS FIG. 1 shows a schematic flowchart of an embodiment of the present invention.

In the present invention, collisions between diatomic TiN sputtered particles and diatomic molecular N ₂ background gas particles are simulated by a TiN sputtering apparatus.
It is calculated by the molecular dynamics method using the et method.

First, as shown in FIG. 2, the change amount ΔU of the two-body potential between the Ti—N and the N—N, The amount of change Δr ^* in the interatomic distance is determined by Δr ^* = ΔU / (dU / dr) so that a constant value determined between −N is obtained. Here, dU / dr is the gradient of the two-body potential at the interatomic distance r.

Further, a table of the change amount Δr of the interatomic distance and the value of dU / dr with respect to the interatomic distance r. At time t = 0, Δt, the particle position, velocity, acceleration vector x, v,
The initial value is set to a, and the time t is advanced to t + Δt.

Next, using the Velet method, the positions, velocities at time t = t−Δt for all the atoms involved in the collision,
Acceleration vector x (t−Δt), v (t−Δt), a
From (t−Δt), x (t) = x (t−Δt) + Δt · v (t−Δt) +1
/ △ · t ² · a (t− △ t) v (t− △ · ｖt) = v (t− △ t) +＋ １ · △
Find t · a (t−Δt).

Next, from the position vector x (t) for all atoms involved in the collision and the position vector x _i (t) of the i-th atom other than x (t), the direction vector r _i (t) =
x _i (t) = x _i (t) −x (t) is obtained, and the gradient (dU / dr) _i of two potentials corresponding to the interatomic distance | r _i (t) | the acceleration vector a (t), a (t ) = (1 / m) · Σ (dU / dr) i · r i (t)
/ | R _i (t) | Here, m is the mass of the atom.

Furthermore, v (t−1 / 2 · Δt) and a (t) for all the atoms involved in collision, v (t) = v (t−1 / 2 · Δt) + 1/2 · △ t ・
a (t) is obtained.

Next, a very large value of 1 × 10 ³² is set for the time step Δt. Further, position vectors x (t) and x (t) for all atoms involved in the collision
From outside of the i-th atom in the position vector x _i (t), determined the direction vector _{r i (t) = x i} (t) -x (t), atomic distance from the table | r _i (t) | to the The corresponding two potential gradients (dU / dr) _i are read, and the acceleration vector a _i (t) in the i-th atomic direction at time t is calculated as a _i (t) = (1 / m) · (dU / dr ) _I · r _i (t)
/ | R _i (t) |, the velocity V _i and the acceleration A _i in the _i -th atomic direction are calculated as follows: V _i = v (t) · a _i (t) / | a _i (t) | A _i = | A _i (t) | Furthermore, | r _i (t) | between relative atomic distance variation △ reads the value of r from the table, △ r = △ r ^{* _{That, △ r * = V i ·}} △ t '+ 1/2
A time step for determination so that the relationship of A _i · {t ′ ² holds}
Find t '. Here, Δr is a change amount of the interatomic distance such that the potential change amount is constant, and thus, by determining as above, a time step for determination in which the potential change amount is constant can be obtained. .

The above calculation is performed for one or more atoms involved in collision. To improve the accuracy, it is preferable to calculate for substantially all atoms involved in the collision. The smallest one of the thus obtained Δt ′ is Δt _tmp .

Next, from 移動 t _tmp and v (t), a temporary moving distance △ x ′ is obtained as △ x ′ = | v (t) | △ t _tmp . Next, △ x ′ and △ x are compared based on a preset moving distance △ x.

When Δx ′ is smaller than Δx, Δt _tmp
= △ t ', and when △ x' is greater than △ x,
For Δx, Δx = {t _tmp (| v (t) | ² + a (t) · v (t)}
The value of the temporary time step Δt _tmp at the next time is obtained by using the Newton method so that t _tmp + ／ · | a (t) | ² · △ t _tmp ² ) ^1/2 is satisfied.

Further, the thus obtained {t _tmp } is
t, and if Δt _tmp <Δt, then Δt = Δt _tmp .

As described above, Δt _tmp is calculated for all the atoms involved in the collision, is compared with Δt, and using the time step Δt at the next time obtained last, the time t = t
+ △ t, and this operation is repeated until the time t reaches the time when the collision ends. The trajectory and the motion state of the sputtered particles are calculated as described above.

When the calculation was performed by the above method with the potential change amount kept constant, the error of the velocity of the sputtered particles after the collision was smaller by about 10% as compared with the conventional method. Note that the estimation of the error was based on a value converging when the time step was approached to infinity.

[0035]

When the time step of the next time is obtained,
Where the potential changes suddenly, the amount of change in potential when the particle moves is kept at a constant value,
The velocity of the sputtered particles after collision is 10 times higher than in the prior art.
The error is reduced by about%. In the case of the conventional method, the calculated value may hunt and the solution may diverge depending on the shape of the potential and the method of dividing the time step, but the simulation method of the present invention causes such a problem. Absent.

[0036]

[Brief description of the drawings]

FIG. 1 is a schematic flowchart of a particle trajectory simulation method of the present invention.

FIG. 2 is a schematic diagram showing a manner of obtaining Δr from a two-body potential in the particle orbit simulation method of the present invention.

FIG. 3 is a schematic flowchart of a conventional particle trajectory simulation method.

