CN102122119A - Improved quickly-propelling method for simulating three-dimensional etching process of photoresist - Google Patents

Abstract

The invention relates to an improved quickly-propelling method for simulating the three-dimensional etching process of a photoresist, which improves the demand of the traditional quickly-propelling algorithm on a storage space under the condition that the error is not greater than 10%, and has practical meanings for realizing high-precision simulation of the three-dimensional photoetching process in a Micro Electronic Mechanical System (MEMS) and an Integrated Circuit (IC); the method comprises the following two conditions that: a, a speed value and a time value are compressed into a same matrix in the format of the scientific notation method, and the position and state values of a grid point corresponding to NarrowBand are compressed into the other matrix; b, a photoresist etching reaction rate matrix calculated according to physical and chemical models and a compressed time speed data matrix occupy the same space; and the quickly-propelling algorithm satisfying the two conditions is the improved quickly-propelling method for simulating the three-dimensional etching process of the photoresist.

Description

The quick propulsion method of improvement of the three-dimensional etching process simulation of a kind of photoresist

Technical field

The invention provides a kind of quick propulsion method of improvement that is used for the three-dimensional etching process simulation of photoresist, belong to microelectronics lithography process simulation technical field.

Background technology

The manufacturing research method of microelectromechanical systems (MEMS) is normally determined the optimum process condition of certain particular device size and structure by a large amount of experiments, done many shortcomings like this, and main problem is cost height and the understanding that is unfavorable for technological process.Utilize computing machine that the technological process that comprises photoetching is simulated, shortened the design cycle, reduced design cost, well remedied the deficiency of methods of experiments, thereby better improve the performance of MEMS product.

Be applied to the wired algorithm of main algorithm (String algorithm), ray casting (Ray-tracing algorithm), cellular automaton algorithm (Cellular automata algorithm) and level set algorithm (Level-set algorithm) in the lithography simulation, and the situation of protecting number for velocity reversal, the level set algorithm derives another kind of numerical method, promptly advances algorithm (Fast Marching Method) fast.Line algorithm and ray algorithm speed are fast, and the required memory space is little, but these two kinds of method less stable, and at reconstructing surface, especially have difficulties during three-dimensional applications.The cellular automaton algorithm speed is very fast, and is stable, and three-dimensional applications is convenient, and use occasion is extensive.Compare with the cellular automaton algorithm, traditional quick propelling algorithm has higher arithmetic speed, can better handle the situation of various change in topology.But along with the raising of precision, to the also raising thereupon of requirement of storage space, traditional method has been difficult to meet the demands.

Summary of the invention

Technical matters: the quick propulsion method of improvement that the purpose of this invention is to provide the three-dimensional etching process simulation of a kind of photoresist, guaranteeing that error is not more than under 10% the prerequisite, improved the requirement of traditional quick propelling algorithm to storage space, this has Practical significance for the high-precision analog of realizing three-dimensional photoetching process among MEMS and the IC.

Technical scheme: to be the material surface that will move be embedded in the level set function of high one dimension as the isoline of level set function the basic thought of this method.In evolutionary process, the evolution curved surface always corresponding to the contour surface of level set function, needs only the evolution position that the certainty equivalents face can be determined the surface.Consider a three-dimension curved surface, this curved surface is divided into two parts with the space, supposes that this curved surface moves along its normal direction with given rate, if speed greater than zero, that is to say speed always on the occasion of, to the curved surface external movement.In this case, the photoresist surface remains outside expansion, under any circumstance promptly, " negative growth " can not occur, is certain so arrive the time of certain point, must be a monodrome.For the point of process, its time corresponding value just can " be freezed ", and does not need to calculate again, and can not influence these time values of the point of process to the calculating through the some time value, and this has also guaranteed the feasibility of algorithm.

A. velocity amplitude and time value are compressed into same matrix according to the form of scientific notation, position and state value among the corresponding NarrowBand of net point then are compressed into another matrix;

B. the time speed data matrix after the photoresist etching reaction rate matrix that calculates according to physics, chemical model and the compression takies the same space;

Promptly this is considered as being used for the quick propulsion method of improvement of the three-dimensional etching process simulation of photoresist to the quick propelling algorithm that satisfies above two conditions;

The basic step of this method is as follows:

1.) according to the lithography simulation accuracy requirement, substrate is subdivided into the array that little square is formed, and adopts three-dimensional matrice to represent array, determine photoetching material, process conditions information and total photoetching time;

2.) point of definition surface boundary is NarrowBand, inner point is Alive, outside point is FarAway, physics, chemical model according to photoetching, the velocity amplitude that initialization is had a few, and the position among the corresponding NarrowBand of time value, state value and net point among difference initialization Alive, NarrowBand and the FarAway;

