JP2655125B2 - Device simulation method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a numerical analysis of a semiconductor device.

[0002]

2. Description of the Related Art An outline of a general device analysis method In a numerical analysis of a semiconductor device, a drift-diffusion model in which carriers (electrons and holes) are approximated by regarding them as a fluid is widely used. A higher order approximation of the energy transport model may be used. Both the technique of the present invention described in this specification and the conventional technique can be applied to both the drift-diffusion model and the energy transport model. In the device simulation of the drift-diffusion model in the steady state, the basic equations are as follows:
A charge storage type, an electron current continuous type, and a hole current continuous type are set as shown below [Process process device simulation technology, edited by Tadashi Ryo, Sangyo Tosho, 1990.
Year].

[0003]

[0004] D: electric flux density [rho: charge density E: electric field epsilon: dielectric constant q: elementary charge p: hole density n: electron density N _D: donor concentration N _A: acceptor density

[0005]

J _n : electron current J _p : hole current R: carrier recombination term G: carrier generation term

[0007]

Μ _n : electron mobility μ _p : hole mobility D _n : electron diffusion coefficient D _p : hole diffusion coefficient

[0009]

K _B : Portzman's constant ：: temperature In the above equation, variables to be solved are potential ψ, electron density n, and hole density p. In general, the bias is sequentially updated with a plurality of specified applied biases as boundary conditions, and the charge conservation formula, the electron current continuous formula, and the hole current continuous formula are calculated.

The charge conservation formula, the electron current continuation formula, and the hole current continuation formula are non-linear equations. Therefore, in general, an iterative calculation called Newton's method is performed to obtain a solution. The Newton method is the following method.

Equation for variable x

[0013]

Is given. Some initial value x ₀
, If the value obtained by adding a certain amount of variation δx ₀ to x ₀ gives a solution

[0015]

## EQU1 ## Then, the derivative of F (x) is given by F '
By performing a first-order Taylor expansion of F (x ₀ + δx ₀ ) with respect to δx ₀ as (x ₀ ),

[0017]

## EQU1 ## So this time,

[0019]

Then, the same calculation is performed for x ₁ . This is sequentially repeated δx _i at the i-th calculation is,
If it becomes smaller than the appropriate minute amount ε (this is called “converged”, this judgment is called “convergence judgment”, and the minute amount ε is called “convergence condition”), x _i at that time is _expressed by the equation (22) )
Is the solution. The processing flow is the procedure shown in FIG. Also,
FIG. 13 schematically shows this procedure. In the one-dimensional case, as shown in FIG. 13, the solution approaches the solution while setting the intersection of the tangent and the x-axis as the next value of x. The closer the given initial value is to the solution, the less the number of iterations required to obtain the solution, and the shorter the computation time required to obtain the solution. That is, an initial value closer to the solution is a better initial value. The above is the Newton method.

In the above description of the Newton method, 1
In the device simulation, a mesh is generated in the entire analysis region, and an equation is set for variables on mesh points, as shown in FIG. That is, since the potential, the electron density, and the hole density are expressed as variables by the number of mesh points N,
3N simultaneous equations will be solved. The above-described charge conservation equation, electron current continuation equation, and hole current continuation equation are expressed as the following equations, with the terms on the right-hand side shifted.

[0022]

Ψ, n, and p in the above equation represent potential, electron density, and hole density, respectively, and represent N variables, respectively. In this case, the charge storage type, the electron current storage type,
Couple method (coupling method) that solves the hole current conservation equation at the same time,
There are the Gummel method (non-coupling method or decoupling method) that separately solves the charge conservation method, electron current conservation method, and hole current conservation method. FIG. 15 shows the procedure of the couple method, and FIG. 16 shows the procedure of the Gummel method. In FIG. 15, ｘ, n,
p represents all. The couple method can obtain a solution with a small number of iterations, but may not converge unless it is calculated with a good initial value. The Gummel method is not strongly dependent on the initial value, but requires a large number of iterations. The calculation time for one iteration is shorter in the Gummel method than in the Couple method, but the number of iterations is smaller in the Couple method than in the Gummel method, so the total calculation time to obtain a solution is The couple method is shorter. Therefore, as long as a good initial value can be given, analysis of a semiconductor device can be performed in a short calculation time by the couple method. That is, it is important to set a better initial value. Edwards et al. As a conventional technique for setting an initial value, Edwards et
al. (SP Edwards, A. et al.).
M. Howland and P.M. J. Mole, NA
SECODE IV, p. 272, 1985. ). In this method, the bias condition analyzed this time is set as the boundary condition between the pseudo-Fermi potential of electrons and holes, and the electron current and hole current using the pseudo-Fermi potential, electron density, and hole density obtained under the previous bias conditions are used. The estimated value of the fluctuation of the pseudo-Fermi potential is determined so that the value of the divergence equation is preserved, and the initial value of the potential is obtained from the estimated value.

