JPH11243089A - Simulation method of stress - Google Patents

Abstract

PROBLEM TO BE SOLVED: To obtain an accurate simulation result by giving the effect of stiffness change when silicon is changed to oxide as an initial stress increment for calculating stress, and by repeating stress calculation until an entire structure including silicon is balanced with force. SOLUTION: An oxidation seed diffusion calculation in an oxide is made based on a diffusion coefficient, a reaction coefficient, and a viscosity coefficient corresponding to a stress distribution at a starting point, and an oxidation flux vector (q) at the interface between silicon 1 and oxide 2 is obtained. Then, the silicon/oxide interface is moved in the direction of oxidation flux by a distance vector dx. Then, a thermal distortion increment, a creep distortion increment, and the like incidental to each material are given to an entire device structure for calculating stress, a displacement increment and a stress are obtained, and a shape is updated according to a displacement increment dv being obtained. Then, while the stress is being updated, force balance is checked, and a calculation is repeated until balancing is reached.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention relates to a method for simulating stress.

[0002]

2. Description of the Related Art With the increase in the degree of integration of LSIs, an oxidizing step under a low-temperature environment has been frequently performed. However, in this low-temperature environment, the oxidative stress calculation method based on the viscous flow model that has been widely used conventionally ignores the elastic effect of the oxide, so that accurate simulation results cannot be obtained.

In such a low-temperature environment, a calculation method using a viscoelastic model without restrictions such as the viscous flow model described above is effective, and diffusion calculation of oxidizing species and stress calculation during the oxidation process based on the viscoelastic model are effective. A method combining the above has already been proposed.

[0004] In this technique combining the calculation of diffusion of oxidizing species and the calculation of stress, it is necessary to forcibly displace the silicon / oxide interface in the oxide direction with respect to the volume expansion caused by the change from silicon to oxide (Reference 1). NO
VEL: A NONLINEAR VISCOELASTIC MODEL FOR
THERMAL OXIDATION OF SILICON, JPPENG eta
l; COMPEL (VOL10, NO.4, PP341-353), Reference 2 Two-D
imensional Simulation of Local Oxidation o
f Silicon: Calibrated Viscoelastic Flow analy
sis Vincent Senez et al; IEEE TRANS.ED
(VOL43, NO5, PP720-731)) or give isotropic strain to this volume expansion (Ref. 3 Development of shape and stress analysis program OXSIM2D in thermal oxidation process of Si) Naoto Saito et al. A) (VOL57,
NO541, PP115-120))
There is a drawback from the viewpoint of simultaneously handling oxidative stress and thermal stress generated in the low-temperature oxidation step.

[0005] Regarding the methods of the above-mentioned references 1 and 2, FIG.
This will be described with reference to FIG. FIG. 3 shows References 1 and 2.
FIG. 4 is a flow chart showing a method of simulating the oxidation of silicon by the method of FIG. 1, and FIG. 4 is a schematic view showing a state of a change of a silicon oxide interface corresponding to the flow chart of FIG.

First, based on the diffusion, reaction coefficient and viscosity coefficient of the oxide at time t0, the diffusion calculation of the oxidized species in the oxide is performed to obtain the oxidation flux q at the interface between silicon 1 and oxide 2 (step a). Then, the amount of movement dx (dx = (α / N) · q · dt of the silicon oxide interface to the silicon side, where α, N
Respectively determine the volume ratio of silicon during volume expansion and the oxidizing species required to form oxides per unit volume) and move through the silicon oxide interface (
Step b).

Next, the interface at time t0 is moved toward the oxide side by d.
u (du = ((1−α) / N) · q · dt) is forcibly displaced and stress calculation is performed on the oxide portion. At this time, an incremental viscoelastic relaxation model corresponding to viscous flow is used as a constitutive rule of the oxide (step c).

Next, the above steps are repeated until the diffusion, reaction coefficient and viscosity coefficient used in steps a and c match the obtained stress. After matching, the shape distribution at time t + dt is obtained, and this is used as the initial shape for the next time step.
(Step d).

However, according to the methods of the above-mentioned references 1 and 2, the volume expansion effect when the oxide is changed to the silicon oxide interface is given as a forced displacement, and as a result, the device structure that should be regarded as an integral body is divided into two. Therefore, not only the rigidity of silicon cannot be taken into account when calculating the oxidative stress, but also it cannot be combined with the thermal stress calculation.

Also, the method of the above-mentioned reference 3 has a drawback that an unusually large stress appears in the interface direction even in the plane oxidation step, as described in the reference. It also has the disadvantage that stresses cannot be combined.

