JP6986810B2 - Calculation method of oxidation time in the thermal oxidation process of semiconductor materials - Google Patents

Description

The present invention relates to controlling a process for producing a silicon oxide film when a silicon oxide film is formed on the surface of a crystal substrate of silicon or silicon carbide by thermal oxidation to produce a semiconductor device.

When manufacturing a MOS (metal-oxide-semiconductor) type semiconductor device as shown in FIG. 1 using a crystal substrate of a semiconductor material such as silicon, an n-type semiconductor crystal (2) and a gate electrode (4) are used. ), A thin film of silicon oxide is formed, and this is used for the gate oxide film (1). Further, a source electrode (5) and a drain electrode (6) are formed on the p-type semiconductor (3). When manufacturing such a MOS type semiconductor device having a low operating voltage and a high degree of integration, it is desirable that the film thickness of the gate oxide film (1) is as thin as possible.

In particular, as shown in FIG. 2, when a semiconductor device is manufactured using a silicon crystal substrate (11), the thickness of the oxide film (13) including the transition layer (12) is 20 nanometers at the thinnest. Semiconductor devices with very fine structures, such as meters or less, have also been created. However, since there is no formula for calculating the thickness with nanometer accuracy required to control the process of creating such ultra-thin films, the work of creating a highly accurate oxide film necessary for manufacturing the latest semiconductor devices. Had the problem of being difficult to implement.

A prominent mathematical model devised to understand the thermal oxidation process of silicon crystals and analyze experimental results is the model proposed by Deal and Grove. In this model, assuming that the thermal oxidation process of the silicon crystal proceeds in a steady state, an ordinary differential equation representing the time change of the film thickness is derived, and by solving this, the time of the film thickness of the oxide film is obtained. It formulates an equation that expresses change. (See Non-Patent Document 1.)

Deal and Grove models have been used for more than half a century to analyze data from thermal oxidation experiments on silicon and silicon carbide crystals. For example, the technical disclosure of process simulation methods relies on Deal and Grove models, data analysis of thermal oxidation experiments of silicon crystals and silicon carbide crystals using high-pressure oxygen molecules for oxidants, and ozone for oxidants. This model is also used in the analysis of experimental results obtained in thermal oxidation experiments of silicon crystals using molecules. (See Patent Document 1 and Non-Patent Documents 2 to 4.)

However, even if the calculation is performed using this model, the very large oxidation rate obtained by the thermal oxidation of silicon crystals in the initial process of dry oxidation is not reproduced. (See Non-Patent Document 1.)

To solve this problem, Massoud et al. Add an exponential term to the differential equation formulated in the Deal and Grove models, and this correction term can explain the initial process of dry oxidation. Insisted. (See Non-Patent Document 5.)

However, in the development of the latest devices using ultra-thin films with a silicon oxide film thickness of 20 nanometers (nm) or less, the results of calculations using differential equations with this correction term are also consistent with the experiments. Is known not to be obtained.

In the interfacial emission model proposed by Uematsu et al. Instead of these models, the strain generated at the interface during thermal oxidation releases silicon atoms from the interface, and these released silicon atoms are further released at the subsequent interface. It was said that it inhibits the oxidation process of silicon. (See Non-Patent Documents 6 and 7.)

In this model, the saturation concentration of silicon atoms dissolved in silicon dioxide is used in the formulation of the model by the reaction diffusion equation, but since the thermal oxidation process is not in the thermal equilibrium state, this equation is a physical phenomenon. It was pointed out by Farjas and Roura that it did not correctly describe the process of thermal oxidation. (See Non-Patent Document 8.)

From the above, mathematical models that correctly describe the process of thermal oxidation with nanoscale accuracy, which are effective for the development of the latest devices, and equations with nanoscale accuracy that express the thickness of the oxide film as a function of oxidation time are , Did not exist until now.

Japanese Unexamined Patent Publication No. 2001-35547

E. Deal and A. S. Grove, J. Appl. Phys. 36, 3770 (1965). L. N. Lie, R. R. Razouk, and B. E. Deal, J. Electrochem. Soc.129, 2828 (1982). Y. Song, S. Dhar, L. C. Feldman, G. Chung, and J. R. Williams, J. Appl. Phys. 95, 4953 (2004). T. Nishiguchi, H. Nonaka, S. Ichimura, Y. Morikawa, M. Kekura, and M. Miyamoto, Appl. Phys. Lett. 81, 2190 (2002). H. Z. Massoud, J. D. Plummer, and E. A. Irene, J. Electrochem.Soc. 132, 2685 (1985). M. Uematsu, H. Kageshima, and K. Shiraishi, Jpn. J. Appl.Phys. 39, L699 (2000). Shinji Uematsu, Hiroyuki Kageshima, Kenji Shiraishi, Surface Science Vol. 23, No. 2, pp. 104 (2002). J. Farjas and P. Roura, J. Appl. Phys. 102, 054902 (2007). J. Crank, The Mathematics of Diffusion (Elsevier / Butterworth Heinemann, London, 1980) 3rd ed. I. Chavel, Riemannian Geometry (Cambridge University Press) (Cambridge, 2006) 2nd ed., P.150. Shigeru Ishihara "Tensol-For Science and Technology" (Shokabo, 1991).

The problem to be solved by the present invention is necessary to specify the oxidation temperature and the partial pressure of the oxidant molecule to form a silicon oxide film having a predetermined film thickness of 20 nanometers or less with an accuracy of nanometers. In addition, there is no reliable calculation method that can calculate the oxidation time of thermal oxidation.

The present invention derives an equation with nanoscale accuracy that expresses the film thickness of a silicon oxide film generated by thermal oxidation of a crystal substrate of silicon or silicon carbide as a function of oxidation temperature, partial pressure of oxidant molecules, and oxidation time. The most important feature is to provide a method for calculating the oxidation time for obtaining a predetermined film thickness by using a computer.

In the calculation method of the present invention, when a semiconductor material such as silicon or silicon carbide is thermally oxidized to form an insulating film used for a gate oxide film of a semiconductor device, the oxidation temperature, the partial pressure of oxidant molecules, and a predetermined film thickness are determined. It has the advantage of allowing the calculation of the oxidation time required to form a silicon oxide film with nanoscale accuracy.