Specifically, first, the equations for making the change in div of the continuous electron current type and the continuous hole current type zero are solved.

[0025]

Here, δψ _n and δψ _p are fluctuation amounts of the electron pseudo-Fermi potential and the hole pseudo-Fermi potential, respectively. These, obtain an estimate of the electron quasi Fermi potential [psi _n and Seiana擬Fermi potential [psi _p.

[0027]

Finally, the initial value of the potential is determined so that the carrier density of majority carriers is not changed from the value of the solution of the previous bias condition.

[0029]

N ₀ : Intrinsic carrier density In this method, the coefficient μ _n · of the equations (30) and (31) is used.
Since there is a large difference in the order of n and μ _p · p, there is a problem that it is difficult to calculate with high precision numerically.

[0031]

As described above, in the prior art, the coefficients of the equations for the electron pseudo-Fermi potential variation and the hole pseudo-Fermi potential variation for obtaining the initial value are determined by the electron density and the hole. Since the density is used as a factor, there is a problem that a change in the order is large and it is difficult to calculate numerically with high accuracy. Further, in this conventional technique, in order to obtain N potential initial values, 2N electron pseudo Fermi potential fluctuation amounts,
Since a hole pseudo-Fermi potential variation is required,
There is a problem that 2N equations have to be solved and the amount of calculation is large.

[0032]

In order to solve the above-mentioned problems, the initial value estimating method according to the present invention employs a Laplace equation for the amount of potential fluctuation, weighted by a coefficient using the reciprocal of the electric field strength.

[0033]

Δψ: Potential fluctuation amount. ψ _i : The potential of the analysis result under the previous bias condition. ψ _{i + 1} : Initial value of potential at the time of this analysis. Ε: Electric field strength. c _s0 : a measure of the maximum length characterizing the device structure. c _s1 : A measure of the minimum length that characterizes the device structure. c _f0 : a measure of the maximum value of the electric field strength. c _f1 : Estimated minimum value of electric field strength. c ₀ : Solving a positive constant to obtain an initial potential value.

Further, according to the present invention, in the numerical analysis of the transient analysis of the semiconductor device by the computer, the estimation of the initial value at the time of updating the bias condition is performed in addition to the coefficient using the reciprocal of the electric field intensity, and the analysis time before the last analysis Laplace's equation for potential variation, with coefficients weighted by a function that is uniformly increasing with respect to potential variation between analysis time

[0036]

Δψ _i : estimated potential fluctuation amount.

Ψ _{i + 1} : Estimated potential for the current analysis time.

Ψ _i : potential at the time of previous analysis.

Ψ _i-1 : potential before analysis time

Δt _i : Time interval between previous analysis time and current analysis time.

C _R : a positive constant.

P _R : constant.

C _s0 : a measure of the maximum length characterizing the device structure.

C _s1 : A measure of the minimum length characterizing the device structure.

C _f0 : a measure of the maximum value of the electric field strength.

C _f1 : Estimated minimum value of electric field strength.

C ₀ : A positive constant is solved to obtain a device simulation method having a procedure for estimating the initial value of the potential.

[0049]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS A first embodiment of the present invention will be described with reference to the drawings. Derivation of Solving Equation of the Present Invention Considering that a series of applied bias conditions are being analyzed, after analyzing the i-th bias condition and assuming that the (i + 1) -th bias condition is the current analysis, in the previous bias, The following charge conservation equation holds.

[0050]

[0051] D: electric flux density [rho: charge density epsilon: dielectric constant E: as [Delta] [phi] _i the field potential variation, representing the potential of this bias and _{ψ i + 1 = ψ + δψ} i. Here, without considering the redistribution of carriers (electrons and holes) (charge density ρ
(Ρ _{i +1} = ρ _i )), that is, without considering the electron current conservation equation and the hole current conservation equation, only the charge conservation equation holds,

[0052]

The following equation can be written. Subtracting these two equations gives

[0054]

The Laplace equation is obtained. Δψ _i obtained by solving the Laplace equation (47) can be regarded as an approximate potential fluctuation amount on the assumption that carrier redistribution is not considered. However, the fact that carrier redistribution is not considered is a rather rough approximation.

