CN102269925A - Phase-shift mask optimizing method based on Abbe vector imaging model - Google Patents

Abstract

The invention provides a phase-shift mask optimizing method based on an Abbe vector imaging model. The method provided by the invention comprises the following steps of: forming a phase difference of 180 DEG through arranging adjacent openings in a three-dimensional phase-shift mask and a phase of a central transmission area; arranging a variable matrix omega and constructing a target function D into a square of an Euler distance between a target figure and an image in optical resist corresponding to the current mask; and guiding the optimization of a phase-shift mask figure by utilizing the variable matrix omega and the target function D. The phase-shift mask which is optimized by using the method is not only suitable for the condition of small NA (Numerical Aperture), but also is suitable for the condition that NA is more than 0.6.

Description

A kind of phase-shift mask optimization method based on Abbe vector imaging model

Technical field

The present invention relates to a kind of phase-shift mask optimization method, belong to photoetching resolution enhancement techniques field based on Abbe (Abbe) vector imaging model.

Background technology

Current large scale integrated circuit generally adopts etching system manufacturing.Etching system mainly is divided into: four parts such as illuminator (comprising light source and condenser), mask, optical projection system and wafer.The light that light source sends is incident to mask, the opening portion printing opacity of mask after focusing on through condenser; Through behind the mask, light is incident on the wafer that scribbles photoresist via optical projection system, so just mask pattern is replicated on the wafer.

The etching system of main flow is the ArF degree of depth ultraviolet photolithographic system of 193nm at present, along with the photoetching technique node enters 45nm-22nm, the critical size of circuit has been far smaller than the wavelength of light source, so interference of light and diffraction phenomena are more remarkable, causes optical patterning to produce distortion and fuzzy.Etching system must adopt resolution enhance technology for this reason, in order to improve image quality.Phase-shift mask (phase-shifting mask PSM) is a kind of important photoetching resolution enhancement techniques.PSM adopts light transmission medium and resistance light medium to make, and the light transmission part is equivalent to opening to light.PSM is by changing the topological structure and the etch depth of mask light transmission part (being opening) in advance, and the amplitude and the phase place of the electric field intensity of modulation mask exit facet are to reach the purpose that improves imaging resolution.

In order further to improve the etching system imaging resolution, industry generally adopts immersion lithographic system at present.Immersion lithographic system enlarges the purpose that numerical aperture (numerical aperture NA) improves imaging resolution for having added refractive index greater than 1 liquid between the lower surface of last lens of projection objective and wafer thereby reach.Because immersion lithographic system has the characteristic of high NA (NA＞1), and when NA＞0.6, the vector imaging characteristic of electromagnetic field can not be out in the cold to the influence of optical patterning, so no longer suitable for its scalar imaging model of immersion lithographic system.In order to obtain the imaging characteristic of accurate immersion lithographic system, must adopt the vector imaging model that the PSM in the immersion lithographic system is optimized.

Pertinent literature (Optics Express, 2008,16:20126～20141) is at the partial coherence imaging system, proposed a kind of comparatively efficiently based on the PSM optimization method of gradient.But therefore above method is not suitable for the etching system of high NA based on the scalar imaging model of etching system.Simultaneously, prior art is not considered the response difference of optical projection system to difference light source incident ray on the surface of light source.But because the incident angle difference of diverse location light on the surface of light source, its effect to optical projection system there are differences, and therefore adopts existing method to obtain imaging and the bigger deviation of physical presence in the air, and then influences the optimization effect of mask.

Summary of the invention

The purpose of this invention is to provide a kind of phase-shift mask optimization method based on Abbe vector imaging model.This method adopts vector model that phase-shift mask is optimized, and it can be applicable to immersion lithographic system with high NA and the dry lithography system with low NA simultaneously.