FIG. 1 is a schematic cross-sectional view of a MOS type FET (metal-oxide-semiconductor field-effect transistor). FIG. 2 is a schematic diagram of the thermal oxidation process of silicon crystals under the condition of dry oxidation. FIG. 3 uses a mathematical model according to the present invention to simulate the growth process of the film thickness of a silicon oxide film formed on a substrate of a silicon crystal having a surface index (100) under dry oxidation conditions at a plurality of temperatures. It is a graph which drew the result obtained by executing the above with respect to the oxidation time. FIG. 4 uses a mathematical model according to the present invention to simulate the growth process of the film thickness of an oxide film formed on a substrate of a silicon crystal having a plane index (100) under the condition of dry oxidation at a plurality of temperatures. It is a graph which divided by the function ζ (T) defined by the mathematical formula (22) of Example 1 and redrawn. FIG. 5 is prepared by a thermal oxidation experiment using a silicon crystal substrate having a plane index (100) under the condition of dry oxidation and when the partial pressure of oxygen is 20 atm. It is a comparison between the measurement result of the film thickness of the oxide film reported in 1 and the calculation result using the equation according to the present invention. FIG. 6 is prepared by a thermal oxidation experiment using a silicon crystal substrate having a plane index (111) under the condition of dry oxidation and when the partial pressure of oxygen is 20 atm. It is a comparison between the measurement result of the film thickness of the oxide film reported in 1 and the calculation result using the equation according to the present invention. FIG. 7 was prepared by a thermal oxidation experiment using a silicon crystal substrate having a plane index (100) under the condition of dry oxidation and when the partial pressure of oxygen was 1 atm. It is a graph which plotted the measurement result of the film thickness of the oxide film reported in 1 and the calculation result using the equation which concerns on this invention on a logarithmic scale. FIG. 8 is prepared by a thermal oxidation experiment using a silicon crystal substrate having a plane index (111) under the condition of dry oxidation and when the partial pressure of oxygen is 1 atm. It is a graph which plotted the measurement result of the film thickness of the oxide film reported in 1 and the calculation result using the equation which concerns on this invention on a logarithmic scale. FIG. 9 shows a thermal oxidation experiment under dry oxidation conditions using a silicon crystal substrate with a plane index (100), and the partial pressure of oxygen was 1 atm to 20 atm while the oxidation temperature was fixed at T = 900 ° C. Regarding the film thickness of the oxide film produced by changing up to It is a comparison of the measurement result reported in 1 and the calculation result using the equation according to the present invention. FIG. 10 shows the case where the oxidation temperature is fixed at T = 900 ° C. and the partial pressure of oxygen is changed from 1 atm to 20 atm using a silicon crystal substrate having a surface index (100) under the condition of dry oxidation. 6 is a graph in which the value of the coefficient K of the equation according to the present invention shown in FIG. 9, which reproduces the measurement result of the film thickness, is plotted on a logarithmic scale with respect to the partial pressure of oxygen. FIG. 11 is prepared by a thermal oxidation experiment using a silicon crystal substrate having a plane index (100) under the condition of wet oxidation when the partial pressure of water vapor is 1 atm. It is a comparison between the measurement result of the film thickness of the oxide film reported in 1 and the calculation result using the equation according to the present invention. FIG. 12 was prepared by a thermal oxidation experiment using a silicon crystal substrate having a plane index (100) when an ozone molecule having a partial pressure of 900 Pa was used for the oxidant. It is a comparison between the measurement result of the film thickness of the oxide film reported in 1 and the calculation result using the equation according to the present invention. FIG. 13 is a block diagram of the arithmetic unit according to the present invention. FIG. 14 is a flowchart showing an algorithm of a computer program created to cause a computer to calculate an oxidation time using the equation according to the present invention. (Example 1)

Calculates the oxidation time of thermal oxidation required to form an oxide film with a predetermined film thickness of 20 nm or less with nanometer accuracy when the oxidation temperature and the partial pressure of the oxidant molecule are specified. , Devised a simple calculation method, and showed the algorithm of a computer program to make a computer execute this calculation.

The method of calculating the oxidation time of a semiconductor material by designating the oxidation temperature, the partial pressure of the oxidant molecule, and the predetermined film thickness will be described below.

In order to execute simulations for molecular diffusion processes and chemical reaction processes in a medium containing strain, a reaction diffusion equation, which is a partial differential equation that can handle both of these processes at the same time, is used. It needs to be extended to a format applicable to media containing strain. To derive this equation, we use Fick's first and second laws.

ここで、φ_αは分子αの濃度を表すものとする。また、∂_iは座標ｘの第i成分であるxⁱに関する偏微分を表す。 According to Fick's first law, the flux generated by the diffusion of a particle α is given by the following equation using _{the diffusion coefficient D α of this particle.} (See Non-Patent Document 9.)

Here, φ _α represents the concentration of the molecule α. Also, ∂ _i represents the partial derivative with respect to ^{x i,} which is the i-th component of the coordinates x.

^{The divergence of the vector v i} in a distorted medium is given by the determinant g = det (g _ij ) of the metric tensor g _ij. (See Non-Patent Documents 10 and 11.)

ここで、二重に用いられた添え字jに関しては、三次元空間の座標軸である、１から３までの和を取るものとする。 Since the flux in equation (1) is a covariant vector, it is converted into a contravariant vector using ^{g ij} , which is a tensor corresponding to the inverse matrix of the _{metric tensor g ij.}

Here, for the subscript j used twice, it is assumed that the sum of 1 to 3, which is the coordinate axis of the three-dimensional space, is taken.

ここで、二重に用いられた添え字 i と j は、１から３までの和を取るものとする。また、ｋ_α,βnは、一方の添え字が示す分子αと、もう一方の添え字のβ_ｎに対応する、シリコン原子やシリコン酸化物との間の化学反応に対する、反応率定数を表す。 Next, Fick's second law is applied to the flux caused by the diffusion of the particle α. At that time, by using the equations (2) and (3), the reaction-diffusion equation of the particle α in the distorted medium is derived as follows. (See Non-Patent Document 9.)

Here, the double subscripts i and j are assumed to be the sum of 1 to 3. Further, k _{α and β n} represent the reaction rate constants for the chemical reaction between the molecule α indicated by one subscript and the silicon atom or silicon oxide corresponding to β n of the other subscript.

右辺の係数V_Si2は、２個のSi原子から成る単位胞Si₂の体積であるが、これは、本来のSi結晶の単位胞であるSi₈ の1/4の数の原子からなる領域である。ここで、Si₈ の体積は 0.543nm³で与えられることを用いると、Si₂の体積はその1/4として計算できる。また、ν_α,βn とE_α,βn は、それぞれ、オキシダント分子αと、添え字β_ｎに対応するシリコン原子または酸化シリコン構造との間の、化学反応の反応率定数の前置因子と活性化エネルギーを表す。また、k_B はボルツマン定数、T は酸化温度を表す。 The reaction rate constant of the chemical reaction included in the reaction term of the formula (4) is given by the following formula.

_{The coefficient V Si} 2 on the right side _{is the volume of the unit cell Si 2} consisting of two Si atoms, which is a region consisting of 1/4 of the number of atoms of _{Si 8} which is the unit cell of the original Si crystal. be. Here, using the fact that the volume of _{Si 8} ^{is given at 0.543 nm 3} _{, the volume of Si 2} can be calculated as 1/4 of that. In addition, ν _{α, β n} and E _α, β n are the precursors and activities of the reaction rate constant of the chemical reaction between the oxidant molecule α and the silicon atom or silicon oxide structure corresponding to the subscript β n, respectively. Represents chemical energy. In addition, k _B represents the Boltzmann constant and T represents the oxidation temperature.