[0011]

As described above, in the conventional simulation method, since the device structure that should be regarded as an integral structure is divided into two parts, the rigidity of silicon cannot be taken into account. In addition, there is a problem in that an unusually large stress appears at the interface. for that reason,
It has been difficult to obtain accurate simulation results.

The present invention has been made to solve the above-mentioned conventional problems, and has as its object to provide a simulation method capable of obtaining an accurate simulation result of stress.

[0013]

According to the present invention, there is provided a method for simulating a stress, which corresponds to a volume expansion when silicon is changed to an oxide in a silicon region oxidized during a time increment dt at a time t0. In addition to the anisotropic initial strain increment, the thermal strain increment of the oxide, and the creep strain increment, the effect of stiffness change when silicon changes to oxide (corresponding to the difference in stiffness between silicon and oxide) is determined by the initial stress increment. , And the stress calculation is repeated until the force is balanced in the entire structure including silicon.

Further, the method for simulating stress according to the present invention is characterized in that the silicon region which is oxidized during the time increment dt at the step of time t0 is anisotropic with respect to volume expansion when silicon changes to oxide. In addition to the initial strain increment, the thermal strain increment of the oxide and the creep strain increment, the effect of the stiffness change when silicon changes to oxide is given as the initial stress increment, and the stress calculation is performed. The stress calculation is repeated until the force is balanced, and the stress calculation is further repeated until the oxide flux calculated under the condition of the balanced force reaches a predetermined value.

According to the present invention, when calculating the stress generated during the oxidation step, the rigidity of the entire device structure including silicon can be considered, and the calculation is repeated until the balance of the internal stress force is maintained. As a result, the oxidative stress in silicon can be determined accurately. In addition, by expressing the oxidative stress generated during the oxidation step by anisotropic one-dimensional initial strain corresponding to the volume expansion, an abnormally large stress field that appears when the isotropic initial strain is given is avoided. be able to.

Further, since the thermal stress generated due to the temperature fluctuation may be simply given as an increment of thermal strain including the amount of oxide development, it is possible to easily integrate the oxidative stress and the thermal stress. Further, since the variation in rigidity when silicon is changed to oxide is taken into consideration, it is possible to perform thermal stress calculation in conjunction with interface development.

Further, in the calculation of the diffusion of oxidizing species, it is possible to take in the shape deformation of the free expansion of the silicon substrate in the oxidation step accompanied by the temperature fluctuation, and to make the consistency of the diffusion and reaction coefficients corresponding to the updated stress. By providing an iterative loop for performing the simulation, it is possible to easily carry out an oxide film growth simulation in consideration of stress dependency.

The simulation method according to the present invention can be similarly performed not only for silicon oxidation but also for silicidation of silicon, and similar effects can be obtained.

That is, in the stress simulation method according to the present invention, the time increment d
The silicon region to be silicided during t is provided with an anisotropic initial strain increment, a thermal strain increment of the silicide, and a creep strain increment corresponding to the volume expansion when the silicon changes to silicide, and the silicon is silicidized. The effect of stiffness change (corresponding to the difference in stiffness between silicon and silicide) is given as an initial stress increment to perform stress calculation, and the stress calculation is performed until the force is balanced for the entire structure including silicon. Is repeatedly performed.

[0020]

Embodiments of the present invention will be described below with reference to the drawings. FIG. 1 is a flowchart showing a simulation method for oxidizing silicon, and FIG.
FIG. 3 is a schematic view showing a state of a change of a silicon oxide interface corresponding to the flowchart of FIG. 1.

First, based on the diffusion coefficient, reaction coefficient, and viscosity coefficient corresponding to the stress distribution at the time t0, calculation of diffusion of oxidizing species in the oxide is performed to determine the relationship between silicon 1 and oxide 2 (see FIG. 2). Interface oxidation flux vector q
(R) is obtained. Here, r is a position vector of the silicon oxide interface at time t0. The oxidation flux is given a reference value at the time t0 when the silicon oxide interface moves afterward, so that it is stored and a new oxidation flux vector q0 (= q) is defined (step a).

Next, the coefficient α / N is set in the direction of the oxidation flux.
(However, α = 0.44, N: oxidizing species necessary to form oxide per unit volume) multiplied by the distance vector dx (dx = (α / N) · q0 · dt). Move the oxide interface. The area between the interface at time t0 and the interface moved by dx is the area where silicon changes to oxide (step b).