Realize that technical scheme of the present invention is as follows:

A kind of phase-shift mask optimization method based on Abbe vector imaging model, concrete steps are:

Step 101, be the targeted graphical of N * N with size

As initial mask pattern M, and set the pairing phase place of each opening on the initial mask, the feasible phase differential that has 180 ° by the light of adjacent apertures;

Step 102, go up the out of phase corresponding opening at initial mask pattern M different transmissivity 1 or-1 are set, it is 0 that resistance light zone transmissivity is set; Set the matrix of variables Ω of N * N: when M (x, y)=1 o'clock,

When M (x, y)=-1 o'clock,

When M (x, y)=0 o'clock,

M (x, y) transmissivity of each pixel correspondence on the expression mask pattern wherein;

Step 103, with objective function D be configured to the Euler's distance between the imaging in the targeted graphical photoresist corresponding with current mask square, promptly

Wherein

Be the pixel value of targeted graphical, Z (x, y) pixel value of imaging in the photoresist of representing to utilize Abbe vector imaging model to calculate current mask correspondence;

Step 104, calculating target function D are for the gradient matrix of matrix of variables Ω

Step 105, utilize steepest prompt drop method to upgrade matrix of variables Ω, Wherein s is predefined optimization step-length; Obtain the mask pattern of corresponding current Ω

Step 106, calculate current mask pattern Corresponding target function value D; When D reaches predetermined upper limit value less than setting threshold or the number of times that upgrades matrix of variables Ω, enter step 107, otherwise return step 104;

Step 107 stops optimizing, with current mask pattern Be defined as through the mask pattern after optimizing.

The concrete steps of utilizing Abbe vector imaging model to calculate imaging in the photoresist of current mask correspondence in the step 103 of the present invention are:

Step 201, mask graph M grid is turned to N * N sub regions;

Step 202, according to the shape of partial coherence light source surface of light source is tiled into a plurality of pointolites, with each grid region center point coordinate (x _s, y _s) represent the pairing pointolite coordinate of this grid region;

Step 203, at a single point light source, utilize its coordinate (x _s, y _s) imaging I (α in the air when obtaining this spot light on the corresponding wafer position _s, β _s);

Step 204, judge whether to calculate imaging in the air on the corresponding wafer positions of all pointolites, if then enter step 205, otherwise return step 203;

Step 205, according to Abbe Abbe method, to imaging I (α in the air of each pointolite correspondence _s, β _s) superpose, when obtaining the partial coherence light illumination, imaging I in the air on the wafer position;

Step 206, based on the photoresist approximate model, calculate the imaging in the photoresist of mask correspondence according to imaging I in the air.

Utilize its coordinate (x at a single point light source in the step 203 of the present invention _s, y _s), imaging I (α in the air when obtaining this spot light on the corresponding wafer position _s, β _s) detailed process be:

The direction of setting optical axis is the z axle, and according to the left-handed coordinate system principle with the z axle set up global coordinate system (x, y, z);

Step 301, according to pointolite coordinate (x _s, y _s), the near field distribution E of the light wave that the calculation level light source sends N * N sub regions on mask; Wherein, E is the vector matrix of N * N, and its each element is one 3 * 1 vector, 3 components of the diffraction near field distribution of mask in the expression global coordinate system;

Step 302, obtain the Electric Field Distribution of light wave at optical projection system entrance pupil rear according near field distribution E

Wherein,

Be the vector matrix of N * N, its each element is one 3 * 1 vector, 3 components of the Electric Field Distribution at entrance pupil rear in the expression global coordinate system;

Step 303, to establish light wave direction of propagation in optical projection system approximate parallel with optical axis, further according to the Electric Field Distribution at entrance pupil rear

Obtain the Electric Field Distribution in optical projection system emergent pupil the place ahead

Wherein, the Electric Field Distribution in emergent pupil the place ahead Be the vector matrix of N * N, its each element is one 3 * 1 vector, 3 components of the Electric Field Distribution in emergent pupil the place ahead in the expression global coordinate system;

Step 304, according to the Electric Field Distribution in optical projection system emergent pupil the place ahead

Obtain the Electric Field Distribution at optical projection system emergent pupil rear

Step 305, utilize Wolf Wolf optical imagery theory, according to the Electric Field Distribution at emergent pupil rear

Obtain the Electric Field Distribution E on the wafer ^Wafer, and according to E ^WaferImaging I (α in the mask air on the corresponding wafer position of acquisition point light source _s, β _s).

Beneficial effect

The present invention utilizes Abbe vector imaging model to describe the imaging process of etching system, has considered the vectorial property of electromagnetic field, and the phase-shift mask after the optimization not only is applicable to the situation of little NA, also is applicable to the situation of NA＞0.6.

Secondly, the present invention utilizes the gradient information of optimization aim function, in conjunction with steepest prompt drop method mask pattern is optimized, and optimizes the efficient height.