Generally, in the PN junction region, electrons and holes are annihilated and the carrier density is low, the potential gradient is steep due to the charge of the impurity ions, the electric field is high, and the resistance is higher than that of the surrounding region. high. Therefore, the fluctuation of the potential due to the applied bias propagates more to the PN junction region. That is, the Laplace equation (4
In 7), in the region where the electric field is high, it can be considered that the dielectric constant is apparently small. Therefore, in the Laplace equation (47), if the coefficient is reduced only in a region where the electric field is high, a larger amount of potential fluctuation can be propagated to that region, and it can be considered that carrier redistribution is considered. . That is,
By giving a coefficient that corresponds to the reciprocal of the electric field strength,
A better approximation potential fluctuation estimate is obtained, and a better potential initial value is obtained.

At this time, the number of Laplace equations to be solved is N, which is the same as the number of potential initial values to be solved. Therefore, while the conventional method has to solve 2N equations, the calculation time for the initial values is longer. Can be shortened.

Based on the above arguments, the present invention specifically calculates the following Laplace equation for the amount of potential fluctuation, weighted by a coefficient using the reciprocal of the electric field strength.

[0059]

Δψ: Potential fluctuation amount. ψ _i : The potential of the analysis result under the previous bias condition. ψ _{i + 1} : Initial value of potential at the time of this analysis. Ε: Electric field strength. c _s0 : a measure of the maximum length characterizing the device structure. c _s1 : A measure of the minimum length that characterizes the device structure. c _f0 : a measure of the maximum value of the electric field strength. c _f1 : Estimated minimum value of electric field strength. c ₀ : Positive constant In the above equation, the constant c ₀ is obtained when the electric field strength is zero (Ε =
0) Set an appropriate value that is a positive constant and large enough to be numerically divided so that the denominator does not become zero and calculation becomes impossible. The constant c _f1 has a role of a normalization constant of the electrolytic strength Ε, and adjusts the order of the coefficient ω.
The ratio of the constant c _s0 to the constant c _s1 (c _s0 / c _s1 ) is a quantity corresponding to the maximum variation ratio of the length quantity related to the device structure appearing in the numerical calculation, which is also in the order of the coefficient ω. Adjust The ratio between the constant c _f0 and the constant c _f1 (c _f0 / c _f1 ) is
This is an amount corresponding to the maximum fluctuation ratio of the electrolytic strength. The logarithmic ratio (ln ( _cs0 / _cs1 ) / ln ( _cf0 /) of the amount corresponding to the maximum fluctuation ratio of the length amount to the maximum fluctuation ratio of the electrolytic strength.
c _f1 )) adjusts the strength of the coupling between the variation in the electrolytic strength and the variation in the coefficient ω.

For these constants, the following values were set in the first and second embodiments.

[0062] _{c s 0: 1.0E + 4 c} s1: 1.0 c f0: 1.0E + 6 c f1: 1.0 c 0: 1.0 coefficient of the Laplace equation, unlike the conventional method, the variation of the large order Since it does not include a factor such as carrier density, the calculation does not cause difficulties in numerical calculation.

FIG. 1 shows a device simulation procedure according to the present invention.
Shown in

In the process 13 in FIG. 1, the Laplace equation (Equations (48) to (52)) weighted by the coefficient based on the reciprocal of the electrolytic strength is solved to obtain an initial potential value. For the carrier density (electron density, hole density), the previous analysis result is set as the current initial value. Using the set initial values, in the process 4 of FIG. 1A, the charge conserving type, the electron current continuous type, and the hole current continuous type are solved by the above-described couple method (FIG. 15). Processing 5
The analysis is repeated as many times as the number of the specified bias conditions.

