JPH0653155A - Method and device for simulating semiconductor element - Google Patents

Abstract

PURPOSE:To provide method and device for simulation of a semiconductor element wherein accurate simulation result can be obtained at a fast speed even a high concentration impurity region exists and time interval is long. CONSTITUTION:Lattice points used for discretization of nonlinear partial difference equation for describing diffusion phenomenon of N-type impurities used for manufacture of a semiconductor element are generated in a lattice point generation part 1. When simultaneous linear equations generated by discretization of partial difference equation are formed, a coefficient matrix and constant terms are formed in a simultaneous linear equation part 3 using linear terms related to impurity concentrations contained in a diffusion term and a non-diffusion term as unknown quantities, and the formed simultaneous linear equations are solved by a simultaneous linear equation solution finding part 5. Thereby, it is possible to restrain generation of negative concentration and oscillation of solution even if time interval is long, to enable accurate simulation and to enable long time interval. Simulation of a thermal treatment process can be carried out rapidly.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a semiconductor element simulation method and apparatus for evaluating a manufacturing method by solving a partial differential equation describing a diffusion phenomenon and simulating a manufacturing state of the semiconductor element.

[0002]

2. Description of the Related Art A process simulation technique (for example, SM Sze, "VLSI TECHNOLO", which is a means for designing, evaluating and analyzing a semiconductor device.
GY ", McGraw-Hill, 1988), an equation (hereinafter referred to as a diffusion equation) describing the diffusion phenomenon of impurities in the heat treatment step shown in the equation (1) is discrete by a finite element method, a finite difference method, or the like. And is solved by reducing to a solution problem of simultaneous linear equations.

[0003]

[Equation 4] C: total concentration (total impurity concentration), t: time, D: diffusion coefficient, N: activation concentration (activated impurity concentration), Z: charge state (P-type impurity: -1, N-type impurity: 1) ), N: electron concentration, and the subscript k is the type of impurity. This process simulation is executed by a simulation device provided with a program for numerical calculation called a process simulator.

Here, the formula (1) is a formula which is considered that electrically activated impurities existing at the lattice position of the semiconductor crystal move due to diffusion and an electric field formed by the impurities themselves, and the first term on the right side. Is the diffusion term, and the second term is the electric field term.

In the process simulator, generally, the total concentration C is an unknown number, the electric field term is a known number, and calculation is performed, and finally the equation (2) is solved numerically.

[0006]

[Equation 5] Hereinafter, an algorithm of the conventional simulation method for solving the equation (2) will be described with reference to FIGS. 6 and 7 which are algorithms regarding impurities forming the N-type diffusion layer.

First, in step S401, the discretization grid points required in the subsequent steps are created, the physical quantities such as the diffusion coefficient are initialized, and the time t of the heat treatment time is set to zero.

Next, after advancing the time t by Δt in step S402, it is judged in step S403 whether or not the gas flowing into the diffusion furnace during the heat treatment is a gas that oxidizes the semiconductor, and the semiconductor is not oxidized. In the case of a non-oxidizing atmosphere, the process directly proceeds to step S405.

In the case of an oxidizing atmosphere in which the semiconductor is oxidized, the process proceeds to step S404, the shape change accompanying the change of the oxide film thickness due to the oxidation and the rearrangement of lattice points in the oxide film are performed, and the process proceeds to step S405.

At step S405, the activation impurity concentration N
Is calculated by equations (3) and (4). Between the total concentration C and the activation concentration N, as the total concentration C increases (about 1 ×
The activation concentration N is smaller than the total concentration C (10 ¹⁹ cm ⁻³ or more). As a physical model for describing this relationship, a cluster model is generally used in which a plurality of impurity atoms are considered to be a cluster (cluster).

Impurities used in semiconductor manufacturing include impurities that form electrically neutral clusters and impurities that form charged clusters in which the clusters are charged. K · N ^m · in calculating the activation concentration (N + (m-i) C cl) i -C cl = 0 (3) N = C-mC cl (4) is used.

Here, K is a cluster formation coefficient, m is the number of impurities contained in the cluster, i is the number of electrons that react with impurities forming a charged cluster during cluster formation, and C _cl is the concentration of the cluster.

Next, in step S406, the activation concentration N is restricted so as not to exceed the solid solubility limit concentration N _sl . The reason for this is to reproduce the phenomenon that when the total concentration C becomes a high concentration exceeding the solid solubility limit of the semiconductor substance, the impurities cannot be completely dissolved in the semiconductor substance and the activation concentration N becomes a constant value irrelevant to the total concentration C. is there.