Once more, the present invention is tiled into a plurality of pointolites with surface of light source, calculates imaging in its corresponding air respectively at the difference light source, has the high advantage of degree of accuracy, this method is applicable to difform light source, and satisfies the lithography simulation demand of 45nm and following technology node.

Description of drawings

Fig. 1 is the process flow diagram based on the PSM optimization method of Abbe vector imaging model.

Fig. 2 sends light wave through form the synoptic diagram of imaging in the air after mask, the optical projection system on wafer position for pointolite.

Fig. 3 is for carrying out the synoptic diagram of rasterizing to circular portion coherent source face in the present embodiment.

The impulse Response Function contrast synoptic diagram that Fig. 4 emits beam for different pointolites for the lithographic projection system.

Fig. 5 turns to the surface of light source grid behind 31 * 31 pointolites in the resulting air imaging and the surface of light source grid is turned to behind 2 * 2 pointolites imaging contrast synoptic diagram in the resulting air for the present invention.

Fig. 6 is the synoptic diagram of imaging in initial phase-shift mask and the corresponding photoresist thereof.

Fig. 7 is the synoptic diagram based on imaging in the photoresist of the phase-shift mask of scalar model optimization and correspondence thereof.

Fig. 8 is the synoptic diagram based on imaging in the photoresist of the phase-shift mask of method optimization of the present invention and correspondence thereof.

Embodiment

Further the present invention is described in detail below in conjunction with accompanying drawing.

Principle of the present invention: when light when imaging is identical with targeted graphical or approximate in photoresist by mask, the figure that then is printed in the etching system on the wafer has very high resolution.Therefore the present invention with the optimization aim function D of PSM be configured to the Euler's distance between the imaging in the targeted graphical photoresist corresponding with mask square; Size as targeted graphical is N * N, then

Be the pixel value of each point in the targeted graphical, Z (x y) is the pixel value of imaging in the photoresist of mask correspondence, Z (x, y) with

Value be 0 or 1.

As shown in Figure 1, the present invention is based on the PSM optimization method of Abbe vector imaging model, concrete steps are:

Step 101, be the targeted graphical of N * N with size

As initial mask pattern M, and set the pairing phase place of each opening on the initial mask, the feasible phase differential that has 180 ° by the light of adjacent apertures.Preferred phase place by adjacent apertures light is 0 ° or 180 ° among the present invention.Setting the pairing phase place of each opening on the initial mask is to realize by the etch depth of setting each opening of mask.

Step 102, go up the out of phase corresponding opening at initial mask pattern M different transmissivity 1 or-1 are set, it is 0 that resistance light zone transmissivity is set.The transmissivity of setting 0 ° of phase place opening correspondence in the present embodiment is that the transmissivity of 1,180 ° of phase place opening correspondence is-1, and the transmissivity of light-blocking part correspondence is 0.

Set the matrix of variables Ω of N * N: when M (x, y)=1 o'clock,

When M (x, y)=-1 o'clock,

When M (x, y)=0 o'clock,

M (x, y) transmissivity of each pixel correspondence on the expression mask pattern wherein.

Step 103, with objective function D be configured to the Euler's distance between the imaging in the targeted graphical photoresist corresponding with current mask square, promptly

Wherein

Be the pixel value of targeted graphical, Z also is the matrix of N * N, Z (x, y) pixel value of imaging in the photoresist of representing to utilize Abbe vector imaging model to calculate current mask correspondence; Wherein Z (x, y) with

Value be 0 or 1.

It is as follows to utilize Abbe vector imaging model to calculate in the photoresist of current mask correspondence imaging method among the present invention:

Variable predefine

As shown in Figure 2, the direction of setting optical axis is the z axle, and according to the left-handed coordinate system principle with the z axle set up global coordinate system (x, y, z); If the world coordinates of any point light source is (x on the partial coherence light source face _s, y _s, z _s), the direction cosine of being sent and be incident to the plane wave of mask by this pointolite are (α _s, β _s, γ _s), then the pass between world coordinates and the direction cosine is:

α _s＝x _s·NA _m，β _s＝y _s·NA _m， ${\mathrm{\&gamma;}}_s=\cos \left[{\sin }^{-1}({\mathrm{NA}}_m\&CenterDot;\sqrt{x_{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+y_s^2})\right]$

Wherein, NA _mBe optical projection system object space numerical aperture.