An analysis example when this procedure is applied to an N-type MOSFET device structure will be described. Target N
FIG. 2 shows a device structure of the MOSFET. The PN junction is indicated by a broken line (25, 26). FIG. 3 shows the impurity distribution in the lateral direction below the gate interface. Black square ■, solid line (d
onor) is a donor, white square □
A dotted line (denoted as acceptor) represents an acceptor. FIG. 4 shows the analysis mesh. The number of mesh points is 1475. FIG. 5 shows a potential initial value and a potential analysis result as a result of applying the method of the present invention to this device structure in the horizontal direction below the gate interface. This is because the gate electrode (22 in FIG. 2)
V, 0.5 V for the drain electrode (23 in FIG. 2) and 0.0 for the source electrode (21 in FIG. 2) and the substrate electrode (24 in FIG. 2).
The bias condition of V is set as the previous analysis bias condition, and the current bias condition is a condition in which only the drain bias is increased by 0.1 V from the previous bias condition. Black square ■
What is indicated by a solid line (described as Result) is the analysis result of the bias condition this time, and a white square □ and a dotted line (Estimate (proposed m)
What is described as (ethod) is the potential initial value according to the method of the present invention.
What is indicated by “imate” (described as Edwards et al.) is the potential initial value according to the above-described conventional method.

As shown in FIG. 5, the potential initial value according to the method of the present invention gives a value closer to the analysis result than the potential initial value according to the conventional method. That is, the potential initial value according to the method of the present invention is a better initial value.

In practice, the number of iterations required to obtain a solution when the initial value of each of the method of the present invention and the conventional method is given, the convergence condition is set to 1.0E-6, and the method of the present invention is Four times and five times in the conventional method, the initial value by the method of the present invention can obtain a solution with a smaller number of iterations.

The calculation time for obtaining the initial value in this case is also 4.35E-3 seconds for the method of the present invention and 6.23E-3 seconds for the conventional method in a computer equipped with a 33MIPS CPU. Yes, the method of the present invention is faster.

Next, the method according to the second embodiment of the present invention is described as NP
A case where the present invention is applied to an N-type bipolar transistor structure will be described with reference to the drawings.

FIG. 6 shows a bipolar transistor structure of this example. 61 and 63 are base electrodes, 62 is an emitter electrode, 64 is a collector electrode, and 65 is a substrate electrode. PN
The junction is represented by a dashed line (66, 67). FIG. 7 shows the impurity distribution in the depth direction below the emitter electrode. Black square mark ■
A solid line (one described as donor) represents a donor, and a white square □ represents a dotted line (one described as acceptor). FIG. 8 shows an analysis mesh in this example. The number of mesh points is 1495. FIG. 9 shows the potential initial value obtained as a result of applying the method of the present invention to this bipolar transistor structure, the potential initial value obtained by the conventional method, and the potential value of the analysis result in the depth direction below the emitter electrode. .
This means that 0.8V is applied to the base electrode (61, 63 in FIG. 6).
1.0 V is applied to the collector electrode (64 in FIG. 6) and 0.0V is applied to the emitter electrode (62 in FIG. 6) and the substrate electrode (65 in FIG. 6).
In this bias condition, only the base bias is increased by 0.02 V from the previous bias condition. Black square mark ■, solid line (R
The result indicated by the current bias condition is indicated by a white square □ and a dotted line (estimate (proposedmetho)).
Those described in d) are potential initial values according to the method of the present invention, and are indicated by white circles.
A broken line (Estimate (Edwards et a
l. ) Are potential initial values according to the above-described conventional method.

As shown in FIG. 9, the potential initial value according to the method of the present invention gives a value closer to the analysis result than the potential initial value according to the conventional method. That is, the potential initial value according to the method of the present invention is a better initial value.

Actually, when the initial value of each of the method of the present invention and the conventional method is given, the number of iterations required to obtain a solution is determined by setting the convergence condition to 1.0E-12 and using the method of the present invention. The number of times is five and the number of conventional methods is six, and the initial value according to the method of the present invention can obtain a solution with a smaller number of iterations.

The calculation time for obtaining the initial value in this case is also 5.26E-3 seconds for the method of the present invention and 5.29E-3 seconds for the conventional method in a computer equipped with a 33 MIPS CPU. Yes, the technique of the present invention is slightly, but faster.

Next, a third embodiment of the present invention will be described with reference to the drawings.

This embodiment is directed to claim 2 of the present invention relating to the case of transient analysis.

