JPH06232231A - Electric properties evaluation device for semiconductor - Google Patents

Abstract

PURPOSE:To obtain a stable solution quickly in a numerical analysis of semiconductor elements. CONSTITUTION:This device is provided with a calculation part 13 based on a decoupled method and a calculation part 17 based on a Newton method and has a change over part 18 which changes over the two calculation parts in automatic mode. In the calculation part 13, a calculation part 14, which solves a Poison equation, an electron and electric current continuous equation and a hole current continuous equation, determines potential, electron concentration and hole concentration while a carrier temperature prediction part 15 provides an initial value of a carrier temperature and a calculation part 16 which solves electron energy conservation law, hole energy conservation law, calculates electron temperature and hole temperature. The calculation part 17 solves the Poison equation, the electron continuous equation, the hole current continuous equation, the electron energy conservation law and the hole energy conservation law based on the Newton method and determines everything about the potential, the electron concentration, the electron concentration and the hole temperature.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for analyzing and evaluating electric characteristics of a semiconductor device, and more particularly to a physical quantity representing electric characteristics inside a semiconductor by numerical calculation, such as a potential distribution, an electron concentration distribution, a hole concentration distribution, and an electron. The present invention relates to an electric characteristic evaluation apparatus for a semiconductor element, which obtains a value at a predetermined position of at least one physical quantity of a temperature distribution and a hole temperature distribution.

[0002]

2. Description of the Related Art As a method for numerically analyzing a potential distribution, an electron concentration distribution, and a hole concentration distribution in a semiconductor, three basic equations of Poisson equation, electron current continuity equation, and hole current continuity equation are numerically solved. Therefore, a method for analyzing the electrical characteristics of a semiconductor is known. Further, the above-mentioned means for predicting prior to the trial manufacture of a semiconductor device is known as a device simulation technique (for example, Analysis and Simulation of Semiconductor Devices, 1984, Publisher Springer Berlag, author S. ．S. Selberherr, 'Analysis
and Simulationof Semiconductor Devices', Springer
-Verlag, 1984)).

In this technique, as shown in FIG. 4, a grid line is drawn vertically and horizontally inside a semiconductor device, and the intersections thereof are used as discretization grid points A. On these grid points A, Poisson's equation, electron The following three simultaneous partial differential equations in total of the current continuity equation and the hole current continuity equation are solved for the unknown potential ψ, electron concentration n, and hole concentration p. Now, when the Poisson equation, the electron current continuity equation, and the hole current continuity equation are respectively expressed by the functionals of F, G, and H, F (ψ, n, p) = 0 G (ψ, n, p) = 0 H It can be expressed as (ψ, n, p) = 0. It is usually implemented in the form of a program that numerically solves these equations on a large computer. In the above equation, ψ is potential, n is electron concentration, p
Is the hole concentration.

In recent years, with the miniaturization of semiconductor elements, the energy of electrons and holes has been increased in order to solve the physical problem of hot carrier phenomenon in which the temperature of carriers (electrons and holes) is apparently higher than the device temperature. By solving the conservation law by solving the above three basic equations at the same time, a more accurate simulation in which the electron temperature and the hole temperature are taken into consideration has been aimed at.

As a result, the above three basic equations are slightly changed, and the following five basic equations can be solved. The electron energy conservation law and the hole energy conservation law are respectively represented by functionals P and Q: F (ψ, n, p, Tn, Tp) = 0 G (ψ, n, p, Tn, Tp) = 0 H ( ψ, n, p, Tn, Tp) = 0 P (ψ, n, p, Tn, Tp) = 0 Q (ψ, n, p, Tn, Tp) = 0 In the above equation, Tn is the electron temperature and Tp is the hole temperature.

Since the above five basic equations are non-linear equations related to unknown numbers ψ, n, p, Tn, and Tp, linearization is performed based on an initial value determined by some method, and a solution is iteratively obtained. The required method is used.

That is, starting from initial values ψ ₀ , n ₀ , p ₀ , Tn ₀ , and Tp _{0 of} ψ, n, p, Tn, and Tp, correction amounts δψ,
δn, δp, δTn, δTp are obtained by the method described below.

Ψ _{k + 1} ← ψ _k + δψ _k n _{k + 1} ← n _k + δn _k p _{k + 1} ← p _k + δp _k Tn _{k + 1} ← T _nk + δT _nk T _{pk + 1} ← T _pk This procedure is repeated until the + δT _pk correction amount becomes sufficiently small. Here, k = 0, 1, 2, ...