In the following, the thermal oxidation process of a semiconductor material having a planar type interface will be considered. In this case, since the diffusion direction of the particles can be limited to the direction perpendicular to the interface, this direction is selected as the x-axis. Accordingly, we limit the metric tensor g _ij to have only diagonal elements (g, 1, 1) and consider only the diffusion of the oxidant molecule along the x-axis. By this simplification, the three-dimensional reaction-diffusion equation of the equation (4) results in the following one-dimensional reaction-diffusion equation.

_{The diffusion coefficient D α} of the molecule α included in the diffusion term of equation (6) is expressed by the following equation using the precursor factor D ⁰ _α and the activation energy E _{α using the Arrhenius form.}

In order to set the boundary conditions for solving the equation (6), the process of incorporating the oxidant molecule into the oxide film will be described with reference to FIG. The oxidant (16) supplied as a gaseous molecule is adsorbed on the surface (15) of the silicon oxide film (13) and becomes a chemisorbent state (17). The concentration of oxidant molecules in this chemically adsorbed state (17) is φ ^*, and the frequency of occurrence of the oxidant molecule uptake process from the surface (15) of the silicon oxide film (13) to the inside of the silicon oxide film (13) is R. _Assuming that it is inc, the frequency with which the chemically adsorbed oxidant molecule (17) changes to the state φ capable of diffusing in the silicon oxide film (13) is given by the following equation.

Here, it is used in the calculation of the diffusion process of oxygen molecules in the oxide film and the silicon crystal and the uptake process of oxygen molecules from the surface of the oxide film into the oxide film, which are included in the mathematical model according to the present invention. The values of the precursor D ₀ and the activation energy E are shown in Table 1. The first row, the second row, and the third row of Table 1 show _{the diffusion process of oxygen molecules in the silicon oxide film (SiO 2} ) and the diffusion process of oxygen molecules in the silicon crystal (c—Si), respectively. The values of the precursor factor and the activation energy for the process of incorporating oxygen molecules from the surface of the silicon oxide film into the oxide film are shown.

Next, the chemical formula of the oxidation reaction according to the present invention will be described with reference to FIG. As the oxidation process of the silicon crystal or the silicon oxide structure that proceeds at the interface (14) or the transition layer (12) in FIG. 2, the following series of chemical formulas are used, assuming that only the primary chemical reaction is used with respect to the reactant. think.

_{The values of the precursor R 0} of the reaction rate constant and the activation energy E of the chemical reactions represented by the chemical formulas (1) to (3) are shown in the first row, the second row, and the third row of Table 2, respectively. Shown in.

The volume ratio of the intermediate product produced by the chemical reaction of the chemical formulas (1) and (2) to the silicon crystal is interpolated by the following formula using _{V Si O2} = 2.25, which is the volume ratio of _{SiO 2 and crystalline silicon.} do.

Similarly, an intermediate product of a chemical reaction formula (1) or (2), the diffusion coefficient when a medium consisting of SiO in or SiO ₃ oxygen molecules diffuse, in Table 1, the silicon oxide film Using the diffusion coefficient of oxygen molecules in the silicon crystal, it is interpolated by the following formula.

Incidentally, the determinant of the square root of the metric tensor, since it represents the expansion of the local volume, and the square root, and density [rho _A of the local structure A, with the structure A and the volume ratio V _A of the crystalline silicon The following relationship holds between the two.

From the above, by executing the numerical integration of the equation (6) with the equation (11) as the constraint condition, the simulation of the thermal oxidation process using the substrate of the silicon crystal having the surface index (100) under the condition of the dry oxidation. Becomes executable.

When executing the numerical integration of the equation (6), the diffusion term of the equation (6) is decomposed into the second-order differential term of the concentration of the molecule α and the other terms as follows, and the former is decomposed into the diffusion term. , The latter will be treated as part of the reaction term.

When executing the numerical integration of the reaction diffusion equation of the equation (6), first, the diffusion term including the derivative of the concentration of the molecule α is rewritten into the form of the difference, and then the difference equation obtained by this conversion is performed. Numerical integration of is performed using the Crank-Nicolson method. (See Non-Patent Document 9.)

ここで、右辺の係数Dは、濃度φを持つ粒子の拡散係数である。 The Crank-Nicolson method is a calculation method devised to stably carry out numerical calculations when performing numerical integration of diffusion equations. To explain this method, consider the following diffusion equation in one-dimensional Euclidean space.

Here, the coefficient D on the right side is the diffusion coefficient of particles having a concentration φ.

ここで、濃度φの上付き添え字のｎとｎ＋１は、差分Δｔを単位として測った、離散的な時間を表すものとする。また、濃度φの下付き添え字は、差分Δｘを単位として測った、離散的な空間座標の座標値を表すものとする。 Equation (13) is differentiated by forward differentiation with respect to time t. Further, for the spatial coordinates x, a quadratic center difference is applied. By these processes, the diffusion equation of the equation (13) is replaced with the following difference equation.

Here, the superscripts n and n + 1 of the concentration φ represent discrete times measured in units of the difference Δt. Further, the subscript superscript of the density φ represents the coordinate values of the discrete spatial coordinates measured in units of the difference Δx.

In the Crank-Nicolson method, the right-hand side of equation (14) is approximated by the arithmetic mean of the equation on the right-hand side at discrete times n and n + 1. With this change, equation (14) is replaced by the following equation. (See Non-Patent Document 9.)

Next, among the terms on the right side, the term at time n + 1 is moved to the left side. Further, among the terms on the left side, the term at time n is moved to the right side. By this process, the following equation is obtained.

右辺の括弧（）に付けられた上付き添え字Tは、ベクトルの転置を表す。このベクトルを用いると、数式（１７）は、行列A、Bを用いて、以下の式に書き換えられる。

Here, the following vector is introduced.

The superscript T in parentheses () on the right-hand side represents the transpose of the vector. Using this vector, the equation (17) can be rewritten into the following equation using the matrices A and B.

If the matrices A and B are N x N matrices, the diagonal elements of these matrices are the coefficients of the second term on the left and right sides of equation (17), respectively. In addition, the matrix elements of (i, i-1) and (i, i + 1) in 1 <i <N are as shown by the coefficients of the first and third terms on the left and right sides of the equation (17). , -1 for matrix A and +1 for matrix B. All other matrix elements in 1 <i <N of matrices A and B are zero. The boundary conditions on the surface of the oxide film represented by the equation (8) and the boundary conditions at the lower end of the semiconductor material are reflected in the matrix elements of the matrices A and B at i = 1 and i = N.