In general, in the case of the transient analysis, the basic equation of the device simulation includes the time partial differential term d of the carrier.
The expression to which n / dt and dp / dt are added is analyzed. Then, in the transient analysis, a displacement current generated due to the time change of the carrier appears. In estimating the initial value during such transient analysis, by considering the displacement current,
Even better initial values can be obtained. According to the method of claim 2 of the present invention, it is assumed that the effect of the displacement current is large in the region where the potential change over time is large, and it is considered that the apparent resistance becomes small in the region where the displacement current is large, and the apparent dielectric constant in the Laplace equation is reduced. We think that the rate has become smaller. That is, in addition to the above-described method of the first aspect of the present invention, the influence of the displacement current is taken into consideration by transmitting the amount of potential fluctuation even in a region where the potential changes over time. Specifically, in addition to the coefficient ω using the reciprocal of the electric field strength of claim 1 as shown below, a function that is uniformly increased with respect to the potential change amount between the analysis time before the previous analysis and the analysis time before the previous analysis is used. Compute the Laplace equation for the amount of potential variation with the weighted coefficient R.

[0077]

Δψ _i : estimated potential fluctuation amount.

Ψ _{i + 1} : Estimated potential for the current analysis time.

Ψ _i : potential at the time of the previous analysis.

Ψ _i-1 : the analysis time potential two times before.

Δt _i : Time interval between previous analysis time and current analysis time.

C _R : a positive constant.

P _R : constant.

C _s0 : a measure of the maximum value of the length characterizing the device structure.

C _s1 : Estimate of the minimum length characterizing the device structure.

C _f0 : a measure of the maximum value of the electric field strength.

C _f1 : Estimated minimum value of electric field strength.

C ₀ : a positive constant. In the above equation, the constant C _R is a constant that determines the weight ratio of the coefficient R relating to the potential time change and the coefficient ω relating to the electric field strength. Constant P _R is to adjust the strength of the coupling between the variation of the potential time variation and the coefficient R. Also, the constant C
₀ is a positive constant for preventing the calculation from being impossible due to the denominator being zero when the electric field strength is zero (E = 0), and sets an appropriate value that can be numerically divided. I do. The constant c _f1 serves as a normalization constant for the electric field strength E, and adjusts the order of the coefficient ω. The ratio of the constant c _{s0 and the} constant c _s1 (c _s0 / c _s1 ) is an amount corresponding to the maximum variation ratio of the amount of length related to the device structure appearing in the numerical calculation, which also has the order of the coefficient ω. Adjust. The ratio of the constant c _f0 and the constant c _f1 (c _f0 / c _f1 ) is an amount corresponding to the maximum fluctuation ratio of the electric field intensity. The logarithmic ratio (l) between the amount corresponding to the maximum variation ratio of the length amount and the amount corresponding to the maximum variation ratio of the electric field intensity
n (c _s0 / c _s1 ) / ln (c _f0 / c _f1 )) adjusts the strength of the coupling between the electric field intensity fluctuation and the coefficient ω fluctuation.

For these constants, in the third embodiment and the fourth embodiment, the following values are set.

C_R： 0.1 ・ ω_max/ ｜ ψ_i−ψ_i-1|_max  P_R: 0.5 c_s0: 1.0E + 4 c_s1: 1.0 c_f0: 1.0E-6 c_f1: 1.0 c₀: 1.0 The coefficient of the Laplace equation is also calculated according to the method of claim 1 of the present invention.
Similar to the conventional method, large order fluctuations
Does not include factors such as carrier density
There is no value difficulty. Specific processing means of the present invention Device simulation at the time of transient analysis in the present invention
1 (b) is shown.

In process 9 in FIG. 1B, the Laplace equation (Equations (53) to (5)) weighted by the coefficient based on the potential time change and the coefficient based on the reciprocal of the electric field strength is used.
8)) to obtain the potential initial value. For the carrier density (electron density, hole density), the previous analysis result is set as the current initial value. Using the set initial values, in the process 10 of FIG. 1B, the charge conserving equation, the electron current continuous equation, and the hole current continuous equation are solved by the above-described couple method (FIG. 15). By the branch of the process 11, the analysis is repeated by the number of the bias conditions at the specified analysis time.