Solving the basic equations, δψ, δn, δp, δ
Tn and δTp are calculated, of which δψ, δn, δ
There are two known methods for obtaining p. They are called here the Newton method and the decoupled method.

The outline of these methods will be described below. In short, the Newton's method has the advantage that a solution may not be obtained in many cases, but in many cases, a solution can be obtained very quickly. The feature of is that it takes a long time to calculate, but a solution can be obtained reliably.

Here, a method for solving the basic equation by an iterative procedure will be described, and the Newton method and the decoupled method will be described. Now, for simplification, a case of solving three basic equations of Poisson equation, electron current continuity equation and hole current continuity equation will be described.

F (ψ ^* , n ^* , p ^* ) = 0 G (ψ ^* , n ^* , p ^* ) = 0 H (ψ ^* , n ^* , p ^* ) = 0 where ψ ^* , n ^* , P ^* is the true solution and the initial values ψ ₀ , n
_{For 0} and p ₀ , ψ ^* = ψ ₀ + δψ n ^* = n ₀ + δn p ^* = p ₀ + δp is satisfied. The correction amounts δψ, δn, δp satisfy the following simultaneous linear equations.

∂F / ∂ψ ・ δψ + ∂F / ∂n ・ δn + ∂F / ∂p ・ δp = −F (ψ ₀ , n ₀ , p ₀ ) ∂G / ∂ψ ・ δψ + ∂G / ∂n ・ δn + ∂G / ∂p ・ δp = −G (ψ ₀ , n ₀ , p ₀ ) ∂H / ∂ψ ・ δψ + ∂H / ∂n ・ δn + ∂H / ∂p ・ δp = − H (ψ ₀ , n ₀ , p ₀ ) When these equations are discretized using the N lattice points A shown in FIG. 4, 3N simultaneous linear equations are obtained. The correction amounts δψ, δn, and δp on each grid point A are obtained from this simultaneous equation, and the operation of adding them to the initial value is repeated. This is the Newton method described earlier.

By the way, when the voltage condition of the element to be analyzed is low and no current is flowing, the carrier density does not change with respect to a minute change of the voltage, and generation / recombination does not occur, so that ∂G / ∂ψ, ∂G / ∂p, ∂H / ∂ψ, ∂H /
The value of ∂n becomes so small that it can be ignored. In this case, correction amounts δψ, δn,
δp can be obtained. That is, ∂F / ∂ψ · δψ = −F (ψ ₀ , n ₀ , p ₀ ) ∂G / ∂n · δn = −G (ψ ₀ , n ₀ , p ₀ ) ∂H / ∂p · δp = -H (ψ ₀ , n ₀ , p ₀ ) 3 equations are solved separately, and the correction amounts δψ, δn, δp
Ask for. This is the decoupled method. In the decoupled method, N simultaneous linear equations are separately solved three times, but the total amount of calculation time is much larger in the Newton method, so the decoupled method is shorter.

On the other hand, when the voltage condition rises to some extent and a current flows, the carrier density changes greatly with respect to a slight change in the voltage, and the generation and recombination is performed easily, so that ∂G / ∂ψ, ∂G / ∂p, ∂H
The values of / ∂ψ and ∂H / ∂n become so large that they cannot be ignored. If you try to solve this by the decoupled method, the number of iterations will be too large, or a convergent solution cannot be obtained.

As described above, the Newton's method and the decoupled method are in a complementary relationship, and the user designates which one is to be used so as to make the best use of the advantages of both, and switches from the decoupled method to the Newton's method when a certain voltage condition is exceeded, When it is judged that the decoupled method does not converge based on some prediction, it is automatically switched to the Newton method.

On the other hand, as a solution of the above-mentioned electron energy conservation law and hole energy conservation law, ∂P / ∂T _n · δT _n = −P (ψ ₀ , n ₀ , p ₀ , Tn ₀ , Tp ₀ ) ∂ Q / ∂T _p · δT _p = −Q (ψ ₀ , n ₀ , p ₀ , Tn ₀ , Tp ₀ ) can be solved independently. This is the same as the above Poisson equation, electron current continuity equation and hole current continuity equation.
Two basic equations well describe the physical phenomena of semiconductors,
There is a reason that it is difficult to solve, for example, the energy conservation law does not converge if the initial value of the solution is not good, while the solution can be obtained quite consistently. Therefore, the three basic equations of Poisson's equation, electron current continuity equation and hole current continuity equation are completely converged, the solution is fixed by the energy conservation law, and the solution is obtained using only δTn and δTp as variables, and then again. The process of solving the above three basic equations is repeated.