If the world coordinates of any point is on the mask (x, y, z), based on diffraction principle, the direction cosine that are incident to the plane wave of optical projection system entrance pupil from mask are (α, beta, gamma), wherein (α, beta, gamma) be that mask (object plane) is gone up global coordinate system (x, y z) are carried out coordinate system after the Fourier transform.

If it is (x that wafer (image planes) is gone up the world coordinates of any point _w, y _w, z _w), the direction cosine that are incident to the plane wave of image planes from the optical projection system emergent pupil are (α ', β ', γ '), and wherein (α ', β ', γ ') be that wafer (image planes) is gone up global coordinate system (x _w, y _w, z _w) carry out the coordinate system after the Fourier transform.

Transformational relation between global coordinate system and the local coordinate system:

Set up local coordinate system (e _⊥, e _P), e _⊥The direction of vibration of axle middle TE polarized light for light source emits beam, e _PThe direction of vibration of axle middle TM polarized light for light source emits beam.Wave vector is

The plane that is made of wave vector and optical axis is called the plane of incidence, and the direction of vibration of TM polarized light is in the plane of incidence, and the direction of vibration of TE polarized light is perpendicular to the plane of incidence.Then the transformational relation of global coordinate system and local coordinate system is:

$\left[\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right]=T\&CenterDot;\left[\begin{array}{c}{E}_{\&perp;}\\ E_P\end{array}\right]$

Wherein, E _x, E _yAnd E _zBe respectively that light source sends the component of light wave electric field in global coordinate system, E _⊥And E _PBe that light source sends the component of light wave electric field in local coordinate system, transition matrix T is:

$T=\left[\begin{array}{cc}-\frac{\mathrm{\&beta;}}{\mathrm{\&rho;}}& -\frac{\mathrm{\&alpha;\&gamma;}}{\mathrm{\&rho;}}\\ \frac{\mathrm{\&alpha;}}{\mathrm{\&rho;}}& -\frac{\mathrm{\&beta;\&gamma;}}{\mathrm{\&rho;}}\\ 0& \mathrm{\&rho;}\end{array}\right]$

Wherein, $\mathrm{\&rho;}=\sqrt{{\mathrm{\&alpha;}}^2+{\mathrm{\&beta;}}^2}.$

The concrete steps of obtaining imaging method in the photoresist of mask correspondence are:

Step 201, mask graph M grid is turned to N * N sub regions.

Step 202, according to the shape of partial coherence light source surface of light source is tiled into a plurality of zones, each zone is approximate with pointolite.Each grid region center point coordinate (x _s, y _s) represent the pairing pointolite coordinate of this grid region.

Because there is multiple shape in the surface of light source of employed partial coherence light source in the etching system, therefore can carry out rasterizing to it according to the shape of surface of light source.As shown in Figure 3, when for example the partial coherence light source is circular, described shape according to the partial coherence light source is carried out grid with surface of light source and turned to: with central point on the surface of light source is the center of circle, k the concentric circless different with the radius of prior setting are divided into k zone with the sphere shape light face, described k zone begun to carry out from inside to outside 1～k numbering from the center circle district, 301 is the center circle district, and 302 is the 3rd zone, and 303 is k zone of outermost.With each area dividing that is numbered 2～k is a plurality of fan-shaped grid region.The present invention can preferably become each area dividing that is numbered 2～k the fan-shaped grid region of same number.

Step 203, at a single point light source, utilize its coordinate (x _s, y _s) imaging I (α in the air when obtaining this spot light on the corresponding wafer position _s, β _s).

Step 204, judge whether to calculate imaging in the air on the corresponding wafer positions of all pointolites, if then enter step 205, otherwise return step 203.

Step 205, according to the Abbe method, to imaging I (α in the air of each pointolite correspondence _s, β _s) superpose, when obtaining the partial coherence light illumination, imaging I in the air on the wafer position.

Step 206, based on the photoresist approximate model, calculate the imaging in the photoresist of mask pattern correspondence according to imaging I in the air.

Below to utilizing single source point coordinate (x in the step 203 _s, y _s) imaging process is further elaborated in the air when obtaining this spot light on the corresponding wafer position:

Step 301, according to pointolite coordinate (x _s, y _s), the calculation level light source sends the near field distribution E of light wave N * N sub regions on mask.

