JP3214108B2 - Simulation method of semiconductor device - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a simulation method for analyzing electrical characteristics of a semiconductor device when designing the semiconductor device.

[0002]

2. Description of the Related Art Conventionally, as a simulation method of this kind, the electrical characteristics of a device have been estimated by solving a Poisson equation and a continuous current equation with a carrier temperature being a constant value equal to a substrate temperature. .

However, in recent submicron devices in which the so-called hot carrier effect greatly affects the electrical characteristics, there has been a problem that accurate simulation results cannot be obtained by the conventional method in which the carrier temperature is kept constant.

In view of the above, in addition to the above two conventional equations, an energy conservation equation for determining the carrier temperature has been introduced, and a method has been proposed in which each of these equations is discretized by the method of Schaeffer-Gummel and computer simulation is performed. (IEEE Transactions On
Computer Aided Design, VOL. 7, N
O2, February 1988 231-P. 242).

Further, as a solution of the above three equations, a so-called non-joining iterative solution has been proposed (simulation).
Of Semiconductor Devices and Processes,
VOL. 4 September 1991 483-P. 48
8), which is shown in FIG.

Referring to the figure, initial conditions are given in step 301, and the discretized Poisson equation and current continuity equation are solved under the initial conditions to obtain the electrostatic potential and carrier concentration as convergent solutions (steps 302 and 303). Subsequently, in steps 304 and 305, the discretized energy conservation equation is solved to obtain a carrier temperature as a convergence solution. If not converged, the process returns to step 302 again, and the above procedure is repeated until the entire convergence determination condition is satisfied.

[0007]

In the above-mentioned iterative non-coupling method, since the energy conservation equation is a rather complicated nonlinear equation with respect to temperature, it is necessary to calculate a steep pn junction or increase the applied bias voltage. This often causes divergence, and in the case of convergence, there is a problem that the convergence speed is slow as a whole.

The present invention solves such a problem, and solves the problem of poor convergence of the energy conservation equation in a semiconductor device model including the energy conservation equation.
It is an object of the present invention to provide a method of simulating a semiconductor device with an improved convergence speed.

[0009]

According to the simulation method of the present invention, a first step of converging a discretized Poisson equation and a current continuity equation to obtain a carrier mobility, a carrier diffusion coefficient, and a carrier current in a semiconductor device is provided.
Using the obtained carrier mobility, carrier diffusion coefficient and carrier current as constants, substituting them into a discretized energy conservation equation and solving the conservation equation in one operation, or
Or a second step of causing convergence by repetition of a plurality of calculations. The first and second steps are performed until the electrostatic potential, carrier concentration, and carrier temperature obtained after completion of the second step satisfy the convergence condition. Are calculated alternately,
It is to obtain the electrostatic potential, carrier concentration and carrier temperature in the semiconductor device as a convergent solution.

Further, in addition to the above configuration, convergence acceleration is performed when returning from the second step to the first step. Alternatively, in addition to the above configuration, convergence acceleration is performed in the second step.

[0011]

The carrier mobility, carrier diffusion coefficient and carrier current are generally complex functions with respect to carrier temperature, which makes it difficult to converge the energy conservation equation. If this is regarded as a constant, the energy conservation equation becomes a linear equation or a weakly nonlinear equation with respect to the carrier temperature, and can be solved by one or several matrix operations. This solves the problem that the energy conservation formula has poor convergence.

Although this method approximately solves the energy conservation equation, the convergence solution coincides with the solution obtained when it is strictly solved, so that there is no problem in accuracy.

According to another configuration of the present invention, since the convergence is accelerated, the convergence speed of the operation is improved.

[0014]

The following equation (1) has been proposed as an energy conservation equation discretized by the method of Schaeffer-Gummel.

[0015]

(Equation 1) In the above equation (1), the subscripts i and j are the node numbers, lij is the distance between the nodes i and j, dij is the cross-sectional area of the boundary traversed by the energy flux, and Ωi is the volume of the element. Here, the carrier temperatures Ti and Tj, the current density * J, the heat conduction coefficient κij, and the energy relaxation time τw are variables depending on the carrier temperature, the electron concentration ni, the electric field * E, the generated recombination rate Ri, the lattice temperature TO, The Boltzmann constant k is a variable or constant independent of the carrier temperature. ωij is a quantity related to * J and κ and therefore depends on the carrier temperature. However, τw is often treated as a constant. * J, * E
Is a vector.

Since * J, κ, and ω are complicated functions with respect to T, strictly solving the above energy conservation equation solves a strong nonlinear equation, which makes it difficult to converge the solution.

Therefore, once the Poisson equation and the current continuity equation are solved, the mobility, diffusion coefficient and current are calculated once, and these are regarded as constants and substituted into the above equation (1) to obtain * J, κ, With ω as a constant, this discrete equation results in a linear equation for T. Therefore, this equation can be solved by one matrix operation, and the convergence problem of the energy conservation equation is solved.

FIG. 1 shows a calculation procedure of the method of the present invention based on such an idea. In step 101, initial conditions are set,
The discretized Poisson equation and current continuity equation are solved (steps 102 and 103), and the mobility, diffusion coefficient, and current are obtained as convergence solutions (step 104). These are substituted into an energy conservation equation to linearize it, and a carrier temperature solution is obtained by one matrix operation (step 105).