An analysis example of a transient analysis when this procedure is applied to an N-type MOSFET device structure will be described. FIG. 2 shows the device structure of the target N-type MOSFET. The PN junction is represented by a broken line (25, 26). FIG. 3 shows the impurity distribution in the lateral direction below the gate interface. Black squares {}, practice (written as donor) indicates a donor, and white squares □, dotted lines (acceptor) indicate acceptors. FIG. 4 shows the analysis mesh. The number of mesh points is 1475. FIG. 10 shows a potential initial value and a potential analysis result as a result of applying the method of claim 2 of the present invention to the transient analysis of the device structure in the horizontal direction below the gate interface. In this figure, the gate electrode (22 in FIG. 2) is set to 2.0 V, the source electrode (21 in FIG. 2) and the substrate electrode (24 in FIG. 2) are set to 0.
At 0 V, only the bias condition of the drain electrode (23 in FIG. 2) was set to 0.
In the transient analysis changed from 4V to 2.2V,
The situation at the time of 0.833 ns (drain bias 1.9 V) is shown. The results indicated by the black squares and solid lines (described as Result) are the analysis results of the bias conditions this time, and the white circles and broken lines (Estim
ate (proposed method No. 1)) is the potential initial value according to the method of claim 1 of the present invention.

As shown in FIG. 10, in the transient analysis, the potential initial value according to the method of the second aspect of the present invention is smaller than the potential initial value according to the method of the first aspect of the present invention in the analysis result. We are giving close values. That is, in the transient analysis, the potential initial value according to the method of claim 2 of the present invention is an even better initial value.

Actually, the method of claim 2 of the present invention, when the respective initial values are given, the number of iterations required to obtain a solution is based on the assumption that the convergence condition is 1.0E-8. The method of item 2 is 5 times, the method of claim 1 of the present invention is 6 times, and the method of the initial value according to the method of claim 2 of the present invention obtains a solution in a transient analysis even with a smaller number of iterations. be able to.

Next, the method according to claim 2 of the present invention is applied to a transient analysis of an NPN-type bipolar transistor structure.
Will be described with reference to the drawings.

FIG. 6 shows the structure of the bipolar transistor of this example. 61 and 63 represent base electrodes, 62 represents emitter electrodes, 64 represents collector electrodes, and 65 represents substrate electrodes. P
The N junction is represented by a broken line (66, 67). FIG. 7 shows the impurity distribution in the depth direction below the emitter electrode. A black square ■ and a solid line (denoted as donor) represent a donor, and a white square □ and a dotted line (denoted as acceptor) represent an acceptor. FIG. 8 shows an analysis mesh in this example. The number of mesh points is 1495. The potential initial value obtained as a result of applying the method of claim 2 of the present invention to the transient analysis of the bipolar transistor structure, the potential initial value according to the method of claim 1 of the present invention, and the potential value of the analysis result, FIG. 11 is an enlarged view of the vicinity of the base in the depth direction below the emitter electrode. This figure shows the emitter electrode (6 in FIG. 6).
2) The bias between the substrate electrode (65 in FIG. 6) and 0.0 V
The bias of the collector electrode (64 in FIG. 6) is fixed at 1.0 V, and the base electrode (61, 63 in FIG. 6) is fixed.
In the transient analysis in which only the bias condition was changed from 0.75 V to 0.85 V in the time of 0.0 s to 1.0 ns, the time was 0.5 ns (the base bias was 0.8 volt).
The situation at the time of V) is shown. Black square mark ■, solid line (Resu
It is the analysis result of the bias condition this time, which is indicated by a white square □ and a dotted line (E
steady (proposed method N
o. 2)) are potential initial values according to the method of claim 2 of the present invention, and are indicated by white circles and broken lines (Estimate (propos)).
ed method No. What is described in 1)) is the potential initial value according to the method of claim 1 of the present invention.

As shown in FIG. 11, the potential initial value according to the method of claim 2 of the present invention has a value closer to the analysis result than the potential initial value according to the method of claim 1 of the present invention. I have. That is, in the transient analysis, the potential initial value according to the method of claim 2 of the present invention is a better initial value.

In practice, when the method of claim 2 of the present invention and the method of claim 1 of the present invention are given their initial values,
The number of iterations required to obtain a solution is determined by setting the convergence condition to 1.0E
As −9, the method of claim 2 of the present invention is four times, the method of claim 1 of the present invention is five times, and the initial value by the method of claim 2 of the present invention is smaller in the number of iterations. A solution can be obtained.

[0100]

As described above, the method of the present invention has no difficulty in numerical calculation, can provide a better initial value, has the effect of speeding up analysis and shortening the calculation time.

[Brief description of the drawings]

FIG. 1 is an explanatory diagram of a procedure of a device simulation process according to the technique of the present invention.

FIG. 2 is an explanatory diagram of a structure of an N-type MOSFET according to the first and third embodiments.

FIG. 3 is an explanatory diagram of a lateral impurity distribution below a gate interface of the N-type MOSFETs of Examples 1 and 3.