When the equation (18) is solved by the Gaussian removal method, the value at each coordinate of the vector φ at the discrete time n + 1 of the time step Δt is obtained at each coordinate of the vector φ at the discrete time n, which is the previous discrete time. An expression expressed by the value in is obtained. By repeating this process, numerical integration of the difference equation represented by the equation (16) is executed for an arbitrary discrete time. (See Non-Patent Document 9.)

Now, let us return to the problem of thermal oxidation of semiconductor materials. When investigating the growth process of a silicon oxide film by computer simulation, it is necessary to calculate the film thickness of the oxide film every time the oxidation time is updated and record the change over time. In executing this calculation, the layer in which the substrate of the silicon crystal is partially oxidized and the occupancy of the crystal portion reaches 90% is regarded as the layer including the interface, and its coordinates are set to X1. The coordinates of the surface layer of the oxide film are X2. Using these, the film thickness of the silicon oxide film is calculated by the following formula that takes into account the volume expansion effect due to oxidation.

As an initial condition for solving mathematical formula (6) by numerical calculation, consider a state in which a natural oxide film is partially formed on the uppermost layer after dividing a semiconductor crystal into layers having a thickness of ΔX. The density of the uppermost layer is set to a weighted average of the ratio of the density of the semiconductor crystal of the silicon oxide structure and the substrate to 2: 8. In the layers below this layer, the density of all layers is set equal to the density of the semiconductor crystals of the substrate. For silicon crystals, this is 5.2x10 ²² cm ^-1 . In addition, the density of the silicon oxide structure can be obtained by a simple calculation by utilizing the fact that the volume ratio of the silicon oxide structure to the silicon crystal is 2.25.

数式（２０）において、左辺の濃度φの下付添え字１は、このφが酸化膜の最上層の濃度であることを表すものとする。また、右辺の濃度φの下付添え字０は、このφが、化学吸着状態にあるオキシダント分子の濃度を表すものとする。さらに、これらの濃度の上付き添え字ｎとｎ＋１は、差分により離散化された、酸化プロセスの経過時間を表す。また、酸化の対象である半導体材料が充分に大きな厚さを有するものとして、その下端部分におけるオキシダントの濃度を０とする。 Boundary conditions are set as follows. First, when the formula (8) is differentiated by the forward difference, the following formula is obtained.

In the formula (20), the subscript 1 of the concentration φ on the left side indicates that this φ is the concentration of the uppermost layer of the oxide film. Further, in the subscript 0 of the concentration φ on the right side, it is assumed that this φ represents the concentration of the oxidant molecule in the chemisorbed state. Further, the superscripts n and n + 1 of these concentrations represent the elapsed time of the oxidation process, discretized by the difference. Further, assuming that the semiconductor material to be oxidized has a sufficiently large thickness, the concentration of oxidant at the lower end portion thereof is set to 0.

Figure 3 shows the results of numerical integration with the differences between the time variables and the spatial coordinates set to Δt = 1.0x10-5 s and ΔX = 1 nm, respectively. In FIG. 3, the calculation results are arranged in ascending order from the bottom, with temperatures ranging from T = 800 ° C to T = 1200 ° C, all at 100 ° C intervals.

Here, referring to FIG. 2, in the process of thermal oxidation of the semiconductor crystal, the oxidant molecule diffuses in the silicon oxide film (13), and the transition layer (12) and the crystal substrate (11) of the semiconductor material are diffused. ), The process of reaching the interface (14) will be described. The oxidant molecule (16) that has reached the surface (15) of the silicon oxide film (13) is initially adsorbed on the surface (15) of the silicon oxide film (13) while maintaining the molecular morphology, and is in a chemisorbed state. It becomes (17). Molecules in this chemisorbed state (17) are taken into the inside of the silicon oxide film (13) from the surface (15) of the silicon oxide film (13) and then diffused in the silicon oxide film (13). , Reach the transition layer (12). Further, among these molecules, the molecule that has passed through the transition layer (12) and reached the interface (14), which is the boundary between the crystal substrate (11) of the semiconductor material and the transition layer (12), and the transition layer. In (12), the molecule that has reacted with the intermediate product generated by the chemical reaction of the chemical formula (1) or (2) is subjected to the chemical reaction of the chemical formulas (1) to (3) to form the silicon oxide film (13). It will contribute to growth.

Therefore, in the thermal oxidation process of the crystal substrate (11) of the semiconductor material, the effective diffusion barrier of the oxidant molecule is that the oxidant molecule (16) undergoes the chemical adsorption state (17) and the surface (15) of the silicon oxide film (13). ), The activation energy of the process of being taken into the silicon oxide film (13), the diffusion barrier when diffusing in the silicon oxide film (13), and the diffusion barrier when diffusing in the transition layer (12). It will be given in Japanese. This is expressed by the following formula.

The left side of the equation (21) is an effective diffusion barrier of oxidant molecules in the thermal oxidation process of the crystal substrate (11) of the semiconductor material. Further, the first term on the right side activates the process in which the chemically adsorbed oxidant molecule (17) adsorbed on the surface (15) of the silicon oxide film (13) is incorporated into the silicon oxide film (13). The second term is energy, the second term is the diffusion barrier associated with the process of the oxidant molecule (16) diffusing in the silicon oxide film (13), and the third term is the oxidant molecule (16) in the transition layer (12). It is a diffusion barrier that accompanies the process of diffusion. Here, the third term on the right side includes contributions from traps and the like of oxidant molecules due to defect structures and the like formed in the transition layer.

The following function is defined using the effective diffusion barrier of the oxidant molecule (16) represented by the formula (21).

When the calculation result of the film thickness of the oxide film obtained in the simulation of FIG. 3 is divided by the function ζ (T) and plotted again, the simulation results at all temperatures converge to one curve in FIG. 4. You can check from. This simulation, the value of each term of the right side of equation (21), are used _{E inc = 0.3eV, E D =} 0.2eV, the ΔE _D = 0.

Therefore, if a one-variable function of time variables that represents this curve can be specified, it should be possible to reproduce the simulation results at any temperature by simple calculation. It is assumed that this function is written in the form of a power function of time t, and that its exponent ν is a function of time t.

From this assumption, the equation expressing the film thickness of the oxide film at an arbitrary time t uses the function ζ (t) of the equation (22), the newly introduced function ν (t), the constant τ, and the coefficient K. , Given by the following equation.

ここで、τは1 h という値を有する、時間の次元を持つ定数である。 From the calculation performed assuming several functional forms of the exponent ν (t) and the comparison with the simulation results, if the functional form of the exponent ν (t) is selected as follows, the experimental results and the simulation results will be obtained. , Was found to match very well.

Where τ is a constant with a dimension of time that has a value of 1 h.

In Example 1 of the present invention, A = 0.20 is used as a value that gives a coincidence with the curve of FIG. 3 for the molecule A of the second term of the formula (24).

Moreover, as seen below, the coefficient K of the equation (23) is not a constant, is a function of the partial pressure P _OX and oxidation temperature T of the oxidant.