[0018]

However, in the above-mentioned conventional configuration, a convergent solution cannot be obtained because the solution method is not good, or a solution having a long calculation time is selected although the solution can be obtained in a short time. There was something that happened.

In the energy conservation law, the model equation itself may be difficult to describe accurately, but the calculation method has poor convergence, and Poisson's equation, electron current continuity equation, and hole current continuity equation are not good. In many cases, the calculations did not diverge due to physical contradictions with the potential, electron concentration, and hole concentration that were the solution of the equation.

The object of the present invention is to quickly and stably implement the Poisson's equation, the electron current continuity equation, the hole current continuity equation, the electron energy conservation law, and the hole energy conservation law, which are the basic equations for conducting the numerical analysis of a semiconductor device. Another object of the present invention is to provide an electric characteristic evaluation apparatus that solves these potentials, the electron concentration, the hole concentration, the electron temperature, and the hole temperature, which are the solutions, without physical contradiction.

Another object of the present invention is that, when the calculation unit of the Newton method and the calculation unit of the decoupled method are switched, a convergent solution cannot be obtained because the solution is not good, or a solution can be obtained in a short time. In other words, it is an object of the present invention to provide an electric characteristic evaluation apparatus capable of surely obtaining a converged solution for any element structure and bias condition in a short time so as not to select a method having a long calculation time.

Another object of the present invention is to predict the carrier temperature, which is the solution of the energy conservation law, in order to solve the energy conservation law quickly and surely. An object of the present invention is to provide an electrical characteristic evaluation device capable of predicting a value that is close to a solution in advance.

[0023]

In order to solve the above-mentioned problems, an apparatus for evaluating electrical characteristics of a semiconductor device according to the present invention comprises an input section, a CPU connected to the input section, and a switch connected to the CPU. A calculation unit, two calculation units connected to the switching unit, and an output unit that outputs the output of the calculation unit,
One of the calculation units is at least a calculation unit of the decoupled method, which solves the Poisson equation, the electron current continuity equation, the hole current continuity equation, and the electron energy conservation law and the hole energy conservation law. A second calculator,
And a carrier temperature prediction unit that predicts the carrier temperature,
The other of the calculation units is a calculation unit of Newton's method, and is configured to include a calculation unit that solves the five basic equations simultaneously and repeatedly.

As a switching unit for switching the calculation method between the decoupled method and the Newton method, the calculation unit of the decoupled method is used to see how the correction amount δT of the carrier temperature T on the lattice point for discretization changes. After making a decision, the calculation unit of the Newton method is switched to, and thereafter the calculation unit of the decoupled method is not used.

As the carrier temperature predicting unit for predicting the carrier temperature in the semiconductor element, the energy conservation law is solved by the values of the electron concentration and the hole concentration before the energy conservation law is calculated by the calculation unit of the decoupled method. It is configured to have a function of predicting the required carrier temperature and setting it as an initial value.

[0026]

When a numerical analysis of a semiconductor device is performed, a convergent solution cannot be obtained because the solution is not good in the above-mentioned conventional configuration, or a method that requires a long calculation time although the solution can be obtained in a short time. I sometimes chose. In addition, although the energy conservation law sometimes makes it difficult to describe the physical phenomenon itself accurately, the convergence of the calculation method is not so good, and the Poisson equation, electron current continuity equation, and hole current continuity equation are solved. In many cases, it did not converge due to a physical contradiction between the potential, the electron concentration, and the hole concentration, or the calculation diverged.

However, the present invention has a function of solving the basic equation by the decoupled method and the Newton's method, and has a switching section for appropriately switching between them and a predicting section for predicting the carrier temperature. You can ask.

Further, in the present invention, the switching unit switches the calculation means to the calculation unit of the Newton method by making a judgment while observing how the correction amount δT of the carrier temperature T changes, and thereafter the calculation unit of the decoupled method is not used. The solution can be obtained quickly and stably.

In the present invention, the predicting unit has a function of predicting the carrier temperature obtained by solving the energy conservation law based on the values of the electron concentration and the hole concentration before calculating the energy conservation law in the decoupling method calculation unit. Since it has, it can obtain a solution quickly and stably.

[0030]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described below with reference to the drawings. FIG. 1 is a schematic configuration diagram showing an electric characteristic evaluation apparatus for a semiconductor device according to an embodiment of the present invention. In FIG. 1, 11 is a CPU for performing various calculations and controls, 12 is an input section for reading the element shape, which is the basic structure of the element, the donor / acceptor distribution in the semiconductor portion, the electrode position and the potential, and 13 is the basic equation by the decoupled method. A calculation unit for solving the Poisson equation, an electron current continuity equation, and a Hall current continuity equation out of the five basic equations, and 15 predicts a carrier temperature solution obtained by solving the energy conservation law, A carrier temperature predicting unit that gives a value, and 16 is a calculating unit that solves the electron energy conservation law and the hole energy conservation law.
6 is included in the calculation unit 13 of the decoupled method.

