JP3152184B2 - Device simulation method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

[0001] The present invention relates to a method for simulating a semiconductor device.

[0002]

2. Description of the Related Art For example, the following literature is referred to as a semiconductor device simulation technique.

(1) Ryo Tan, edited by "Process / Device Simulation Technology", Industrial Books, pp. 99 ~
104, 1990. (2) R. Thoma et al., “Hydrodynamic Equatio
ns for Semiconductors with Nonparabolic Band
Structure ”, IEEE Transactions on Electro
n Devices, Vol. 38, No. 6, pp. 1343-135
3, June, 1991. (3) Yoichi Hayashi, Ryo Tan, "Adaptive mesh-type device simulation-Comparison of evaluation criteria-", IEICE Technical Report, SDM90-106, pp. 139-143. 41-4
5, 1990.

[Outline of General Device Simulation] In the numerical analysis of a semiconductor device, a carrier (electrons, holes) is regarded as a fluid and a drift is approximated.
Diffusion models and higher-order approximation energy transport models are widely used. In the device simulation of the drift-diffusion model in a steady state, a charge conservation formula, an electron current continuous formula, and a hole current continuous formula as shown below are set as basic equations [see the above-mentioned document (1)]. ].

[0005] divD = ρ (charge conservation equation) ... (1) D = εE ... (2) E = -gradψ ... (3) ρ = q (p-n + N D -N A) ... (4) D: the 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

DivJ _n = q · (RG) (continuous electron current) (5) divJ _p = −q · (RG) (continuous hole current) (6) J _n : electron current J _p : hole current R: carrier recombination term G: carrier generation term

J _n = q · n · μ _n · E + q · D _n · grad n ··· (7) J _p = q · p · μ _p · E−q · D _p · grad p · (8) μ _n : Electron mobility μ _p : hole mobility D _n : electron diffusion coefficient D _p : hole diffusion coefficient

[0008]

(Equation 1)

K _B : Boltzmann constant T: lattice temperature

In the above equation, the variables to be solved are the potential ψ, the electron density n, and the hole density p.

The steady-state energy transfer model is set as an equation of the following form in which the energy conservation equation of carriers (electrons and holes) is added to the equation of the drift-diffusion model described above [the above-mentioned literature ( 2)].

[0012] divD = ρ (charge conservation equation) ... (11) D = εE ... (12) E = -gradψ ... (13) ρ = q (p-n + N D -N A) ... (14)

[0013]

(Equation 2)

[0014] T _n ^*: electron temperature T _p ^*: hole temperature τ _in: electron momentum relaxation time τ _ip: hole momentum relaxation time

[0015]

(Equation 3)

[0016] m _n ^*: electron effective mass m _p ^*: hole effective mass M _n ^-1: Electronic reverse effective mass tensor M _p ^-1: inverse effective mass tensor of the hole <>: in k-space Average operation

[0017]

(Equation 4)

S _n : electron energy flow density S _p : hole energy flow density T _{n eq} : electron equilibrium temperature T _{p eq} : hole equilibrium temperature

[0019]

(Equation 5)

<Ε _n >: average electron energy <ε _p >: average hole energy <ε _{n ep} >: electron equilibrium energy <ε _{p ep} >: hole equilibrium energy τ _wn : electron energy relaxation time τ _wp : positive Pore energy relaxation time

[0021]

(Equation 6)

[0022] tau _sn: electron energy flux density S relaxation time corresponding to the _n tau _sp: hole energy flux density S relaxation time corresponding to the _p v _n: electron velocity v _p: hole Speed

In the above equation of the energy transfer model, variables to be solved are the potential ψ, the electron density n, the hole density p, the electron temperature T _n ^* , and the hole temperature T _p ^* . Here, * is attached to the symbol of the carrier temperature, which means the thermodynamic temperature definition as shown in the following equation.

[0024]

(Equation 7)

This is for the purpose of distinction. To avoid complexity, the following description omit *.

In general, biases are sequentially updated with a plurality of specified applied biases as boundary conditions, and these charge conservation type, electron current continuous type, hole current continuous type, electron energy conservation type, hole energy conservation type The five equations of the equation are calculated.