数式（２４）で与えられる指数νが常に１より小さいことと、時間ｔの対数関数が数式（２５）に含まれる事が、シリコン結晶の熱酸化の初期に観察される非常に大きな酸化速度の原因になっている。 By time-differentiating the equation (23), the following equation representing the oxidation rate of the semiconductor material is obtained.

The fact that the exponent ν given by equation (24) is always less than 1 and that the logarithmic function of time t is included in equation (25) is the very large oxidation rate observed in the early stages of thermal oxidation of silicon crystals. It is the cause.

In the following, multiple experiments show that the calculation result of the oxide film thickness obtained by using the mathematical formula (23) is very good with the measurement result of the oxide film thickness generated by the thermal oxidation of the silicon crystal. Illustrated by comparison with the results.

The assumptions for the analysis of the experimental results are described with reference to FIG. Under the condition of dry oxidation, oxygen molecules are used for the oxidant. After the oxygen molecules are adsorbed on the surface (15) of the silicon oxide film (13), if the oxidation temperature is high, a part of the oxygen molecules is oxidized to silicon. It evaporates from the surface (15) of the film (13). Therefore, in the following description, it is assumed that the effect of evaporation of oxygen molecules from the surface (15) of the silicon oxide film (13) is negligibly small when the oxidation temperature is 850 ° C. or lower.

FIG. 5 shows the measurement results of the film thickness of the silicon oxide film obtained in the thermal oxidation experiment under the dry oxidation condition of _{partial pressure POX} = 20 atm using a silicon crystal substrate in the plane index (100) direction. , Comparison with the calculation result by the mathematical formula (23). The calculation of the value of the effective diffusion barrier, with _{E D} ^eff = _2.8eV. The oxidation temperature in FIG. 5 is as follows. (a) T = 1000 ° C, (b) T = 950 ° C, (c) T = 900 ° C, (d) T = 850 ° C, (e) T = 800 ° C. In order to obtain a quantitative agreement with the equation (23), the following values were used for the coefficient K. (a) K = 9.6x10 ⁷ nm, (b) K = 1.14x10 ⁸ nm, (c) K = 1.26x10 ⁸ nm, (d) and (e) K = 1.4x10 ⁸ nm. (See experimental data Non-Patent Document 2. As for the method of determining the E _D ^eff and K, see Example 2.)

FIG. 6 shows the measurement results of the film thickness of the silicon oxide film obtained in the thermal oxidation experiment under the dry oxidation condition of _{partial pressure POX} = 20 atm using the crystal substrate of the silicon crystal in the plane index (111) direction. And the calculation result by the mathematical formula (23). The calculation of the value of the effective diffusion barrier, with _{E D} ^eff = 2.6 eV. The oxidation temperature in FIG. 6 is as follows. (a) T = 1000 ° C, (b) T = 950 ° C, (c) T = 900 ° C, (d) T = 850 ° C, (e) T = 800 ° C. In order to obtain a quantitative agreement with the equation (23), the following values were used for the coefficient K. (a) K = 4.8x10 ⁷ nm, (b) K = 5.7x10 ⁷ nm, (c) K = 6.3x10 ⁷ nm, (d) and (e) K = 7.0x10 ⁷ nm. (See experimental data Non-Patent Document 2. As for the method of determining the E _D ^eff and K, see Example 2.)

From the analysis results of the experimental data shown in FIGS. 5 and 6, it was found that the value of the coefficient K did not change when the temperature was 850 ° C. or lower. This indicates that the desorption of oxygen molecules adsorbed on the surface of the oxide film is not activated at T = 850 ° C. or lower. Therefore, if there is a measurement result of the film thickness of the oxide film obtained in the thermal oxidation experiment in this temperature region under the dry oxidation condition of _{partial pressure POX = 1 atm, the coefficient K included in the equation (23) is included.} It becomes possible to obtain the value of. (Refer to Example 2 for the method of determining the coefficient K.)

FIG. 7 shows the measurement results of the film thickness of the silicon oxide film obtained in the thermal oxidation experiment under the dry oxidation condition of _{partial pressure POX} = 1 atm using the crystal substrate of the silicon crystal in the plane index (100) direction. And the calculation result by the mathematical formula (23). The calculation of the value of the effective diffusion barrier, with _{E D} ^eff = _2.8eV. The oxidation temperature is as follows. (a) T = 1000 ° C, (b) T = 900 ° C, (c) T = 800 ° C. Further, the following values were used for the coefficient K of the mathematical formula (23). (a) K = 2.0x10 ⁷ nm, (b) K = 2.4x10 ⁷ nm, (c) K = 3.2x10 ⁷ nm. (Refer to Non-Patent Document 2 for experimental data.)

FIG. 8 shows the measurement of the film thickness of the silicon oxide film obtained in the thermal oxidation experiment under the condition of dry oxidation with _{partial pressure POX} = 1 atm using a crystal substrate of silicon crystals in the plane index (111) direction. It is a comparison between the result and the calculation result by the mathematical formula (23). The calculation using the _{E D} ^eff = 2.6 eV in value of the effective diffusion barrier. The oxidation temperature is as follows. (a) T = 1000 ° C, (b) T = 900 ° C, (c) T = 800 ° C. Further, the following values were used for the coefficient K of the mathematical formula (23). (a) K = 1.0x10 ⁷ nm, (b) K = 1.2x10 ⁷ nm, (c) K = 1.6x10 ⁷ nm. (Refer to Non-Patent Document 2 for experimental data.)

Here, the growth rate of the film thickness of the oxide film by thermal oxidation, relative to the partial pressure P _OX oxidant molecules, that have a sub-linear dependence is confirmed from the analysis of the experimental data.

FIG. 9 shows a thermal oxidation experiment under the condition of dry oxidation using a silicon crystal substrate in the plane index (100) direction. The oxidation temperature is fixed at T = 900 ° C., and the partial pressure is P _OX = 1 atm to P. It is a comparison between the measurement result of the film thickness of the silicon oxide film and the calculation result by the mathematical formula (23) when the temperature is changed to _{OX = 20 atm.} The calculation of the value of the effective diffusion barrier, with _{E D} ^eff = _2.8eV. The partial pressure used for the measurement is as follows. (a) P _OX = 20 atm, (b) P _OX = 10 atm, (c) P _OX = 5 atm, (d) P _OX = 1 atm. The following values were used for the coefficient K in the equation (23). (a) K = 1.3x10 ⁸ nm, (b) K = 8.4x10 ⁷ nm, (c) K = 5.9x10 ⁷ nm, (d) K = ^{2.5x10 7} nm. (Refer to Non-Patent Document 2 for experimental data.)

FIG. 10 shows the result of plotting the value of the coefficient K obtained by the data analysis of FIG. 9 on the _{logarithmic scale with respect to the partial pressure POX.} The slope of the straight line in FIG. 10 is 0.54. This result is dependent on the partial pressure P _OX coefficient K and by being represented by the power, that the index is less than 1, the partial pressure dependence of the sub linear is confirmed. (See Non-Patent Document 8.)