Reference numeral 17 is a calculator for solving Poisson's equation, electron current continuity equation, hole current continuity equation, electron energy conservation law, and hole energy conservation law by the Newton method, and 18 is a computation section 13 for solving by the decoupled method and the Newton method. A switching unit having a function of switching between the calculation unit 17 and the calculation unit 17, and 19 is an output unit that outputs the finally obtained converged solution.

Here, the characteristic part of the present apparatus is such that, in addition to the calculation unit 13 that solves the five basic equations by the decoupled method, the calculation unit 17 that solves by the Newton method is added, and they are switched so that they are calculated by an appropriate one. Switching unit 18 that can
Then, the carrier temperature predicting unit 15 for predicting the carrier temperature which is the solution of the energy conservation law is provided.

The characteristic part of the present apparatus is the switching unit 1 described above.
In step 8, while observing the change of the correction amount δT of the carrier temperature T, it is judged whether to continue the calculation with the decoupled method or to use the Newton method, and when it is judged that the Newton method is better, the Newton method is solved. To avoid returning to the decoupled method again. In addition, if it is determined that the calculation will diverge as it is, a function of changing the voltage condition in the input unit 12 and performing the calculation again is provided.

The characteristic part of this device is that the carrier temperature predicting section 15 solves the Poisson's equation, the electron current continuity equation and the hole current continuity equation by the decoupled method, and determines the depletion layer from the electron concentration and the hole concentration. The point is that a function for predicting the carrier temperature distribution inside is provided. The function of this prediction is to give a rising gradient of the carrier temperature from the edge of the depletion layer to the inside of the depletion layer by changing the carrier temperature under the previous voltage condition, and predict the carrier temperature of the entire device to be the initial value. That is.

Next, a method for evaluating the electrical characteristics of a semiconductor device using the above apparatus will be described with reference to the flowchart of FIG. 2 and the diagram for explaining the concept of prediction in FIG. FIG. 2 is a flowchart showing a processing procedure for evaluating electrical characteristics of a semiconductor device using the above apparatus. In FIG. 2, first, the terminal voltage for calculation is determined. The following treatments are all performed under one voltage condition, and then the following treatments are performed again under the next voltage condition. Under the current voltage conditions, the initial values of the electric potential, the electron concentration, and the hole concentration are obtained, and the Poisson equation, the electron current continuity equation, and the hole current continuity equation are solved to obtain the electric potential ψ, the electron concentration n, and the hole concentration p. . At this time, either a decoupled method for solving the three basic equations separately or a Newton's method for solving them simultaneously may be used. After this, a convergence judgment is made. If the convergence condition is satisfied, the process proceeds to the next step, and if not, this calculation is repeated. As the convergence determination condition, for example, the correction amounts δψ, δn,
A method such as determining whether or not each δp is a predetermined value or less can be considered. Next, the prediction unit predicts the electron temperature and the hole temperature calculated thereafter.

Next, using the predicted electron temperature and hole temperature as initial values, the electron energy conservation law and the hole energy conservation law are solved to obtain the electron temperature Tn and the hole temperature Tp. After this, a convergence judgment is made. If the convergence condition is satisfied, the process proceeds to the next step, and if not, this calculation is repeated. As the convergence determination condition, for example, a method of determining whether or not the correction amounts δTn and δTp are each less than or equal to a predetermined value can be considered.

Next, the switching unit determines whether to continue the calculation by the decoupled method or to switch to the Newton method. Here, it is determined whether to use the decoupled method or the Newton method while observing the changes in the maximum values of δTn and δTp. If it is determined that the calculation may diverge if the calculation is continued as it is, the voltage condition in the input unit is changed and the calculation is performed again.

Next, the calculation unit of the Newton method simultaneously calculates five basic equations to simultaneously obtain the potential ψ, the electron concentration n, the hole concentration p, the electron temperature Tn and the hole temperature Tp, and update the solution at the same time. . After that, the convergence judgment is performed, and if the convergence condition is satisfied, the calculation under the voltage condition is ended, and the above procedure is repeated under the next voltage condition.