Since these equations are nonlinear equations,
Generally, a solution is obtained by performing an iterative calculation called Newton's method. The Newton method is the following method.

It is assumed that the equation F (x) = 0 (31) is given for a variable x. Given an initial value x ₀ ,
If a value obtained by adding a certain amount of variation δx ₀ to x ₀ gives a solution, then F (x ₀ + δx ₀ ) = 0 (32). When the differential coefficient of F (x) is F ′ (x ₀ ) and F (x ₀ + δx ₀ ) is subjected to first-order Taylor expansion for δx ₀ , the following equations (33) and (34) are obtained.

[0029]

[0030]

(Equation 8)

Therefore, this time, the same calculation is performed for x ₁ , where x ₁ = x ₀ + δx ₀ (35).

This is sequentially repeated to calculate δ in the i-th calculation.
If x _i 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 at that time _i is the solution of equation (31).

The processing flow is the procedure shown as a flowchart in FIG.

FIG. 9 schematically shows this procedure. In the one-dimensional case, as shown in FIG. 9, 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 more
The number of iterations required to obtain the solution is small, and the calculation time to obtain the solution is short. 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 the mesh points. FIG. 7 shows an example of the analysis mesh. That is, since the potential, the electron density, the hole density, the electron temperature, and the hole temperature are expressed as variables by the number of mesh points N, 5N simultaneous equations are solved. The above-described charge conservation equation, electron current continuation equation, hole current continuation equation, electron energy conservation equation, and hole energy conservation equation are expressed as follows by transposing the terms on the right side.

[0036]

(Equation 9)

Ψ, n, p, T _n , and T _{p in the} above equation represent potential, electron density, hole density, electron temperature, and hole temperature, respectively, and represent N variables, respectively. in this case,
Coupled method (coupling method) that simultaneously solves charge conservation, electron current continuous, hole current conservation, electron energy conservation, and hole energy conservation, and charge conservation, continuous electron current, hole current conservation There is a Gummel method (non-bonding method or decoupled method) that separately solves the equation, the electron energy conservation equation, and the hole energy conservation equation.

FIG. 10 shows the procedure of the coupled method, and FIG. 11 shows the procedure of the Gummel method. In the formula of the matrix of the procedure 1002 of FIG. 10, the symbol Fψ _n 'represents the following partial differential.

[0039]

(Equation 10)

The same applies to other subscripts. The coupled method solves all variables simultaneously. In the Gummel method, each equation is solved with the variables other than the variables of interest fixed. For example, in the processing procedure for solving the electron energy conservation equation, potentials other than the electron temperature, the electron density, the hole density, and the hole temperature are fixed.

A matrix calculation is performed to solve the equation at each iteration. One iteration, 5 for coupled method
One N × 5N matrix is solved. In the Gummel method, five N × N matrices are solved.

The coupled 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 Coupled method, but the number of iterations is smaller in the Coupled method than in the Gummel method.

In many cases, it is known that the total calculation time required to obtain a solution is shorter in the coupled method. Therefore, if a good initial value can be given,
By the coupled method, analysis of a semiconductor device can be performed in a short calculation time.

A control volume method is used for discretizing the basic equation to be solved into an equation expressed on an analysis mesh.

Taking a part of the triangular mesh represented by the solid line in FIG. 12 as an example, the polygon formed by the bisector of the mesh edge connected to the mesh point as shown by the broken line in FIG. Volume. The vertices of the polygon of the control volume are the circumcenters (centers of circumscribed circles) of the triangular elements of the mesh.

In the control volume method, the flow (for example, current) of a physical quantity on the mesh edge IJ￣ (inversion) depends on the flow density (for example, current density) on the edge, and the side OP 辺 (generally 2
Even when it is a dimension, it is multiplied by the cross section.