FIG. 4 is an explanatory diagram of an analysis mesh according to the first and third embodiments.

FIG. 5 is an explanatory diagram of a potential initial value according to the method of the first embodiment of the present invention, a potential initial value according to a conventional method, and a potential value as an analysis result.

FIG. 6 is an explanatory diagram of the structure of the NPN-type bipolar transistors of Examples 2 and 4.

FIG. 7 is an explanatory diagram of the impurity distribution in the depth direction below the emitter electrode of the NPN-type bipolar transistors of Examples 2 and 4.

FIG. 8 is an explanatory diagram of an analysis mesh according to the second and fourth embodiments.

FIG. 9 is an explanatory diagram of a potential initial value according to the method of the present invention in Embodiment 2, a potential initial value according to a conventional method, and a potential value as an analysis result.

FIG. 10 is an explanatory diagram of a potential initial value according to the method of the second embodiment of the present invention, a potential initial value according to the first embodiment of the present invention, and a potential value of an analysis result.

FIG. 11 is an explanatory diagram of a potential initial value according to the method of claim 2 of the present invention, an initial potential value according to the method of claim 1 of the present invention, and a potential value as an analysis result.

FIG. 12 is an explanatory diagram of the Newton method.

FIG. 13 is a schematic explanatory view of the Newton method.

FIG. 14 is an explanatory diagram of an example of an analysis mesh.

FIG. 15 is an explanatory diagram of the couple method.

FIG. 16 is an explanatory diagram of the Gummel method.

[Explanation of symbols]

11 to 12 processing steps 21 source electrode 22 gate electrode 23 drain electrode 24 substrate electrode 25, 26 PN junction 61, 63 base electrode 62 emitter electrode 64 collector electrode 65 substrate electrode 66, 67 PN junction 101 Initial value setting processing step 102 repetition Processing step of initializing the number counter 103 Processing step of obtaining a fluctuation amount 104 Processing step of convergence determination 105 Processing step of updating value 106 Processing step of ending 111 Initial value 112 Solution 131 Processing step of initial value setting 132 Charge storage formula, A processing step for simultaneously solving the electron current continuous equation and the hole current continuous equation to obtain fluctuation amounts of potential, electron density, and hole density 133 Processing step of convergence determination 134 Processing step of updating value 135 Processing step of ending 141 Initial value Processing settings Step 142 A process for obtaining the potential variation by solving the charge conservation equation. 143 A process for obtaining the variation in the electron density by solving the continuous electron current formula. 144 A process for obtaining the variation in the hole density by solving the hole current continuous formula. Step 145 Convergence determination processing step 146 Value update processing step 147 End processing step

Claims (2)

(57) [Claims] In a numerical analysis of a semiconductor device by a computer, an initial value estimation at the time of updating a bias condition is weighted by a coefficient using a reciprocal of an electric field intensity.
Laplace equation for potential fluctuations δψ: Potential fluctuation amount. ψ _i : The potential of the analysis result under the previous bias condition. ψ _{i + 1} : Initial value of potential at the time of this analysis. Ε: Electric field strength. c _s0 : a measure of the maximum length characterizing the device structure. c _s1 : A measure of the minimum length that characterizes the device structure. c _f0 : a measure of the maximum value of the electric field strength. c _f1 : Estimated minimum value of electric field strength. c ₀ : A device simulation method having a procedure of solving a positive constant and estimating an initial value of potential. 2. In a numerical analysis of a transient analysis of a semiconductor device by a computer, in addition to a coefficient using a reciprocal of the electric field strength, an initial value estimation at the time of updating a bias condition is performed between an analysis time two times before and a previous analysis time. Laplace's equation for potential variation with coefficients weighted by a uniformly increasing function for the potential variation of δψ _i : estimated potential fluctuation amount. ψ _{i + 1} : Estimated potential for the current analysis time. ψ _i : Last analysis time potential. ψ _i-1 : Pre-analysis time potential. δt _i : Time interval between previous analysis time and current analysis time. C _R : a positive constant. P _R : constant. c _s0 : a measure of the maximum length characterizing the device structure. c _s1 : A measure of the minimum length that characterizes the device structure. c _f0 : a measure of the maximum value of the electric field strength. c _f1 : Estimated minimum value of electric field strength. c ₀ : a positive constant. And a device simulation method having a procedure for estimating the initial value of the potential.