Wherein, E is that the vector matrix of N * N is (if all elements of a matrix is matrix or vector, then be called vector matrix), each element in this vector matrix is one 3 * 1 vector, 3 components of the diffraction near field distribution of mask in the expression global coordinate system.E represents that two matrix corresponding elements multiply each other.

Be the vector matrix of one N * N, each element is the electric field intensity of electric field in global coordinate system that pointolite sends light wave; As establish the electric field that a pointolite on the partial coherence light source sends light wave and in local coordinate system, be expressed as

Then this electric field is expressed as in global coordinate system:

The diffraction matrices B of mask is the scalar matrix of one N * N, and each element is scalar in the scalar matrix, and approximate according to Hopkins (Thelma Hopkins), each element of B can be expressed as:

$B(m,n)=\exp \left(\frac{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}{\mathrm{\&beta;}}_{s}x}{\mathrm{\&lambda;}}\right)\exp \left(\frac{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}{\mathrm{\&alpha;}}_{s}y}{\mathrm{\&lambda;}}\right)$

$=\exp \left(\frac{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}{\mathrm{\&beta;}}_{s}m\&times;\mathrm{pixel}}{\mathrm{\&lambda;}}\right)\exp \left(\frac{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}{\mathrm{\&alpha;}}_{s}n\&times;\mathrm{pixel}}{\mathrm{\&lambda;}}\right),$ m，n＝1，2，...，N

Wherein, pixel represents the length of side of all subregion on the mask graph.

Step 302, obtain the Electric Field Distribution of light wave at optical projection system entrance pupil rear according near field distribution E $E_b^{\mathrm{ent}}(\mathrm{\&alpha;},\mathrm{\&beta;}).$

The detailed process of this step is:

Because each subregion on the mask can be regarded a secondary sub-light source as,,, the Electric Field Distribution in optical projection system entrance pupil the place ahead can be expressed as the function of α and β according to the Fourier optics theory with the center of subregion coordinate as this subregion:

$E_l^{\mathrm{ent}}(\mathrm{\&alpha;},\mathrm{\&beta;})=\frac{\mathrm{\&gamma;}}{\mathrm{j\&lambda;}}\frac{e^{-\mathrm{jkr}}}{r}F\left\{E\right\}---\left(2\right)$

Wherein, owing to have N * N sub regions on the mask, so the Electric Field Distribution in entrance pupil the place ahead

Be the vector matrix of N * N, each element in this vector matrix is one 3 * 1 vector, 3 components of the Electric Field Distribution in entrance pupil the place ahead in the expression global coordinate system.F{} represents Fourier transform, and r is the entrance pupil radius, Be wave number, λ is the wavelength that pointolite sends light wave, n _mBe the object space medium refraction index.

Because the reduction magnification of optical projection system is bigger, is generally 4 times, this moment, the numerical aperture of object space was less, caused entrance pupil the place ahead Electric Field Distribution

Axial component can ignore, so optical projection system entrance pupil the place ahead is identical with the Electric Field Distribution at entrance pupil rear, promptly

$E_b^{\mathrm{ent}}(\mathrm{\&alpha;},\mathrm{\&beta;})=E_l^{\mathrm{ent}}(\mathrm{\&alpha;},\mathrm{\&beta;})=\frac{\mathrm{\&gamma;}}{\mathrm{j\&lambda;}}\frac{e^{-\mathrm{jkr}}}{r}F\left\{E\right\}---\left(3\right)$

Wherein, owing to have N * N sub regions on the mask, so the Electric Field Distribution at entrance pupil rear

Be the vector matrix of N * N, each element in this matrix is one 3 * 1 vector, 3 components of the Electric Field Distribution at entrance pupil rear in the expression global coordinate system.

Step 303, to establish light wave direction of propagation in optical projection system approximate parallel with optical axis, further according to the Electric Field Distribution at entrance pupil rear

Obtain the Electric Field Distribution in optical projection system emergent pupil the place ahead

The detailed process of this step is:

For aberrationless preferred view system, the mapping process of entrance pupil rear and emergent pupil the place ahead Electric Field Distribution can be expressed as the form of a low-pass filter function and a modifying factor product, that is:

$E_l^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;})=\mathrm{cUe}{E}_b^{\mathrm{ent}}(\mathrm{\&alpha;},\mathrm{\&beta;})---\left(4\right)$