[Analysis of Difficulty Converging Due to Isolated High-Density Carrier Region] In a device simulation, if a situation where a high-density carrier region is isolated in a region under analysis appears (that is, a floating region appears), The convergence of the solution becomes extremely difficult, and the solution may not be obtained. However, in such a situation, it may be possible to prevent the appearance of an isolated high-density carrier region by adjusting the analysis region. For example, FIG.
An analysis for applying a reverse bias to the PN junction as shown in FIG. 3 is such a case.

When a positive bias is gradually applied to the electrode TN for analysis, the depletion layer spreads in the P-type region (see FIG. 14). This is higher than the donor density in the N-type region by P
This is because the acceptor density of the mold region is low. If the distribution of the acceptor density has a difference in height as in this example,
The spread of the depletion layer at a place where the acceptor density is high becomes relatively small.

When a higher positive bias is applied, the depletion layer further expands and reaches the boundary of the analysis region (see FIG. 15). In the case of this example, since the expansion of the depletion layer at a high acceptor density is small, isolated high-density carrier regions are generated at the upper right and upper left portions of the analysis region.

In the analysis of a state where such a high-density carrier region is isolated (that is, a floating state), it is difficult to converge a solution, and a solution may not be obtained.

As a conventional technique in the case where convergence is difficult, a technique for making a mesh denser is known [refer to the above reference (3)]. This method is an adaptive mesh method for automatically increasing the mesh density. FIG. 17 shows an outline of the processing procedure.

By setting a bias and solving the basic equation,
If not converged (No branch in step 1703 in FIG. 17), the potential of the unconverged solution is checked using a potential to determine whether or not the potential difference between both end points of the mesh edge is larger than a predetermined reference value. The mesh edges are subdivided to make the mesh dense (step 170 in FIG. 17).
5). Using the regenerated mesh,
Solve basic equations.

However, when an isolated high-density carrier region exists, the convergence is not improved because the isolated region is not eliminated even if the mesh becomes dense.

As another conventional technique in the case where convergence is difficult, there is a technique in which the analysis bias is finely divided and analyzed. FIG. 18 shows an outline of the processing procedure.

When the bias is set, the basic equation is solved, and the convergence is not achieved (N in step 1803 in FIG. 18).
(o branch), an intermediate value between the previously analyzed bias and the currently analyzed bias is set as the bias (step 1805 in FIG. 18), and the basic equation is solved again.

However, eventually, the analysis proceeds to a high bias, and a situation appears in which an isolated high-density carrier region exists. Therefore, even with this method, the convergence is not improved.

[0057]

As described above, the conventional technique cannot solve the problem of difficulty in convergence when there is an isolated high-density carrier region as in the example shown in FIG. Have.

However, in the case of the example shown in FIG. 13, the appearance of an isolated high-density carrier region can be prevented by widening the boundary between the left and right analysis regions as shown in FIG.

Normally, the user performing the analysis does not check until the expansion of the depletion layer, so that the user may give up the analysis without noticing the isolated high-density carrier region as in this example.

That is, there is a problem that the device simulator detects such an isolated high-density carrier region, alerts the user, and sets an appropriate analysis region to avoid difficulty in convergence. ing.

Accordingly, the present invention has been made in view of the above problems and problems, and an object of the present invention is to provide a device simulation method that can obtain a solution even when an isolated high-density carrier region exists. Is to do.

[0062]

Means for Solving the Problems In order to achieve the above object, the first invention of the present application provides the present invention, in which convergence is difficult.
And detecting the isolated high-density carrier region and notifying the user.

Further, the second invention of the present application includes a step of detecting an isolated high-density carrier region and notifying the user even when convergence is obtained.

[0064]

BEST MODE FOR CARRYING OUT THE INVENTION In a preferred embodiment of the device simulation method according to the present invention, in a numerical analysis of a semiconductor device by a computer, a bias is set and a basic equation is solved.
There is a processing procedure for detecting an isolated high-density carrier region using the unconverged solution and notifying the user. According to the embodiment of the present invention, it is possible to improve convergence and obtain a solution in an analysis in which an isolated high-density carrier region exists and convergence is difficult.

[0065]

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a block diagram showing an embodiment of the present invention;

[Embodiment 1] FIG. 1 is a flowchart for explaining the processing flow of an embodiment of the present invention.