3.) set up rickle, in order to the coordinate figure and the corresponding time value of preserving net point among the NarrowBand with search minimum value fast according to the size of the time value of NarrowBand mid point;

4.) time value and velocity amplitude are represented with scientific notation, and are compressed in the matrix that rickle subscript value of net point correspondence and state value then are compressed into another matrix, reduce storage space under the prerequisite that guarantees error requirements;

5.) search the point of time value minimum among the NarrowBand, and it is removed from NarrowBand, be added into Alive, can think that the current time is the time value of this point, the non-Alive adjoint point that adds it is to NarrowBand, upgrade the time value of these adjoint points according to the Hamilton-Jacobi equation of speed and time, arrive the default photoetching time with cocycle until time value, then the representation of a surface formed by the point among the NarrowBand at the default surface topography of photoetching during the time of material.

Make a general survey of overall process of the present invention, its maximum characteristics have been effectively to utilize the requirement of photoresist etching to precision, have reached the effect of saving memory headroom by the compression to time value and velocity amplitude.

The present invention has realized that to a certain extent three-dimensional photoetching simulation trial speed is fast, little requirement takes up room.The improvement of the three-dimensional etching process simulation of the photoresist that this paper proposes advances algorithm mainly to have following feature fast: one, velocity amplitude and time value are compressed into same matrix according to the form of scientific notation, and the subscript value and the state value of the rickle of net point correspondence are compressed into another matrix; Two, the time speed data matrix after the photoresist etching reaction rate matrix that calculates according to physics, chemical model and the compression takies the same space.Thereby the memory headroom that takies can save computing time of this method.

The quick propelling algorithm that satisfies above two conditions i.e. this improvement that is considered as being used for the three-dimensional etching process simulation of photoresist advances algorithm fast.

Beneficial effect: the present invention has improved the computing memory headroom requirement of the quick propelling algorithm of conventional three-dimensional lithography process simulation to a certain extent, has precision height, fast, the little advantage that takes up room of speed, can simulate the three-dimensional photoetching process quickly and accurately.Successfully realized the simulation of SU-8 glue photoetching process on the computer of Core2/2GHz, analog result is consistent with experimental result, and the algorithm time complexity is O (C * N ³Log (N))+O (N ²), C is an a small amount of, N is the square number on every limit of little square array, and the aspect that takes up room, grid with 50 * 50 * 50 is an example, and the quick propelling of tradition algorithm takes up room and is 1.5MByte, is 1.1Mbyte and improve algorithm, for the grid of N * N * N, the quick propelling of tradition algorithm takes up room and is about 3 * 4N ³=12N ³Byte, improving algorithm is (1+0.1) * 2 * 4N ³=8.8N ³Byte has saved 26% than the quick propelling algorithm of tradition, and this has Practical significance for the high-precision analog of realizing three-dimensional photoetching among MEMS and the IC.

Description of drawings

Fig. 1 is the arrowband synoptic diagram that advances algorithm fast.A represents Alive, i.e. the point of curved surface inside; N represents NarrowBand, i.e. the point of surface boundary, and F represents FarAway, i.e. the point of curved surface outside.

Fig. 2 advances the algorithm initialization synoptic diagram fast.A represents Alive, i.e. the point of curved surface inside; N represents NarrowBand, i.e. the point of surface boundary, and F represents FarAway, i.e. the point of curved surface outside.

Fig. 3 advances algorithm to be worth synoptic diagram update time fast.(i, j+1 k) represent the coordinate figure of this net point, and B represents the time value of this net point, and other net point is similar.

Fig. 4 improves to advance algorithm data storage format synoptic diagram fast, Fig. 4 (a) is the data layout of data array Data, data segment A represents the velocity amplitude truth of a matter, refer to preceding 4 position effective digitals of net point velocity amplitude F, the data segment B express time truth of a matter refers to preceding 5 position effective digitals of net point time value T; Fig. 4 (b) is the data layout of state array State, data segment C represents that net point is corresponding to the position among the NarrowBand, mean the subscript of this net point in rickle array, D and E represent the mould of velocity amplitude exponential sum time value index respectively, i.e. the absolute value of 10 power exponent in the floating-point of velocity amplitude and the time value counting form.S is the comprehensive state position, 10 power exponent symbol in expression floating-point counting form velocity amplitude and the time value, and the status attribute of this net point.

Fig. 5 is that application enhancements advances algorithm to carry out SU-8 glue lithography simulation result fast, and wherein Fig. 5 (a) is a domain, and Fig. 5 (b) is an analog result.