In step S407, for example, in the case of impurities forming charge clusters, the differential coefficient of the activation concentration used in the calculation of the coefficient matrix of the simultaneous linear equations generated by discretization of the diffusion equation is calculated by the equation (5).

[0015]

[Equation 6] Further, in step S408, the electron concentration n _cl due to the charged clusters is calculated, and in step S409, the electron concentration n used in the calculation of the constant term of the simultaneous linear equations generated by the discretization of the diffusion equation is calculated.

The electron concentration n _cl relating to the impurities forming the charged clusters is that electrons are emitted from the impurities and the clusters at the lattice position.

[Equation 7] Becomes Since the process simulator handles diffusion of multiple types of impurities at the same time, the electron concentration n per unit volume is

[Equation 8] Calculated by Here, k is a subscript indicating the type of impurities, and kmax is the total number of types of impurities. However, the subscript k of the impurities forming the charged clusters was set to 1.

Next, step S410 and step S41
In step 1, the coefficient matrix and the constant term of the simultaneous linear equation (2) generated by discretization of the diffusion equation are calculated from the values calculated in steps S405 to 409, and then in step S412, the matrix of the simultaneous linear equation is calculated. And calculate the numerical solution of the diffusion equation.

After that, in step S413, it is determined whether or not the designated heat treatment time has been reached. If not, the process returns to step S402 and the above loop is repeated. When the designated heat treatment time has been reached, the calculation ends.

In the process simulation method of such a heat treatment step, generally, in the case of impurities forming charge clusters, the electric field generated in the high concentration region is stronger than in the case of impurities forming neutral clusters. Further, in the high concentration region exceeding the solid solubility limit, the activation concentration N becomes constant and the differential coefficient becomes zero, so that diffusion does not occur. Therefore, when the impurity concentration is a high concentration exceeding the solid solubility limit, the impurities move only by the electric field.

By the way, as a method for handling the time differential included in the diffusion equation, there are an explicit method and an implicit method. Since the explicit method does not need to solve simultaneous linear equations, the calculation time per hour is short, but if the time is not shortened, the solution will oscillate and accurate calculation will not be possible. Especially, when the above-mentioned impurities move only by the electric field, the vibration of the solution becomes a serious problem.

On the other hand, since the implicit method needs to solve simultaneous linear equations, the calculation time per hour step is longer than that of the explicit method, but the solution oscillation does not occur even if the time step is lengthened.

In the conventional simulation method, a linear term (equation (7)) relating to the impurity concentration included in the non-diffusion term (electric field term) (right side of equation (5)) is used in creating the coefficient matrix and constant term of simultaneous linear equations. Since the first term of the molecule) is treated as a known number, the explicit method is used in the high concentration region. Therefore, in the conventional simulation method, when a region having a high impurity concentration (about 1 × 10 ²⁰ cm ⁻³ or more) exists, vibration of the solution and a negative concentration occur when the time step is long, and an accurate simulation cannot be performed. . Figure 8
An example (according to high-concentration diffusion of arsenic) in which an accurate simulation cannot be performed by a conventional simulation method is shown. (A) is an initial impurity distribution before simulation, and (b) is an impurity distribution after heat treatment simulation. . The heat treatment condition during simulation is 1000 ° C.
20 minutes and the time step was 1 minute.

Solution oscillation and negative concentration occur in the region where the impurity concentration is about 1 × 10 ²⁰ cm ^-3 or more (FIG. 8).
(Ellipse part of (b)). (In the following figures, the axis relating to concentration is a logarithmic scale, and in the case of negative concentration, the absolute value is shown.) However, as shown in FIG. In some cases, it is possible to obtain an accurate solution by making it finer, but it is necessary to make the time step very small, which increases the calculation time. In the example of FIG. 9, the calculation time is set to 1/10, so that the calculation time is about 10 times as long.

Generally, most of the time required for simulation of the entire semiconductor manufacturing process is spent for heat treatment simulation. Therefore, the above-mentioned increase in calculation time significantly reduces the practicability of the simulation. Especially, it is a serious problem in the two-dimensional problem and the three-dimensional problem, which originally have a long calculation time.