すでに述べたように、数式（２３）の係数Ｋは、オキシダントの分圧Ｐ_OXと酸化温度Ｔとの関数であった。これに対して係数Ｋ_１は、酸化温度Ｔのみの関数である。 Therefore, if the value of the coefficient K when the partial pressure P _{OX = 1 atm is written as K 1} and the slope in FIG. 10 is written as α _{, the dependence of the coefficient K on the partial pressure P OX} is written by the following equation. expressed.

As already mentioned, the coefficient K in equation (23) _{was a function of the oxidant's partial pressure POX} and the oxidation temperature T. On the other hand, the coefficient K ₁ is a function of the oxidation temperature T only.

Substituting the mathematical formula (26), the mathematical formula (23) is rewritten as follows.

From the above, by substituting the oxidation temperature, oxidation time, and partial pressure of oxygen molecules into the formula (27) under the condition of dry oxidation, the film thickness of the oxide film formed by the thermal oxidation of silicon can be calculated with high accuracy. It was shown that it can be done. Therefore, next, the validity of the mathematical formula (23) or the mathematical formula (27) when the type of the oxidant is changed is verified.

FIG. 11 shows the measurement results of the film thickness of the silicon oxide film obtained in the thermal oxidation experiment under the condition of wet oxidation with _{partial pressure POX} = 1 atm using a crystal substrate of silicon crystals in the plane index (100) direction. And the calculation result by the mathematical formula (23). The calculation of the value of the effective diffusion barrier, with _{E D} ^eff = 1.0 eV. The oxidation temperature in FIG. 11 is as follows. (a) T = 1200 ° C, (b) T = 1100 ° C, (c) T = 1000 ° C, (d) T = 920 ° C. For the coefficient K, K = 4.2 × 10 ⁴ nm was used regardless of the oxidation temperature. (For the method of experimental data of the reference .E _D ^eff Non-Patent Document 1 determines, see Example 2.)

FIG. 12 shows the measurement results of the film thickness of the silicon oxide film obtained by the thermal oxidation experiment using ozone having a partial pressure of 900 Pa with respect to the crystal substrate of the silicon crystal in the plane index (100) direction, and the mathematical formula ( It is a comparison with the calculation result by 23). The calculation of the value of the effective diffusion barrier, with _{E D} ^eff = 0.84 eV. The oxidation temperature in FIG. 6 is as follows. (a) T = 830 ° C, (b) T = 600 ° C, (c) T = 500 ° C, (d) T = 400 ° C, (e) T = 330 ° C, (f) T = 260 ° C. The following values were used for the coefficient K. (a) K = 4.0x10 ² nm, (b) K = 7.0x10 ² nm, (c) K = 1.1x10 ³ nm, (d) K = ^{1.9x10 3} nm, (e) and (f) ) K = 2.5x10 ³ nm. (For the method of experimental data of the reference .E _D ^eff Non-Patent Document 4 determines, see Example 2.)

The conversion of the pressure unit is 1 atm = 101325 Pa.

From the above, it was confirmed that the calculation results using the mathematical formula (23) show very good agreement with the experimental results in all of the dry oxidation, the wet oxidation, and the thermal oxidation using ozone. This demonstrates that the formula (23) is effective in calculating the thickness of the silicon oxide film produced by thermal oxidation of the semiconductor material regardless of the type of oxidant.

Therefore, using Equation (23) or formula (27), by creating a computer program, the coefficient K ₁ corresponding to the type and plane index and the type of oxidation temperature and oxidant semiconductor material subjected to thermal oxidation, the semiconductor material By specifying the type, surface index, and effective diffusion barrier corresponding to the type of oxidant, and further specifying the oxidation temperature, the partial pressure of the oxidant molecule, and the predetermined film thickness, an oxide film having a predetermined film thickness can be obtained. It will be possible to perform the calculation of the oxidation time required for formation with nanometer accuracy.

Hereinafter, a method for calculating the oxidation time of the semiconductor material according to the first embodiment of the present invention will be described with reference to FIGS. 13 and 14.

FIG. 13 is a block diagram of the arithmetic unit according to the first embodiment of the present invention. When the command to start the calculation is input from the input device (21), the arithmetic unit (22) reads out the computer program recorded in the recording device (23), and then the input device (21) waits for input. Next, when various conditions necessary for the calculation are input through the input device (21), the arithmetic unit (22) executes the calculation and outputs the obtained result to the output device (24).

FIG. 14 is a flowchart showing a method of calculating the oxidation time of a semiconductor material according to the first embodiment of the present invention.

First, the partial pressure of the oxidation temperature T and oxidant molecules P _OX and predetermined thickness X1 is, is input from the input device of FIG. 13 (21), these values, in the computer program, the corresponding variable Alternatively, it is set to a constant value (step S1).

Next, when the effective diffusion barrier E and the coefficient K are input through the input device (21) of FIG. 13, these values are set to the values of the corresponding variables or constants in the computer program (step S2). ).

When the film thickness of the oxide film at the start time of thermal oxidation, the increase Δt of the oxidation time, and the value of the tolerance of the film thickness are input to the second film thickness X2 through the input device (21) of FIG. , These values are set to the values of the corresponding variables or constants in the computer program (step S3).

Initialize the first oxidation time t1. (Step S4).

The second oxidation time t2 is set to the sum of the first oxidation time t1 and the increase Δt of the oxidation time. _{In addition, the value of the first function f (P OX , α) is calculated using the partial pressure P OX} of the oxidant molecule and the exponent α, and the second function is calculated from the ratio of the effective diffusion barrier E and the oxidation temperature T. The value of g (E / T) is calculated, and the value of the third function g (t2) is calculated using the second oxidation time t2. Further, the increase in film thickness ΔX is _{calculated using the product of the coefficient K 1} and the function values of the first to third functions (step S5).

The third film thickness X3 is set to the sum of the second film thickness X2 and the increase in film thickness ΔX (step S6).

It is determined whether or not the absolute value of the difference between the third film thickness X3 and the predetermined film thickness X1 is smaller than the margin of error (step S7).

When the determination result in step S7 is YES, the second oxidation time is output to the output device (24) of FIG. 13 (step S8).

When the determination result in step S7 is NO, it is determined whether or not the third film thickness X3 is smaller than the predetermined film thickness X1 (step S9).

If the determination result in step S9 is YES, the first oxidation time t1 is set to the second oxidation time t2, and the process returns to step S5 (step S10).

If the determination result in step S9 is NO, the increase Δt is divided by the real number B satisfying the condition B> 1, a new Δt is set, and the process returns to step S5 (step S11).

In Example 1 of the present invention, the chemical reactions represented by the chemical formulas (1) to (3) were considered, but the oxidation reaction was carried out by oxygen reaching the interface or transition layer between the crystal substrate of the semiconductor material and the oxide film. The atoms may be configured to contribute to the formation of an oxide film through a chemical reaction, and the formulas (21) to (26) are not limited to the use of the chemical formulas (1) to (3).