At the stage when the carrier temperature solution is obtained, the overall convergence is determined (step 106). , Carrier concentration and carrier temperature.

If not converged, Aitken acceleration is performed (step 107). This is calculated from the initial value in the energy conservation equation, the sequence of carrier temperatures T0, T1, and T2 of the first solution and the second solution by calculating the convergence prediction value TE by the following equation (2), and calculating TE by Poisson equation. And the initial value of the current continuity equation, the overall convergence speed can be greatly improved. TE = T2 + (T2-T1) / (R-1) (2) where R = (T1-T0) / (T2-T1)

Since the Aitken acceleration may not give a good predicted value for a sequence of monotonically increasing or decreasing values, if the Aitken acceleration is performed only at the node where the carrier temperature converges while oscillating, the convergence speed is further increased. Can be raised.

Further, other acceleration methods can be used instead of or in addition to the Aitken acceleration.

In order to further improve the convergence speed, a coefficient matrix for solving the energy conservation equation as shown in the following equation (3) is obtained by adding a value proportional to the carrier temperature differential term of the joule heating term. It may be replaced.

[0024]

(Equation 2)

That is, since the mobility generally decreases as the carrier temperature increases, the second term on the right side of the above equation is always positive in the heat generation region, and the value of the replaced coefficient matrix increases, resulting in an overestimation of the carrier temperature. Suppression is suppressed, and the width of vibration at the time of convergence is reduced, so that convergence is further stabilized at high speed.

The same effect is obtained by increasing the value of the coefficient matrix as shown in the following equation (4), that is, appropriately selecting a function G that always becomes positive even if the carrier temperature changes, and adding the function G to the coefficient matrix. It is also achieved by:

[0027]

(Equation 3)

For example, G can be represented by the following equations (5) and (6).

(Equation 4)

[0029]

(Equation 5)

FIG. 2 shows the effect of the present invention.
7 is a graph showing the convergence error of the electron temperature and the number of operation repetitions when the simulation method of the present invention is applied to a one-dimensional PN junction and the reverse bias is set to 4 V in a 1 V step-up step. In the figure, the mark x is based on the conventional example, the triangle is based on one embodiment of the present invention in which Aitken acceleration was performed, and the square is further obtained by adding the electron temperature derivative term of the Joule heating term to the energy conservation equation coefficient. According to another embodiment of the present invention.

As can be seen from the figure, when the convergence error of the electron temperature is set to 10 ^-4 eV, it is necessary to repeat 13 calculation repetitions in the conventional example, whereas it is 8 when the Aitken acceleration is used. Times, when the electron temperature derivative term is added, 6
The number of calculation iterations is sufficient, and the convergence speed is greatly improved.

FIG. 3 shows another embodiment. As described above, when an appropriate function G is added to the coefficient matrix in order to suppress the overestimation of the carrier temperature, it is better to repeat the operation several times rather than to solve the energy conservation equation with only one matrix operation. May be obtained. As in the first embodiment, after obtaining the carrier temperature solution in step 205, convergence determination is performed (step 208). If not converged, Aitken acceleration is performed (step 209) and the process returns to step 204. In the figure, steps 201 to 207 correspond to step 101 in FIG.
To 107.

[0033]

As described above, according to the semiconductor device simulation method of the present invention, the energy conservation equation of the discretized fluid dynamic model is linearized to solve the problem of difficulty in convergence and accelerate convergence. By applying the method together, the overall convergence speed can be drastically improved.

[Brief description of the drawings]

FIG. 1 is a flowchart of a simulation calculation showing one embodiment of the present invention.

FIG. 2 is a graph showing the effect of the present invention.

FIG. 3 is a flowchart of a simulation calculation showing another embodiment of the present invention.

FIG. 4 is a flowchart of a simulation calculation showing a conventional example.

──────────────────────────────────────────────────続 き Continuation of front page (72) Inventor Toyoharu Kamiya 1-1-1, Showa-cho, Kariya-shi, Aichi Japan Inside Denso Co., Ltd. (56) References JP-A-4-127552 (JP, A) (58) Survey Field (Int.Cl. ⁷ , DB name) H01L 21/66 G06F 17/00 H01L 29/00

Claims (3)

(57) [Claims] 1. A first step of converging a discretized Poisson equation and a current continuity equation to obtain a carrier mobility, a carrier diffusion coefficient and a carrier current in a semiconductor device, and the obtained carrier mobility, carrier diffusion coefficient and Assuming that the carrier current is a constant, these are substituted into a discrete energy conservation equation to solve the conservation equation in one operation,
Or a second step of converging by repetition of a plurality of operations, and alternately performing the first and second steps until the electrostatic potential, carrier concentration, and carrier temperature obtained after the completion of the second step satisfy the convergence condition. And calculating a convergent solution to obtain an electrostatic potential, a carrier concentration, and a carrier temperature in the semiconductor device. 2. The method according to claim 1, wherein convergence acceleration is performed when returning from the second step to the first step. 3. The semiconductor device simulation method according to claim 1, wherein convergence acceleration is performed in the second step.