Next, the entire device structure is subjected to stress calculation by giving an increment of thermal strain, an increment of creep strain, etc., associated with each material, and a displacement increment and stress are obtained. A free thermal expansion increment vector dv of the silicon substrate is given as a forced displacement to a lateral boundary of the device in accordance with the temperature fluctuation within the dt time. The shape is updated based on the obtained displacement increment.

For a region other than the region that changes from silicon to oxide obtained in step b (for example, a region of oxide, silicon, etc.), the following stress-strain relational expression dσ = D ^ox (dε−dεc ^ox −dεth) ^ox ) dσ = D ^s (dε−dεth ^s ). Where dσ is a stress increment tensor, dε is a strain increment tensor, dεc ^ox is a creep strain increment tensor representing the viscous flow effect of oxide, dε
th ^ox is the thermal strain increment tensor of oxide, D ^ox is the fourth order elastic tensor of oxide, D ^s is the fourth order elastic tensor of silicon, and dεth ^s is the thermal strain increment tensor of silicon.

[0025] Step for the region changes oxide of silicon obtained b, the following stress-strain relationship ^{dσ = D ox (dε-dεc} ox -dεth ox -dεv) + d
g. Here, dεv is an anisotropic initial strain increment tensor corresponding to one-dimensional volume expansion strain when silicon changes to oxide, and dg corresponds to a difference in rigidity between silicon and oxide when silicon changes to oxide. Initial stress increment tensor.

Further, R is a transformation matrix from the global coordinate system to the local orthogonal coordinate system set at the interface between silicon and oxide, R ^T is a transposed matrix of R, and dεv ^L is a volume expansion increment tensor in the local coordinate system. Diipushironv = when expressed as R ^T · dεv ^L · R, in the case where the silicon volume expansion rate when the direction of the oxidizing flux at the interface between silicon and oxide and the local first coordinate component Diipushiron'enu (typically changes to oxides Since the volume increases 2.2 times, the volume expansion rate d
εn is 1.2)

[0027]

(Equation 3) Can be expressed as

The thermal expansion tensor εth ^{ox of the} oxide generated between the reference temperature Tref and the temperature T0 at time t0 is defined as α ^ox (T0), where δ _ij is the delta of Kronecker, where α is the linear expansion coefficient at the temperature T0 of the oxide. the when expressed as ^{εth ox = α ox (T0)} (T0 -Tref) δ ij, the total strain tensor in the silicon at time t0 the Ipushironti0, as the stress tensor of silicon Shigumati0 ^s time t0, the dg dg = it can be expressed as ^{D ox (εt0-εth ox)} -σt0 s ( step c).

In the above stress calculation, since there is a remarkable stress dependency such as a remarkable deformation or creep strain, the balance of the force of the entire device structure cannot be guaranteed by one calculation. Therefore, the balance of the force is checked while updating the shape and the stress, and the calculation is repeated until the balance is obtained (the number of repetitions is usually about three times). If the forces are not balanced, an equilibrium repetition calculation regarding the residual force is performed, and the process returns to step c.

Next, diffusion and a reaction coefficient are calculated based on the oxide shape and the oxide stress obtained in the above steps. The diffusion flux of the oxidizing species is calculated using the diffusion and reaction coefficients to obtain an oxidation flux vector Q at the silicon-oxide interface, and an average value qn (qn = (Q + q) / 2) of the oxidation flux vector q at time t0 is calculated. Ask
(Step d).

The average value qn is equal to the value q assumed in step b.
Steps b-d are repeated until the value becomes equal to 0. q
When n becomes the same as q0, it is considered that the convergence has been reached, the process proceeds to the next time step, and the above steps a to d are performed again.

The present invention is not limited to the above embodiment. In the above embodiment, the simulation for the oxidation of silicon has been described.
By repeating the steps similar to the steps a to c shown in (1), a similar simulation can be performed for silicidation instead of oxidation. Others
The present invention can be implemented with various modifications without departing from the scope of the invention.

[0033]

According to the present invention, an oxidative stress (or a silicidation stress) and a thermal stress are considered at the same time, and the calculation is repeatedly performed until the force is balanced for the entire device structure, thereby accurately calculating the stress. Can be requested.

[Brief description of the drawings]

FIG. 1 is a view showing an embodiment of the present invention, and is a flow chart showing a simulation method in oxidation of silicon.

FIG. 2 is a schematic view showing a state of a change of a silicon oxide interface corresponding to the flowchart of FIG. 1;

FIG. 3 is a view showing a conventional technique, and is a flow chart showing a simulation method in oxidation of silicon.