The above-mentioned problems also occur under conditions often used in actual semiconductor manufacturing processes. As an example, FIG. 10 shows a simulation result when the source / drain portions of the N-type MOSFET are formed by ion implantation of arsenic and then oxidized. The arsenic ion implantation condition at this time is that the oxide film thickness at the time of implantation is 100Å, the incident energy is 40 keV, and the dose amount is 5 × 10 ¹⁵ cm ⁻³ . The heat treatment condition is dry oxidation at 900 ° C. for 20 minutes. It was

In this case, the time step is 0.01 minutes (dotted line)
Even so, solution vibration and negative concentration occur near the oxide film / silicon interface, and an accurate simulation cannot be performed.
Further, when the time step is set to 0.001 minutes (solid line), the vibration of the solution and the negative concentration do not occur, but the impurity distribution in the oxide film is overestimated, and the simulation accuracy is significantly reduced.

[0027]

As described above, the conventional simulation method has a problem in that a high concentration impurity region exists, and when the time step is long, vibration of the solution and negative concentration occur, and an accurate simulation cannot be performed. There is. Further, even when an accurate solution is required, it is necessary to shorten the time step, which causes a problem that the calculation time is remarkably increased.

The object of the present invention is to solve the above problems.
It is an object of the present invention to provide a semiconductor element simulation method and apparatus capable of obtaining an accurate solution at high speed even when a high-concentration impurity region exists and a time step is long.

[0029]

In order to achieve the above object, the present invention generates lattice points used for discretization of a non-linear partial differential equation which describes the diffusion phenomenon of N-type impurities used in the manufacture of semiconductor devices. When creating a simultaneous linear equation generated by discretization of the partial differential equation and the lattice point generation means, a simultaneous linear equation that creates a coefficient matrix and a constant term with the linear term relating to the impurity concentration contained in the diffusion term and the non-diffusion term as unknowns It is composed of equation creating means and simultaneous linear equation solving means for solving the created simultaneous linear equations.

[0030]

With the above structure, the present invention generates lattice points used for discretization of the nonlinear partial differential equation describing the diffusion phenomenon of N-type impurities used in the manufacture of semiconductor devices by the lattice point generating means, and the partial differential is generated. When the simultaneous linear equations generated by the discretization of the equations are created, the coefficient matrix and the constant term are created by the simultaneous linear equation creating means with the linear term regarding the impurity concentration contained in the diffusion term and the non-diffusion term as unknowns The simultaneous linear equations are solved by the simultaneous linear equation solving means.

When the simultaneous linear equations are created by the simultaneous linear equation creating means, the coefficient matrix and the constant term are created with the linear terms relating to the impurity concentrations contained in the diffusion term and the non-diffusion term as unknowns, so that the high concentration impurities It is possible to increase the portion using the implicit method when the region exists.

[0032]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of a semiconductor element simulation method and apparatus according to the present invention will be described below with reference to the drawings.

FIG. 1 is a block diagram showing the configuration of an embodiment of the semiconductor device simulation apparatus of the present invention.

In the figure, a lattice point generating section 1 is for generating lattice points used for discretization of a non-linear partial differential equation describing the diffusion phenomenon of N-type impurities used in the manufacture of semiconductor elements, and simultaneous linear equations are used. The creating unit 3 has a function of creating a coefficient matrix and a constant term by using a linear term related to the impurity concentration contained in the diffusion term and the non-diffusion term as an unknown number when creating a simultaneous linear equation generated by discretization of a partial differential equation. ing. Further, the simultaneous linear equation solving section 5 solves the simultaneous linear equations created by the simultaneous linear equation creating section 3.

FIGS. 2 and 3 are flowcharts of the simulation method of the present invention. The difference from the conventional simulation method shown in FIGS. 6 and 7 is step S110 and step S111. Steps S101 to S
After performing the same calculation as the conventional method up to 109, in step S110, a linear term for the concentration n _cl of electrons generated from the impurities forming the charge clusters included in the calculation formula of the electron concentration n on the right side of Expression (2). To the left side, and the equation (6) is substituted:

[Equation 9] Is used to calculate the coefficient matrix of the simultaneous linear equations.

Next, in step S111, equation (8)
The electron concentration n _cl generated from the impurities forming the charge clusters included in the calculation formula of the electron concentration n transferred to is subtracted from the right side of the equation (2):

[Equation 10] Is used to calculate the constant term of the simultaneous linear equations. After that, heat treatment simulation is performed by the same calculation procedure as the conventional method.

FIG. 4 shows simulation results when heat treatment was performed at 1000 ° C. for 20 minutes from the initial impurity distribution shown in FIG. The solid line indicates the result calculated by the conventional simulation method, and the triangular mark indicates the result calculated by the simulation method of the present invention in a time step of 1 minute.