Further, steps S1 to S4 of the first embodiment of the present invention do not need to be set in this order, and these orders may be changed to another order.

Further, in the present invention, A = 0.20 is used for the molecule of the second term of the formula (24), but this value may be in the range of 0.18 <A <0.22, and A = 0 is not always required. It does not have to be .20.

Hereinafter, the oxidant molecule represented by the mathematical formula (21) of Example 1 of the present invention is used by using the measurement results of the film thickness of the silicon oxide film obtained by the measurement at a plurality of oxidation temperatures and a plurality of oxidation times. The method of analyzing the measurement result, which determines the effective diffusion barrier of No. 2 and the coefficient K of the mathematical formula (23), will be described with reference to FIG.

In the thermal oxidation process of the crystal substrate (11) of a semiconductor material such as silicon, the oxidant molecule (16) is first adsorbed on the surface (15) of the silicon oxide film (13), and then the silicon oxide film (13). ) Is desorbed from the surface (15) by evaporation or is incorporated into the silicon oxide film (13) at a finite frequency. Further, among the oxidant molecules incorporated into the silicon oxide film (13), the molecule that diffuses in the silicon oxide film (13), passes through the transition layer (12), and reaches the interface (14), and the transition. In the layer (12), the molecule that has reacted with the intermediate product generated by the chemical reaction of the chemical formula (1) or (2) contributes to the growth of the silicon oxide film (13).

The frequency of desorption of the oxidant molecule (17) chemically adsorbed on the surface (15) of the oxide film (13) from the surface (15) of the silicon oxide film (13) increases as the temperature increases. Therefore, if the measurement result of the film thickness of the silicon oxide film (13) at a high temperature is used to determine the effective diffusion barrier of the diffusion process of the oxidant molecule, the correct result may not be obtained.

Therefore, among the measurement results of the film thickness of the silicon oxide film (13) at a plurality of oxidation temperatures, the process of desorption of the oxidant molecule (17) from the surface (15) of the silicon oxide film (13) is activated. in a temperature region lower than the temperature at which the partial pressure of oxidant molecules was measured under the condition that the same, to the use of measurement results of two sets of different oxidation temperatures, noted these temperature T _a and T _B To distinguish. In addition, it is assumed that the film thickness of the silicon oxide film (13) was measured multiple times at each temperature, and each of the oxidation times was t _i (i = 1,2,3,…) using the subscript i. It is expressed as. Similarly, the thickness of the silicon oxide film (13) in the thermal oxidation of the oxidation temperature T _A and T _B, the measurement results of oxidation time t _i, respectively, to be referred to as Xi ^A and Xi ^B.

Same oxidation time _{t i (i = 1,2,3, ...} ) in the ratio of the measurement results at two different temperatures T _A and T _B is the first embodiment of the present invention and Equation (22) (24 ) is used, as a function of the effective diffusion barrier and oxidation temperature T _a and T _B of oxidant molecules, are expressed as follows.

From this, the effective diffusion barrier of the diffusion process of the oxidant molecule is calculated using the following equation using the measurement results of the film thickness of the silicon oxide film (13) at two different oxidation temperatures measured at the same oxidation time. You will be able to do it.

Furthermore, the same obtained by the oxidation experiment using the crystal substrates (11) having two different plane orientations of A and B, which were carried out by setting the oxidation temperature and the partial pressure of the oxidant molecule to the same value. If there are two sets of measurement results of _{{X i} ^A } and {X _i ^B }, which are the measurement results of the film thickness of the silicon oxide film (13) at the measurement time t _{i, the measurement results of these film thicknesses.} On the other hand, consider the ratio of the measurement results of the film thickness at the same time. Further, after applying the equation (23) to this ratio, the logarithm of this ratio is taken to obtain the following equation.

ここで、数式（３１）と数式（３２）の右辺は、それぞれ、数式（３０）に最小二乗法を適用して得られる直線の、傾きと縦軸の切片から得られる定数である。 Focusing on the fact that the equation (30) is a linear equation in which the result of dividing the reciprocal of temperature by 2 is a variable, applying the least squares method to this equation results in the following two relational expressions. can get.

Here, the right-hand side of the formula (31) and the formula (32) are constants obtained from the slope and the intercept of the vertical axis of the straight line obtained by applying the least squares method to the formula (30), respectively.

By combining the formula (29), the formula (31), and the formula (32), the analysis of the measurement result of the film thickness of the silicon oxide film (13) in both the two surface indices A and B can be simultaneously advanced. However, it is possible to improve the accuracy of the analysis result by removing a part of the systematic error included in these measurement results.

For example, the data of the silicon oxide film growth experiment by thermal oxidation obtained in the dry oxidation experiment on the Si (100) plane and Si (111) plane substrates shown in FIGS. 5 and 6 is a mathematical formula ( When 30) is applied, 0.2 eV and 2.0 are obtained as the values of the ratio of the difference between the effective diffusion barriers and the coefficient K of the Si (100) plane to the Si (111) plane, respectively. By combining this with mathematical formula (29), 2.8 eV and 2.6 eV are used as effective diffusion barriers for oxygen molecules in thermal oxidation on the Si (100) plane and Si (111) plane crystal substrates, respectively. Is obtained. Further, the result obtained by dividing the coefficient K obtained by the fitting performed in FIG. 5, which is the data of the Si (100) plane, by 2.0, which is the ratio of the coefficient K to the two sets of experimental data, is obtained by Si (111). It can be confirmed from FIG. 6 that by using it for the surface coefficient K, a very good agreement with the measurement result can be obtained by the calculation using the mathematical formula (23) for the data of the Si (111) surface. ..

The effective diffusion barrier determined by combining the mathematical formula (29) and the mathematical formulas (31) and (32) is the effective diffusion barrier used in step S2 of FIG. 14 of the first embodiment of the present invention. Barrier E. (See step S2 of Example 1.)

In addition, the superscripts A and B attached to the thickness variable on the left side of the formula (30) are not limited to the plane orientation of the crystal substrate, but also the difference in oxidation conditions such as dry oxidation and wet oxidation, and silicon. The composition of the crystal substrate may be different, such as a crystal and a silicon carbide crystal.

In the development of semiconductor devices including a silicon oxide film, it becomes possible to control the formation of the film thickness of the silicon oxide film by thermal oxidation of a substrate of a silicon crystal or a silicon carbide crystal with nanoscale accuracy.