Embodiment

Fig. 1 is the synoptic diagram that advances algorithm fast, left-half is represented a known two-dimentional closed curve, the black arrowband is the profile of curve, right half part is a partial enlarged drawing, and black color dots is the point (representing with A) of process, the i.e. point of curved surface inside, white point is for going back the point of no show, be the point (representing with F) of curved surface outside, figuratum point is the arrowband point, i.e. the point (representing with N) of surface boundary.Advance algorithm only the point of setting up on the surface perimeter arrowband to be calculated fast, this arrowband has only the width of a point.At three-dimensional problem, carry out 3D grid by cartesian coordinate system on the space and divide, h is the space step-length, (i, j k) represent pairing grid.According to

$F|\▿T|=1---\left(1\right)$

Formula (1) is also referred to as Hamilton-Jacobi " form " equation.

It is approximate to the space derivation of formula (1) to use upstreame scheme,

$F_{\mathrm{ijk}}{\left[\begin{array}{c}\max {(\max (D_{\mathrm{ijk}}^{-x}T,0),-\min (D_{\mathrm{ijk}}^{+x}T,0))}^2\\ +\max {(\max (D_{\mathrm{ijk}}^{-y}T,0),-\min (D_{\mathrm{ijk}}^{+y}T,0))}^2\\ +\max {(\max (D_{\mathrm{ijk}}^{-z}T,0),-\min (D_{\mathrm{ijk}}^{+z}T,0))}^2\end{array}\right]}^{1/2}=1---\left(2\right)$

Wherein

$D_{\mathrm{ijk}}^{+x}T=\frac{T(i,j+1,k)-T(i,j,k)}{h}$ $D_{\mathrm{ijk}}^{-x}T=\frac{T(i,j,k)-T(i,j-1,k)}{h}$

$D_{\mathrm{ijk}}^{+y}T=\frac{T(i+1,j,k)-T(i,j,k)}{h}$ $D_{\mathrm{ijk}}^{-y}T=\frac{T(i,j,k)-T(i-1,j,k)}{h}$

$D_{\mathrm{ijk}}^{+z}T=\frac{T(i,j,k+1)-T(i,j,k)}{h}$ $D_{\mathrm{ijk}}^{-z}T=\frac{T(i,j,k)-T(i,j,k-1)}{h}$

The concrete steps of this method are as follows:

(1) according to the lithography simulation accuracy requirement, substrate is subdivided into the array that little square is formed, and adopts three-dimensional matrice to represent array, determine photoetching material, process conditions information and total photoetching time.

(2) initialization: the point of definition surface boundary is NarrowBand, and inner point is Alive, and outside point is FarAway.According to physics, the chemical model of photoetching, the velocity amplitude that initialization is had a few is set up Alive, NarrowBand, FarAway array.As shown in Figure 2, A represents Alive, and N represents NarrowBand, and F represents FarAway.The point of Dark grey, promptly the point of k=0 leaves in the Alive array, represents the point of process, makes its T=0, and their time value is with frozen; Grayish, promptly the point of k=1 is put into the NarrowBand array, and the adjoint point of expression Alive is the point of curved surface just, makes T (i, j, 1)=h/F (i, j, 1); The point of white leaves in the FarAway array, and establishing its initial value is ∞.

(3) set up rickle according to the size of time value, in order to the coordinate figure and the corresponding time value of preserving net point among the NarrowBand with search minimum value fast.

(4) packed data: setting up structure Array array and preserve data, two member variables are wherein arranged, is respectively data array Data and state array State.Data array Data save mesh spot speed value F and time value T, be decimal system floating-point counting form, form such as Fig. 3 (a), wherein data segment A represents the velocity amplitude truth of a matter, refer to preceding 4 position effective digitals of net point velocity amplitude F, and the data segment B express time truth of a matter refers to preceding 5 position effective digitals of net point time value T.For instance, if Data[i]=123456789, when its implication was represented for adopting decimal system floating-point counting form, the truth of a matter of i net point velocity amplitude was 1.234, the truth of a matter of time value is 5.6789.Each state value of state array State save mesh point, array form such as Fig. 3 (b), wherein, data segment C represents that net point is corresponding to the position among the NarrowBand, mean the subscript of this net point in rickle array, and D and E represent the mould of velocity amplitude exponential sum time value index respectively, i.e. the absolute value of 10 power exponent in the floating-point of velocity amplitude and the time value counting form.S is the comprehensive state position, 10 power exponent symbol in expression floating-point counting form velocity amplitude and the time value, and negative sign is 0, positive sign is 1, and the status attribute of this net point, promptly it is to be positioned at also right and wrong Alive array (equaling 1) of Alive array (equaling 0), is used for the process of auxiliary judgment iteration.For instance, if F=5, its binary form is shown 101, and implication is that 10 power exponent is for just in the velocity amplitude of this net point, and 10 power exponent is for bearing in the time value, and it does not belong to the Alive array.Comprehensive two arrays, we just can represent the numerical value that each net point is required.For instance, if Data[i]=123456789, State[i]=123456125, then its implication is i net point velocity amplitude F[i]=1.234 * 10 ¹=1.234, time value T[i]=5.6789 * 10 ^-2=0.056789, the position that this point is arranged in rickle array is 123456 (remembering from 0), does not belong to the Alive array, obviously belongs to the NarrowBand array.