In the conventional method, as described with reference to FIG. 9, accurate calculation cannot be performed unless the time step is set to 0.1 minutes.
In the simulation method according to the present invention, accurate calculation can be performed even if the time step is 1 minute. Furthermore, the heat treatment time is 20
Since it is a minute, the conventional method requires 200 iterations, whereas the method according to the present invention requires 20 iterations, and the computation time is reduced by about 10 times.

FIG. 5 uses the method according to the invention and FIG.
This is the result of calculation with a time step of 1 minute under the same conditions as those described in. With the conventional method, the time step is 0.00
Although the accurate calculation could not be performed even for 1 minute, the method according to the present invention enables accurate calculation even if the time step is 1 minute. That is, the calculation time can be shortened by about 1000 times as compared with the conventional method, and accurate calculation can be performed.

[0040]

As described above, in the semiconductor element simulation method and apparatus of the present invention, simultaneous linear equations generated by discretization of the nonlinear partial differential equation describing the diffusion phenomenon of impurities used in the manufacture of semiconductor elements are created. In, the coefficient matrix and the constant term are created by using the linear terms related to the impurity concentration contained in the diffusion term and the non-diffusion term as unknowns.

As a result, even if the time step is long, it is possible to suppress the occurrence of negative concentration and the vibration of the solution, and it is possible to perform accurate simulation. Further, since the time step can be lengthened, the heat treatment process can be simulated at high speed.

[Brief description of drawings]

FIG. 1 is a block diagram showing the configuration of an embodiment of a semiconductor device simulation apparatus according to the present invention.

FIG. 2 is a flowchart showing a semiconductor element simulation method of the present invention.

FIG. 3 is a flowchart showing a method for simulating a semiconductor device of the present invention, which is subsequent to FIG.

FIG. 4 is a graph comparing the calculation results of the simulation method of the present invention and the conventional simulation method.

FIG. 5 is a diagram showing calculation results of conditions used in actual semiconductor manufacturing using the simulation method of the present invention.

FIG. 6 is a flowchart showing a conventional simulation method.

7 is a flowchart showing a conventional simulation method following FIG.

FIG. 8 is a graph showing an initial distribution before simulation and a calculation result in which a solution vibration and a negative concentration are generated by a conventional simulation method.

FIG. 9 is a graph showing a calculation result in which the vibration of the solution and the negative concentration do not occur in the conventional simulation method.

FIG. 10 is a graph showing calculation results of conditions used in actual semiconductor manufacturing using a conventional simulation method.

[Explanation of symbols]

 1 Lattice point generation part 3 Simultaneous linear equation creation part 5 Simultaneous linear equation solving part

Claims (3)

[Claims] 1. A lattice point used for discretization of a non-linear partial differential equation describing the diffusion phenomenon of N-type impurities used for manufacturing a semiconductor device is generated, and simultaneous linear equations generated by the discretization of the partial differential equation are created. A method of simulating a semiconductor device, characterized in that a linear matrix relating to the impurity concentration contained in the diffusion term and the non-diffusion term is used as unknowns to create a coefficient matrix and a constant term, and the created simultaneous linear equations are solved. 2. A coefficient matrix of the simultaneous linear equations, wherein C is total impurity concentration, t is time, D is diffusion coefficient, N is electrically activated impurity concentration, n is electron concentration, and m is N-type impurity. Is the number of N-type impurities contained in the charged cluster, which is a charged mass, and i is N constituting the charged cluster.
Type electrons, the number of electrons reacting with the impurity, i-m is the number of charges of the charged cluster, and n _cl is the N that constitutes the charged cluster.
As the electron concentration related to the type impurities, And the constant term of the simultaneous linear equations is given by [Equation 3] The method for simulating a semiconductor device according to claim 1, wherein the simulation is performed by using. 3. Lattice point generating means for generating lattice points used for discretization of a non-linear partial differential equation describing the diffusion phenomenon of N-type impurities used in the manufacture of semiconductor devices, and said partial differential equation. When a simultaneous linear equation is created, a linear matrix relating to the impurity concentration contained in the diffusion term and the non-diffusion term is used as unknowns to create a coefficient matrix and a constant term, and a simultaneous linear equation creating means is created to solve the created simultaneous linear equation. A semiconductor device simulation apparatus comprising: a linear equation solving means.