When the final voltage conditions are completed, the final potential ψ, electron concentration n, hole concentration p, electron temperature Tn, and hole temperature Tp are used to output them as graphic data, current density, The distribution of terminal current, electric field, mobility, generation / disappearance speed, etc. is output as graphic data.

The switching unit of the above apparatus will be described with reference to FIG. If the maximum values of the correction amounts δTn and δTp obtained by solving the electron energy conservation law and the hole energy conservation law in the decoupling method calculation are smaller than the preset values, proceed to the Newton's calculation section. For example, return to the calculation section of the decoupled method. In the outer loop of the decoupled method, the maximum value δTnk of the correction amounts δTnk and δTpk at the k-th iteration in the entire analysis region.
When m and δTp km do not decrease and do not change or increase, the voltage is changed back to the input section so that the current voltage approaches the previous voltage, and the calculation is performed again. When this method is used, the convergence is improved, and the above-mentioned apparatus can automatically select the calculation method that can perform the fastest calculation.

The prediction unit of the above apparatus will be described with reference to FIG. The distribution of the depletion layer is obtained from the electron concentration n and the hole concentration p obtained by obtaining the Poisson equation, the electron current continuity equation and the hole current continuity equation in the calculation unit of the decoupled method. The carrier temperature is almost equal to the device temperature in the non-depletion layer, but increases from the edge of the depletion layer to the inside of the depletion layer, decreases after a certain peak, and becomes almost equal to the device temperature at the end of the depletion layer. . As shown in FIG. 3A, the rising angles of the electron temperature and the hole temperature obtained under the immediately previous voltage condition are approximated to a straight line, and as shown in FIG.
The carrier temperature of the entire depletion layer is predicted by shifting to the edge of the depletion layer under the current voltage conditions, and is used as the initial value when solving the electron energy conservation law and the hole energy conservation law.

The calculations of the electron energy conservation law and the hole energy conservation law often diverge when no prediction is made. However, if the prediction is performed by this method, both equations can be solved stably. it can.

The present invention is not limited to the above-described embodiments, but can be carried out in various modified forms without departing from the scope of the invention. For example, the device for carrying out the present invention is not limited to the configuration shown in FIG. 1, and the functions of the calculation unit, the output unit, the switching unit, etc. can be realized by software.

[0044]

As described above, according to the present invention, it is possible to obtain a stable and high-speed solution in numerical analysis of a semiconductor device, including the carrier temperature, which has been particularly difficult to solve conventionally.

[Brief description of drawings]

FIG. 1 is a schematic configuration diagram showing an electric characteristic evaluation apparatus for a semiconductor element according to an embodiment of the present invention.

FIG. 2 is a flowchart showing a processing procedure in the embodiment.

FIG. 3 is a conceptual diagram of prediction in the prediction unit of the same embodiment.

FIG. 4 is a schematic diagram showing a discretization lattice for device simulation.

[Explanation of symbols]

11 CPU 12 Input Unit 13 Decoupling Method Calculation Unit 14 Poisson's Equation, Electron Current Continuity Equation, Hole Current Continuity Equation Calculation Unit 15 Carrier Temperature Prediction Unit 16 Electron Energy Conservation Law, Hole Energy Conservation Law Calculation Unit 17 Newton's method calculation unit 18 Switching unit 19 Output unit

Claims (3)

[Claims] 1. An input unit and a CP connected to the input unit
U, a switching unit connected to the CPU, two calculation units connected to the switching unit, and an output unit that outputs the output of the calculation unit, one of the calculation units being at least the decoupled method. Of the Poisson's equation, the electron current continuity equation, and the hole current continuity equation, the second calculation unit for solving the electron energy conservation law, the hole energy conservation law, and the carrier temperature Of the semiconductor device having a carrier temperature predicting unit for predicting an initial value of, and the other of the calculating units is a calculating unit of Newton's method, and the calculating unit solves the five basic equations simultaneously and repeatedly. Characteristic evaluation device. 2. A switching unit for switching between the calculation units of the decoupled method and the Newton's method, wherein the correction unit δT of the carrier temperature T on the lattice points for discretization using the calculation unit of the decoupled method.
2. The electrical characteristic evaluation apparatus for a semiconductor device according to claim 1, wherein the calculation unit of the Newton's method is switched by making a judgment while observing the change of the condition, and the calculation unit of the decoupled method is not used thereafter. 3. A carrier temperature predicting unit for predicting a carrier temperature in a semiconductor device predicts a carrier temperature obtained by solving the energy conservation law based on the values of electron concentration and hole concentration before calculating the energy conservation law. The electrical characteristic evaluation apparatus for a semiconductor element according to claim 1, wherein the apparatus has an initial value function.