(5) push ahead: pushing ahead part is the iterative process of algorithm, mainly is the operation to the NarrowBand array.Alive is removed and put into to the point of time value minimum among the NarrowBand, we claim this point to be the min point, the neighbor that is labeled as that does not belong to Alive in the adjoint point that min is ordered, the point that is among the neighbor among the FarAway is put into NarrowBand, upgrades the time value of neighbor mid point again according to equation (2).As shown in Figure 4, the symbol on the grid is denotation coordination value and time value respectively, and the coordinate of establishing certain point among the neighbor is (i, j, k), and T after upgrading (i, j, k)=x, about this point, up and down, the time value of front and back point is respectively A, B, F, E, C, D, rewrite equation (2)

$\max {(\max (x-A,0),-\min (B-x,0))}^2$

$+\max {(\max (x-C,0),-\min (D-x,0))}^2=h^2/F_{\mathrm{ijk}}^2=f_{\mathrm{ijk}}^2---\left(3\right)$

$+\max {(\max (x-E,0),-\min (F-x,0))}^2$

Might as well establish A=min (A, B, C, D, E, F), promptly A be (i, j, the k) minimum value of time value T in the adjoint point, min (C, D)＜min (E F), can prove, A＜x＜A+f, then

1) (C, in the time of D), equation (3) becomes (x-A) to A+f≤min ²=f ²

2) min (C, D)＜(E, in the time of F), equation (3) becomes (x-A) to A+f≤min ²+ (x-min (C, D)) ²=f ²

3) (E, in the time of F), equation (3) becomes (x-A) to A+f＞min ²+ (x-min (C, D)) ²+ (x-min (E, F)) ²=f ²Solve x, obtain T (i, j, updating value k).

(6) to arrive the default photoetching time be that progradation finishes to the min time value of ordering, and the curved surface that the point among the NarrowBand is formed is material at the default surface topography of photoetching during the time.

We have successfully realized the simulation of the thick SU-8 glue of 100um photoetching process on the computer of Core2/2GHz, analog result as shown in Figure 5, Fig. 5 (a) is four linear patterns, every line line length 200um, live width 20um, distance between centers of tracks are 40um, Fig. 5 (b) is at time shutter t ₁=130s, back baking time t ₂=2400s, photoresist etching time t ₃The analog result that obtains under the condition of=600s is consistent with experimental result.For the grid of N * N * N, the quick propelling of tradition algorithm takes up room and is about 3 * 4N ³=12N ³Byte, improving algorithm is (1+0.1) * 2 * 4N ³=8.8N ³Byte has saved 26% memory headroom than the quick propelling algorithm of tradition, and this has Practical significance for the high-precision analog of realizing three-dimensional photoetching among MEMS and the IC.

Claims (1)

1. the quick propulsion method of improvement of the three-dimensional etching process simulation of a photoresist is characterized in that:

A. velocity amplitude and time value are compressed into same matrix according to the form of scientific notation, position and state value among the corresponding NarrowBand of net point then are compressed into another matrix;

B. the time speed data matrix after the photoresist etching reaction rate matrix that calculates according to physics, chemical model and the compression takies the same space;

The quick propelling algorithm that satisfies above two conditions i.e. this improvement that is considered as being used for the three-dimensional etching process simulation of photoresist advances algorithm fast;

The basic step of this method is as follows:

1.) according to the lithography simulation accuracy requirement, substrate is subdivided into the array that little square is formed, and adopts three-dimensional matrice to represent array, determine photoetching material, process conditions information and total photoetching time;

2.) point of definition surface boundary is NarrowBand, inner point is Alive, outside point is FarAway, physics, chemical model according to photoetching, the velocity amplitude that initialization is had a few, and the position among the corresponding NarrowBand of time value, state value and net point among difference initialization Alive, NarrowBand and the FarAway;

3.) set up rickle, in order to the coordinate figure and the corresponding time value of preserving net point among the NarrowBand with search minimum value fast according to the size of the time value of NarrowBand mid point;

4.) time value and velocity amplitude are represented with scientific notation, and are compressed in the matrix that rickle subscript value of net point correspondence and state value then are compressed into another matrix, reduce storage space under the prerequisite that guarantees error requirements;

5.) search the point of time value minimum among the NarrowBand, and it is removed from NarrowBand, be added into Alive, can think that the current time is the time value of this point, the non-Alive adjoint point that adds it is to NarrowBand, upgrade the time value of these adjoint points according to the Hamilton-Jacobi equation of speed and time, arrive the default photoetching time with cocycle until time value, then the representation of a surface formed by the point among the NarrowBand at the default surface topography of photoetching during the time of material.

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