1 It is an insulating film made of a silicon oxide film.
It is a 2 n-type Si or SiC semiconductor.
It is a 3p type Si or SiC semiconductor.
4 Gate terminal.
5 Source terminal.
6 Drain terminal.
11 It is a crystal substrate of a semiconductor material.
12 It is a transition layer.
13 Silicon oxide film.
14 Planer type interface.
15 The surface of the silicon oxide film.
16 It is an oxidant molecule of the gas phase.
17 It is an oxidant molecule chemically adsorbed on the surface of a silicon oxide film.
21 It is an input device.
22 Arithmetic logic unit.
23 It is a storage device.
24 Output device.
S1, which is the partial pressure P _OX and setting a predetermined thickness X1 of the oxidation temperature T and oxidant molecules.
S2 as effective diffusion barriers E and setting the coefficients K ₁ and exponent alpha.
S3 This is a step of setting an initial value for the second film thickness X2, and setting the increase Δt of the oxidation time and the tolerance of the film thickness.
S4 This is a step of setting the first oxidation time t1.
S5 The second oxidation time t2 is set to the sum of the first oxidation time t1 and the increase Δt of the oxidation time, and the second oxidation time t2, the oxidation temperature T, the partial pressure _{OX of the} oxidant molecule, and the effective diffusion barrier are set. From E, exponent α, and coefficient K ₁ , the product of coefficient K ₁ , first function f ( _POX , α), second function g (E / T), and third function h (t2) is used. Then, it is a step to calculate the increase ΔX of the film thickness in the second oxidation time t2.
S6 This is a step of calculating the third film thickness X3 = X2 + ΔX.
This is a step of comparing the absolute value of the difference between S7 X3 and X1 and the magnitude of the error.
S8 This is a step of outputting the second oxidation time t2.
This is a step of comparing the size of S9 X3 and X1.
This is a step of substituting the second oxidation time t2 for the first oxidation time t1 when S10 X3 is smaller than X1. Further, after this, the processing process returns to step S5.
When S11 X3 is larger than X1, this step is to reduce the increase Δt of the oxidation time while maintaining a positive value. Further, after this, the processing process returns to step S5.

Claims (8)

In a method for calculating the oxidation time of a semiconductor material, which is performed by designating a predetermined thickness of an oxide film formed by thermal oxidation of a semiconductor material, an oxidation temperature, and a partial pressure of an oxidant molecule.
The first film thickness, which is the predetermined film thickness, the oxidation temperature, the partial pressure, the activation energy of the process of incorporating the molecule from the surface of the oxide film into the inside of the oxide film, and the oxide film. An effective diffusion barrier consisting of the sum of the activation energy of the diffusion process of the molecule inside the above and the activation energy of the diffusion process of the molecule in the transition layer formed near the interface between the oxide film and the semiconductor material. The first increase, which is the increase in the oxidation time, and the tolerance of the film thickness of the oxide film are set, the first oxidation time is initialized, and the second film thickness is the thermal oxidation. The first step of setting the film thickness at the start time and
The second oxidation time is set to the sum of the first oxidation time and the first increase, and the film thickness formed by the thermal oxidation from the start time to the second oxidation time. The second increase, which is the increase in A second step calculated using the product of the function 2 and the third function, which is the function of the second oxidation time, and
It is determined whether or not the absolute value of the difference between the third film thickness, which is the sum of the second film thickness and the second increase, and the first film thickness is smaller than the margin of error. The third step to do and
In the third step, when it is determined that the absolute value is smaller than the permissible error, the second step of outputting the second oxidation time and the fourth step.
In the third step, when it is determined that the absolute value is larger than the permissible error, the fifth step of determining whether or not the third film thickness is smaller than the first film thickness.
When it is determined in the fifth step that the third film thickness is smaller than the first film thickness, the first oxidation time is set to the second oxidation time and the second oxidation time is set. The sixth process, which returns to the process, and
A method for calculating the oxidation time of a semiconductor material, which comprises. When it is determined in the fifth step that the third film thickness is larger than the first film thickness, the value is reduced while keeping the first increase at a positive value to reduce the second. The method for calculating the oxidation time of a semiconductor material according to claim 1, further comprising a seventh step, which returns to the step of the above. The fourth function, which is a function of the second oxidation time, in which the third function monotonically decreases from an initial value of less than 1 to 0.5 with an increase in the second oxidation time, is used as an exponent. The method for calculating the oxidation time of a semiconductor material according to claim 1 or 2, further characterized by having the power function of the second oxidation time. The method for calculating the oxidation time of a semiconductor material according to any one of claims 1 to 3, further characterized in that the first function comprises a power function of the partial pressure. When a predetermined thickness of the oxide film formed by thermal oxidation of the semiconductor material, the oxidation temperature, and the partial pressure of the oxidant molecule are specified, a computer for causing the computer to calculate the oxidation time of the semiconductor material. In the program
The first film thickness, which is the predetermined film thickness, the oxidation temperature, the partial pressure, the activation energy of the process of incorporating the molecule from the surface of the oxide film into the inside of the oxide film, and the oxide film. An effective diffusion barrier consisting of the sum of the activation energy of the diffusion process of the molecule inside the above and the activation energy of the diffusion process of the molecule in the transition layer formed near the interface between the oxide film and the semiconductor material. The first increase, which is the increase in the oxidation time, and the tolerance of the film thickness of the oxide film are set, the first oxidation time is initialized, and the second film thickness is the thermal oxidation. The first step of setting the film thickness at the start time and
The second oxidation time is set to the sum of the first oxidation time and the first increase, and the film thickness formed by the thermal oxidation from the start time to the second oxidation time. The second increase, which is the increase in A second step calculated using the product of the function 2 and the third function, which is the function of the second oxidation time, and
It is determined whether or not the absolute value of the difference between the third film thickness, which is the sum of the second film thickness and the second increase, and the first film thickness is smaller than the margin of error. The third step to do and
In the third step, when it is determined that the absolute value is smaller than the permissible error, the second step of outputting the second oxidation time and the fourth step.
In the third step, when it is determined that the absolute value is larger than the permissible error, the fifth step of determining whether or not the third film thickness is smaller than the first film thickness.
When it is determined in the fifth step that the third film thickness is smaller than the first film thickness, the first oxidation time is set to the second oxidation time and the second oxidation time is set. The sixth process, which returns to the process, and
A computer program for causing a computer to perform calculations by a method for calculating the oxidation time of a semiconductor material, which is characterized by being composed of. When it is determined in the fifth step that the third film thickness is larger than the first film thickness, the value is reduced while keeping the first increase to a positive value, and the second. A computer program for causing a computer to perform a calculation by the method for calculating an oxidation time of a semiconductor material according to claim 5, further comprising a seventh step, which returns to the step of. The fourth function, which is a function of the second oxidation time, in which the third function monotonically decreases from an initial value of less than 1 to 0.5 with an increase in the second oxidation time, is used as an exponent. A computer program for causing a computer to perform a calculation by the method for calculating an oxidation time of a semiconductor material according to claim 5 or 6, which is a power function of the second oxidation time having. The computer is made to perform the calculation by the calculation method of the oxidation time of the semiconductor material according to any one of claims 5 to 7, further characterized in that the first function comprises the power function of the partial pressure. Computer program for.

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