Referring to FIG. 1, after setting each bias condition (step 101), a basic equation is solved (step 102).

When a solution is obtained by convergence (step 10
If the bias condition that has not been analyzed remains (No branch in step 104), the process returns to step 101 and proceeds to the analysis of the next analysis bias condition.

On the other hand, if the solution does not converge and a solution is not obtained (No branch in step 103), an isolated high-density carrier region is detected by using an unconverged solution (step 105). Is output (step 106). By processing the coordinate data of the detected area or the like so that it can be displayed by a graphics tool and outputting the processed data, it becomes easy for the user to understand.

The user performing the analysis looks at the detection result, and if there is an isolated high-density carrier region as in the above-described analysis example, the user adjusts the analysis region to an appropriate one so that the convergence is reduced. It is possible to avoid difficulties and obtain a solution.

Next, the detailed processing procedure of the detection processing will be described with reference to FIG.
This will be described with reference to the flowchart of FIG.

First, one electrode is selected from the electrodes of the device to be analyzed (step 201). As schematically shown in FIG. 4, the starting point is a mesh node on the electrode. Then, a search is made for nodes connected by mesh edges from the start point. Nodes having no searched flag set are selected (step 203 in FIG. 2).

If the carrier of the selected node is of the same type as the starting point and has a high density, the next starting point is set (step 204). Here, an appropriate value is determined in advance as a reference value for determining the level of the carrier density.

If the carrier of the selected node is of a type different from the starting point, an opposite type flag and a searched flag are set (step 205).

If the carrier at the selected node has a low density, a low density flag and a searched flag are set (step 206).

At the original starting point, a searched flag is set (step 207). FIG. 5 schematically shows the processing up to this point.

Next, it is checked whether or not there is still a starting point. If there is still a starting point, the above-described search processing in steps 203 to 207 is repeated (step 208).

By repeating steps 203 to 207,
When there are no more starting nodes, the opposite type flag and the searched flag are cleared for the node where the opposite type flag is set (step 209).

By the processing up to this point, a search completion flag can be set for a mesh node in a high density carrier region that is not isolated and connected to one electrode.

Further, the above steps 201 to 209
Repeat until there are no unprocessed electrodes (step 21)
0).

In this manner, for all the electrodes, the search completed flag can be set for the nodes of the non-isolated high-density carrier region connected to the respective electrodes.

As a result of the above processing, the node where the searched flag is not set is checked, and the node where the carrier has a high density is searched for, and the isolated high density carrier region in a floating state not connected to the electrode is searched. It can be detected (steps 211 and 212). FIG. 6 schematically shows the processing procedure so far.

[Embodiment 2] FIG. 3 is a flow chart for explaining the processing flow of the second embodiment of the present invention.

Referring to FIG. 3, in the present embodiment,
In the processing procedure of the first embodiment described with reference to FIG. 1, even in the case of convergence, an isolated high-density carrier region is detected using the obtained solution (step 304), and the result is output (step 305). ).

As a result, even if the convergence is deteriorated due to the existence of the isolated high-density carrier region, although the convergence is made, the number of repetitive calculations required for the convergence is increased. By recognizing the existence of the high-density carrier region and setting an appropriate analysis region, convergence can be improved.

[0086]

As described above, according to the present invention, it is possible to obtain a solution in an analysis in which convergence is difficult due to the presence of an isolated high-density carrier region.

[Brief description of the drawings]

FIG. 1 is a flowchart illustrating a processing flow according to an embodiment of the present invention.

FIG. 2 is a flowchart illustrating a detailed flow of a detection process according to an embodiment of the present invention.

FIG. 3 is a flowchart illustrating a processing flow according to a second embodiment of the present invention.

FIG. 4 is a diagram schematically illustrating setting of a start point of a detection process according to an embodiment of the present invention.

FIG. 5 is a diagram schematically showing a state of searching for a mesh node in a high-density carrier region in one embodiment of the present invention.