[Explanation of symbols]

 1 Two-body potential curve 2 Two-body potential gradient dU / dr

──────────────────────────────────────────────────続 き Continued on the front page (58) Fields surveyed (Int. Cl. ⁷ , DB name) C23C 14/00-14/58 G06F 17/00 G06F 17/50 H01L 21/203 H01L 21/285 JICST file ( JOIS)

Claims (6)

(57) [Claims] In a particle trajectory simulation method for analyzing a motion state of a system including a plurality of particles colliding with each other and calculating a trajectory of each particle, the following steps (A) to (C) are performed at a predetermined time. A particle trajectory simulation method, wherein the trajectory of each particle is calculated by sequentially repeating the trajectory until the particle trajectory is reached. (A) Step of obtaining acceleration, velocity, and position at time t for one or two or more two-particle systems involved in the collision. (B) The amount of change in the two-body potential of the two-particle system becomes equal to a preset value. Step of calculating time step Δt _tmp (C) Δ calculated for each two-particle system involved in collision
selecting the smallest Δt _tmp from t _tmp as Δt, and replacing the value of t with the value of t + Δt 2. In a particle trajectory simulation method for analyzing a motion state of a system including a plurality of particles colliding with each other and calculating a trajectory of each particle, the following steps (A) to (C) are performed at a predetermined time. A particle trajectory simulation method, wherein the trajectory of each particle is calculated by sequentially repeating the trajectory until the particle trajectory is reached. (A) Step of obtaining acceleration, velocity, and position at time t for one or two or more two-particle systems involved in collision. (B) The amount of change in the two-body potential of the two-particle system is substantially equal to a preset value. Calculating the time step Δt _tmp so that the moving distance of each particle constituting the two-particle system becomes smaller than a preset reference value. (C) For each two-particle system involved in collision Calculated Δ
selecting the smallest Δt _tmp from t _tmp as Δt, and replacing the value of t with the value of t + Δt 3. A system comprising a plurality of particles colliding with each other.
Particle trajectory system that analyzes the motion state and calculates the trajectory of each particle
In the simulation method, the following (A) to (D)
Steps are repeated sequentially until the specified time is reached.
Calculating the trajectory of each particle.
Road simulation method. (A) One or more two-particle systems involved in collisions
To obtain the distance r (t) between particles at time t,
Two-body potential of the two-particle system corresponding to the distance r (t)
Particles whose change amount is equal to a preset value
Distance movement amount Δr ^*(B) calculating the (t) (B) the two-body pote corresponding to the inter-particle distance r (t)
The differential value dU / dR of the differential is obtained, and the differential value dU / dR is obtained.
From R, acceleration a of each particle constituting the two-particle system
(T) Step of calculating velocity v (t) (C) Acceleration a (t), velocity v (t) and time step
Distance between particles expressed as a function of
r = Δr^*Set the time step value so that
(D) Δ calculated for each two-particle system involved in collision
t ′, the smallest Δt ′ is selected as Δt,
Step of replacing the value of t with the value of t + Δt 4. A system comprising a plurality of particles colliding with each other.
Particle trajectory system that analyzes the motion state and calculates the trajectory of each particle
In the simulation method, the following (A) to (F)
Steps are repeated sequentially until the specified time is reached.
Calculating the trajectory of each particle.
Road simulation method. (A) One or more two-particle systems involved in collisions
To obtain the distance r (t) between particles at time t,
Two-body potential of the two-particle system corresponding to the distance r (t)
Particles whose change amount is equal to a preset value
Distance movement amount Δr ^*(B) calculating the (t) (B) the two-body pote corresponding to the inter-particle distance r (t)
The differential value dU / dR of the differential is obtained, and the differential value dU / dR is obtained.
From R, acceleration a of each particle constituting the two-particle system
(T) Step of calculating velocity v (t) (C) Acceleration a (t), velocity v (t) and time step
Distance between particles expressed as a function of
r = Δr^*Set the time step value so that
(D) the moving distance of the first particles constituting the two-particle system is
Time step and second particle with a preset value
Time step when the travel distance of the
Step of setting the smaller one to Δt ″ (E) If Δt ′ ≦ Δt ″, Δt ″_tmp= Δt ', Δ
Δt if t ′> Δt ″_tmp= Δt ″
(F) Δ calculated for each two-particle system involved in collision
t_tmpThe smallest Δt_tmpIs selected as Δt, and
Replacing the value of t with the value of t + Δt 5. A system comprising a plurality of particles colliding with each other.
Particle trajectory system that analyzes the motion state and calculates the trajectory of each particle
In the simulation method, the following (A) to (G)
Steps are repeated sequentially until the specified time is reached.
Calculating the trajectory of each particle.
Road simulation method. (A) One or more two-particle systems involved in collisions
To obtain the distance r (t) between particles at time t,
Two-body potential of the two-particle system corresponding to the distance r (t)
Particles whose change amount is equal to a preset value
Distance movement amount Δr ^*(B) calculating the (t) (B) the two-body pote corresponding to the inter-particle distance r (t)
The differential value dU / dR of the differential is obtained, and the differential value dU / dR is obtained.
From R, acceleration a of each particle constituting the two-particle system
(T) Step of calculating velocity v (t) (C) Acceleration a (t), velocity v (t) and time step
Distance between particles expressed as a function of
r = Δr^*Set the time step value so that
(D) Δ calculated for each two-particle system involved in collision
t ′, the smallest Δt ′ is selected, and Δt_tmpTo be
Step (E): The moving distance of the first particles constituting the two-particle system is
Time step and second particle with a preset value
Time step when the travel distance of the
Step of setting smaller one to Δt ″ (F) Δt_tmpIf <Δt ″, Δt = Δt_tmp, Δt
_tmpIf> Δt ″, the step of setting Δt = Δt ″ (G) The step of replacing the value of t with the value of t + Δt 6. A method for analyzing a motion state and a trajectory of a molecule composed of atoms emitted from a sputter target and a molecule contained in a background gas by using the particle orbit simulation method according to claim 1. The sputtering simulation method characterized by the above-mentioned.