Wherein, the Electric Field Distribution in emergent pupil the place ahead

Be the vector matrix of N * N, each element in this vector matrix is one 3 * 1 vector, 3 components of the Electric Field Distribution in emergent pupil the place ahead in the expression global coordinate system; C is the constant correction factor, and low-pass filter function U is the scalar matrix of N * N, and the numerical aperture of expression optical projection system is to the limited receiving ability of diffraction spectrum, promptly and the value in pupil inside be 1, the value of pupil outside is 0, specifically is expressed as follows:

$U=\left\{\begin{array}{cc}1& \sqrt{{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">f</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="g"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">g</figure-callout>}}^2}\&le;1\\ 0& \mathrm{elsewhere}\end{array}\right.$

Wherein, (f g) is normalized world coordinates on the entrance pupil.

Constant correction factor c can be expressed as:

$c=\frac{r}{r^{\&prime;}}\sqrt{\frac{{\mathrm{\&gamma;}}^{\&prime;}}{\mathrm{\&gamma;}}}\frac{n_w}{R}$

Wherein, r and r ' are respectively optical projection system entrance pupil and emergent pupil radius, n _wBe the refractive index of etching system picture side immersion liquid, R is the reduction magnification of preferred view system, is generally 4.

Because the approximate optical axis that is parallel in the direction of propagation of light wave between optical projection system entrance pupil and emergent pupil, therefore for arbitrarily (α ', β '), the entrance pupil rear is identical with phase differential between emergent pupil the place ahead.Because the constant phase difference that finally will find the solution between imaging in the air (being light distribution) so entrance pupil rear and emergent pupil the place ahead can be ignored.The Electric Field Distribution that can obtain emergent pupil the place ahead thus is:

$E_l^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;})=\frac{1}{{\mathrm{\&lambda;r}}^{\&prime;}}\sqrt{{\mathrm{\&gamma;}}^{\&prime;}\mathrm{\&gamma;}}\frac{n_w}{R}\mathrm{Ue\; F}\left\{E\right\}---\left(5\right)$

Step 304, according to the Electric Field Distribution in optical projection system emergent pupil the place ahead

Obtain the Electric Field Distribution at optical projection system emergent pupil rear

According to the rotation effect of TM component between emergent pupil the place ahead and rear of electromagnetic field, to establish in the global coordinate system, the forward and backward side's of emergent pupil electric field is expressed as: the vector matrix of N * N

With

With

Each element as follows:

$E_l^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;},m,n)={[E_{\mathrm{lx}}^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;},m,n);E_{\mathrm{ly}}^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;},m,n);E_{\mathrm{lz}}^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;},m,n)]}^T$

$E_b^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;},m,n)={[E_{\mathrm{bx}}^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;},m,n);E_{\mathrm{by}}^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;},m,n);E_{\mathrm{bz}}^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;},m,n)]}^T$

Wherein, m, n=1,2 ..., N, α '=cos φ ' sin θ ', β '=sin φ ' sin θ ', γ '=cos θ ', promptly the optical projection system emergent pupil is incident to the direction cosine (wave vector) of the plane wave of image planes and is

φ ' and θ ' are respectively the position angle and the elevations angle of wave vector, then

With

Relational expression be:

$E_b^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;})=\mathrm{Ve}{E}_l^{\mathrm{ext}}({\mathrm{\&alpha;}}^{\&prime;},{\mathrm{\&beta;}}^{\&prime;})---\left(6\right)$

Wherein, V is the vector matrix of a N * N, and each element is one 3 * 3 matrix:

$V(m,n)=\left[\begin{array}{ccc}\cos {\mathrm{\&phi;}}^{\&prime;}& -\sin {\mathrm{\&phi;}}^{\&prime;}& 0\\ \sin {\mathrm{\&phi;}}^{\&prime;}& \cos {\mathrm{\&phi;}}^{\&prime;}& 0\\ 0& 0& 1\end{array}\right]\&CenterDot;\left[\begin{array}{ccc}\cos {\mathrm{\&theta;}}^{\&prime;}& 0& {\sin \mathrm{\&theta;}}^{\&prime;}\\ 0& 0& 1\\ -\sin {\mathrm{\&theta;}}^{\&prime;}& 0& \cos {\mathrm{\&theta;}}^{\&prime;}\end{array}\right]\&CenterDot;\left[\begin{array}{ccc}\cos {\mathrm{\&phi;}}^{\&prime;}& \sin {\mathrm{\&phi;}}^{\&prime;}& 0\\ -\sin {\mathrm{\&phi;}}^{\&prime;}& \cos {\mathrm{\&phi;}}^{\&prime;}& 0\\ 0& 0& 1\end{array}\right]$