FIG. 6 is a diagram schematically illustrating a state in which a search for a mesh node in a high-density carrier region in one embodiment of the present invention has been completed.

FIG. 7 is a diagram showing an example of a mesh used in a device simulation.

FIG. 8 is a flowchart of a processing procedure of the Newton method.

FIG. 9 is a diagram schematically illustrating a processing procedure of the Newton method.

FIG. 10 is a flowchart of a processing procedure of the coupled method.

FIG. 11 is a flowchart of a processing procedure of the Gummel method.

FIG. 12 is an explanatory diagram schematically showing a control volume.

FIG. 13 is a schematic diagram of an analysis example.

FIG. 14 is a schematic diagram showing a state where a bias is applied in the analysis example of FIG. 13;

FIG. 15 is a schematic diagram showing a state where a higher bias is applied in the analysis example of FIG. 13;

FIG. 16 is a schematic diagram showing a state where an analysis area is widened in the analysis example of FIG. 13;

FIG. 17 is a flowchart showing an outline of a processing procedure of the first conventional technique.

FIG. 18 is a flowchart showing an outline of a processing procedure of a second conventional technique.

[Equation 4]

Claims (5)

(57) [Claims] 1. A method for performing device simulation of a semiconductor device in an information processing apparatus using numerical analysis, comprising a step of detecting an isolated high-density carrier region and notifying a user when convergence is difficult. Characteristic device simulation method. 2. A method of performing device simulation of a semiconductor device in an information processing apparatus using numerical analysis, comprising a step of detecting an isolated high-density carrier region and notifying a user even when convergence is obtained. , A device simulation method. 3. A method for performing device simulation of a semiconductor device in an information processing apparatus using numerical analysis, wherein a bias is set and a basic equation is solved. If convergence is difficult and a solution cannot be obtained, an unconverged solution is used. A device simulation method comprising: detecting a high-density carrier region which is isolated in a separate manner; and outputting a detection result. 4. A method for detecting an isolated high-density carrier region, comprising the steps of: (a) setting a mesh node on one electrode as a starting point, and connecting the starting point with a mesh edge and having no searched flag set; (B) If the carrier of the selected node is of the same type as the starting point and has a high density, the next starting point is used. If the carrier of the selected node has a low density, the low density flag and the searched flag are set. Stand, or the carrier of the selected node is different from the starting point
If so, raise the opposite flag and the searched flag. (C) Raise the searched flag at the original starting point. (D) If there are no more starting nodes, Clearing the opposite flag and the searched flag, thereby setting a searched flag for the mesh node of the non-isolated high-density carrier region connected to the one electrode, and (e) setting each electrode The process of setting a searched flag is performed for all the nodes of the non-isolated high-density carrier region connected to the above, for all the electrodes. (F) As a result of the above process, the node where the searched flag is not set is checked By searching for a node where the carrier is dense, the floating state where the carrier is not connected to the electrode,
4. The device simulation method according to claim 3, wherein an isolated high-density carrier region is detected. 5. When performing a device simulation of a semiconductor device by an information processing apparatus, a bias is set and a basic equation is solved. In detecting a region, (a) a step of starting from a mesh node on one electrode as a starting point, and searching for a node that is connected from the starting point by a mesh edge and has no searched flag set, (b) a selected node If the carrier of the selected node is of the same type as the starting point and the density is high, the next starting point is used. If the carrier of the selected node is low density, the low density flag and the searched flag are set, or the carrier of the selected node Is different from the starting point
If so, setting the opposite type flag and the searched flag, and setting the searched flag at the original starting point. (C) When there are no more starting nodes, the opposite node is set with the opposite type flag set. Clearing a type flag and a searched flag, thereby setting a searched flag for a mesh node of the non-isolated high-density carrier region connected to the one electrode; A process of setting a searched flag is performed for all the electrodes of the connected non-isolated high-density carrier region. (E) As a result of the above process, a node where the searched flag is not set is checked. By searching for nodes where the carrier is dense, a floating state that is not connected to the electrode,
A step of detecting an isolated high-density carrier region, wherein the program is executed by an information processing apparatus.