FIG. 4 is a schematic diagram showing a state of a change of a silicon oxide interface corresponding to the flowchart of FIG. 3;

[Explanation of symbols]

 1. Silicon 2. Oxide

Claims (5)

[Claims] 1. A time increment dt in a step at time t0.
In addition to providing anisotropic initial strain increments corresponding to the volume expansion of silicon as it changes to oxide, thermal oxide and creep strain of oxide, and the silicon being converted to oxide, A stress simulation method characterized in that the effect of a change in stiffness is given as an initial stress increment, stress calculation is performed, and the stress calculation is repeated until the forces are balanced with respect to the entire structure including silicon. 2. A stress-strain relational expression for a region where silicon changes to an oxide, dσ is a stress increment tensor, d
ε is the strain increment tensor, D ^ox is the fourth order elasticity tensor of oxide, dεc ^ox is the creep strain increment tensor of oxide, dεth ^ox is the thermal strain increment tensor of oxide, d
Anisotropic initial strain increment tensor .epsilon.v, as initial stress increment tensor ^{dg, dσ = D ox (dε} -dεc ox -dεth ox -dεv) + d
where R is a transformation matrix from the global coordinate system to the local rectangular coordinate system set at the interface between silicon and oxide, R ^T is a transposed matrix of R, and dεv ^L is a volume expansion increment tensor in the local coordinate system. the Diipushironv when expressed as ^{dεv = R T · dεv L ·} R, the volume expansion rate when the direction of the oxidizing flux at the interface between silicon and oxide and the local first coordinate component as Diipushiron'enu, Equation 1] ^Where α ^ox (T 0) is the linear expansion coefficient of the oxide at the temperature T 0, δ _ij is the Kronecker delta, and the reference temperature Tre
When representing the thermal distortion tensor Ipushironth ^ox of oxide occurring between the temperature T0 at time ^{t0 εth ox = α ox (T0} ) and (T0 -Tref) δ _ij from f, the total strain in the silicon εt0 at time t0 tensor,
The Shigumati0 ^s as the stress tensor of the silicon at the time t0, the stress simulation method according to claim 1, characterized in that representing the dg and ^{dg = D ox (εt0-εth} ox) -σt0 s. 3. A time increment dt in a step at time t0.
In addition to providing anisotropic initial strain increments corresponding to the volume expansion of silicon as it changes to oxide, thermal oxide and creep strain of oxide, and the silicon being converted to oxide, The effect of the change in stiffness is given as an initial stress increment, stress is calculated, and the above-described stress calculation is repeated until the force is balanced with respect to the entire structure including silicon, which is calculated under the condition where the force is balanced. A stress simulation method wherein the stress calculation is repeated until the oxidation flux reaches a predetermined value. 4. A time increment dt in a step at time t0.
In addition to providing anisotropic initial strain increments corresponding to volume expansion when silicon changes to silicide, thermal silicide increments, and creep strain increments for the silicon region to be silicided, silicon is added to silicide. A stress simulation method, wherein a stress calculation is performed by giving an effect of a change in rigidity as an initial stress increment when the change is made, and the stress calculation is repeated until the force is balanced with respect to the entire structure including silicon. 5. A stress-strain relational expression for a region where silicon changes to silicide, dσ is a stress increment tensor, d
ε is a strain increment tensor, D ^sc is a fourth order elastic tensor of silicide, dεc ^sc is a creep strain increment tensor of silicide, dεth ^sc is a thermal strain increment tensor of silicide, dεv is an anisotropic initial strain increment tensor, dg is As the initial stress increment tensor, dσ = D ^sc (dε−dεc ^sc −dεth ^sc −dεv) + d
where R is the transformation matrix from the global coordinate system to the local rectangular coordinate system set at the interface between silicon and silicide, R ^T is the transposed matrix of R, and dεv ^L is the volume expansion increment tensor in the local coordinate system. the Diipushironv when expressed as ^{dεv = R T · dεv L ·} R, volume expansion rate dε when the direction of the silicidation flux at the interface between silicon and silicide was localized first coordinate component
As n, Where the linear expansion coefficient at the temperature T0 of the silicide is α ^sc (T0), δ _ij is the Kronecker delta, and the reference temperature Tre
When representing the silicide thermal strain tensor Ipushironth ^sc occurring between the temperature T0 at time ^{t0 εth sc = α sc (T0} ) and (T0 -Tref) δ _ij from f, the total strain in the silicon εt0 at time t0 tensor,
The Shigumati0 ^s as the stress tensor of the silicon at the time t0, the stress simulation method according to claim 4, characterized in that representing the dg and ^{dg = D sc (εt0-εth} sc) -σt0 s.