$=\left[\begin{array}{ccc}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}^{\&prime;}\cos {\mathrm{\&theta;}}^{\&prime;}+{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}^{\&prime;}& \cos {\mathrm{\&phi;}}^{\&prime;}\sin {\mathrm{\&phi;}}^{\&prime;}(\cos {\mathrm{\&theta;}}^{\&prime;}-1)& \cos {\mathrm{\&phi;}}^{\&prime;}\sin {\mathrm{\&theta;}}^{\&prime;}\\ \cos {\mathrm{\&phi;}}^{\&prime;}\sin {\mathrm{\&phi;}}^{\&prime;}(\cos {\mathrm{\&theta;}}^{\&prime;}-1)& {\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}^{\&prime;}\cos {\mathrm{\&theta;}}^{\&prime;}+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}^{\&prime;}& \sin {\mathrm{\&phi;}}^{\&prime;}\sin {\mathrm{\&theta;}}^{\&prime;}\\ -\cos {\mathrm{\&phi;}}^{\&prime;}\sin {\mathrm{\&theta;}}^{\&prime;}& -\sin {\mathrm{\&phi;}}^{\&prime;}\sin {\mathrm{\&theta;}}^{\&prime;}& \cos {\mathrm{\&theta;}}^{\&prime;}\end{array}\right]$

$=\left[\begin{array}{ccc}\frac{{\mathrm{\&beta;}}^{\&prime;2}+{\mathrm{\&alpha;}}^{\&prime;2}{\mathrm{\&gamma;}}^{\&prime;}}{1-{\mathrm{\&gamma;}}^{\&prime;2}}& -\frac{{\mathrm{\&alpha;}}^{\&prime;}{\mathrm{\&beta;}}^{\&prime;}}{1+{\mathrm{\&gamma;}}^{\&prime;}}& {\mathrm{\&alpha;}}^{\&prime;}\\ -\frac{{\mathrm{\&alpha;}}^{\&prime;}{\mathrm{\&beta;}}^{\&prime;}}{1+{\mathrm{\&gamma;}}^{\&prime;}}& \frac{{\mathrm{\&alpha;}}^{\&prime;2}+{\mathrm{\&beta;}}^{\&prime;2}{\mathrm{\&gamma;}}^{\&prime;}}{1-{\mathrm{\&gamma;}}^{\&prime;2}}& {\mathrm{\&beta;}}^{\&prime;}\\ -{\mathrm{\&alpha;}}^{\&prime;}& -{\mathrm{\&beta;}}^{\&prime;}& {\mathrm{\&gamma;}}^{\&prime;}\end{array}\right]$ m，n＝1，2，...，N

Step 305, utilize the optical imagery theory of Wolf, according to the Electric Field Distribution at emergent pupil rear

Obtain the Electric Field Distribution E on the wafer ^WaferAs formula (7), and further imaging I (α in the mask air on the corresponding wafer position of acquisition point light source _s, β _s).

$E^{\mathrm{wafer}}=\frac{2\mathrm{\&pi;\&lambda;}{r}^{\&prime;}}{{\mathrm{jn}}_w^2}{e}^{j{k}^{\&prime;}{r}^{\&prime;}}{F}^{-1}\left\{\frac{1}{{\mathrm{\&gamma;}}^{\&prime;}}{E}_b^{\mathrm{ext}}\right\}---\left(7\right)$

Wherein,

F ^-1{ } is inverse Fourier transform.In (5) and (6) formula substitutions (7) formula, and ignore the constant phase item, can get:

$E^{\mathrm{wafer}}=\frac{2\mathrm{\&pi;}}{n_{w}R}{F}^{-1}\left\{\sqrt{\frac{\mathrm{\&gamma;}}{{\mathrm{\&gamma;}}^{\&prime;}}}\mathrm{Ve\; Ue\; F}\right\{E\left\}\right\}---\left(8\right)$

(1) formula is updated in (8) formula, can obtains pointolite (x _s, y _s) light distribution of image planes when throwing light on, that is:

$E^{\mathrm{wafer}}({\mathrm{\&alpha;}}_s,{\mathrm{\&beta;}}_s)=\frac{2\mathrm{\&pi;}}{n_{w}R}{F}^{-1}\left\{\sqrt{\frac{\mathrm{\&gamma;}}{{\mathrm{\&gamma;}}^{\&prime;}}}\mathrm{Ve\; Ue}F\right\{{E_i}^{\&prime;}\mathrm{e\; Be\; M}\left\}\right\}---\left(9\right)$

Because E _i' middle element value and mask coordinate are irrelevant, so following formula can be write as:

$E^{\mathrm{wafer}}({\mathrm{\&alpha;}}_s,{\mathrm{\&beta;}}_s)=\frac{2\mathrm{\&pi;}}{n_{w}R}{F}^{-1}\left\{V^{\&prime;}\right\}\&CircleTimes;\left(\mathrm{Be\; M}\right)$

Wherein,

The expression convolution,

Be the vector matrix of N * N, each element is 3 * 1 vector (v _x', v _y', v _z') ^T

E then ^Wafer(α _s, β _s) three components in global coordinate system are

$E_P^{\mathrm{wafer}}({\mathrm{\&alpha;}}_s,{\mathrm{\&beta;}}_s)=H_p\&CircleTimes;\left(\mathrm{Be\; M}\right)---\left(10\right)$

Wherein,

P=x, y, z, wherein V _p' be the scalar matrix of N * N, formed by the x component of each element of vector matrix V '.

$I({\mathrm{\&alpha;}}_s,{\mathrm{\&beta;}}_s)=\underset{p=x,y,z}{\mathrm{\&Sigma;}}{\left|\right|H_p\&CircleTimes;\left(\mathrm{Be\; M}\right)\left|\right|}_2^2$

Wherein,

Expression is to the matrix delivery and ask square.H wherein _pBe (α with B _s, β _s) function, be designated as respectively

With

Therefore following formula can be designated as:

$I({\mathrm{\&alpha;}}_s,{\mathrm{\&beta;}}_s)=\underset{p=x,y,z}{\mathrm{\&Sigma;}}{\left|\right|H_p^{{\mathrm{\&alpha;}}_s{\mathrm{\&beta;}}_s}\&CircleTimes;\left(B^{{\mathrm{\&alpha;}}_s{\mathrm{\&beta;}}_s}\mathrm{e\; M}\right)\left|\right|}_2^2$

Following formula obtains is that imaging distributes in the air of mask correspondence under the spot light, then in the step 205 under the partial coherence light illumination in the air of mask correspondence imaging can be expressed as

$I=\frac{1}{N_s}\underset{{\mathrm{\&alpha;}}_s}{\mathrm{\&Sigma;}}\underset{{\mathrm{\&beta;}}_s}{\mathrm{\&Sigma;}}\underset{p=x,y,z}{\mathrm{\&Sigma;}}{\left|\right|H_p^{{\mathrm{\&alpha;}}_s{\mathrm{\&beta;}}_s}\&CircleTimes;\left(B^{{\mathrm{\&alpha;}}_s{\mathrm{\&beta;}}_s}\mathrm{e\; M}\right)\left|\right|}_2^2---\left(11\right)$

Wherein, N _sIt is the sampling number of partial coherence light source.

Step 206, the photoresist approximate model that provides based on pertinent literature (Trans.Image Process., 2007,16:774～788), by adopting the sigmoid approximation to function photoresist effect is described:

$\mathrm{sigmoid}\left(I\right)=\frac{1}{1+\exp [-a(I-t_r)]}$

Wherein, α represents the slope of photoresist approximate model, t _rThe threshold value of expression photoresist approximate model;

Calculate being imaged as in the photoresist of mask pattern correspondence according to imaging I in the air:

$Z=\frac{1}{1+\exp [-a(I-t_r)]}---\left(12\right)$

Step 104, calculating target function D are for the gradient matrix of matrix of variables Ω

Among the present invention, objective function D is for the gradient matrix of matrix of variables Ω Can be calculated as:

Wherein, ^*Conjugate operation is got in expression; ° expression is with matrix equal Rotate 180 degree on horizontal and vertical.

The present invention can adopt following two kinds of algorithm speed technologies, improves PSM and optimizes speed, reduces the complexity of optimizing.

First method is electric field intensity caching technology (electric field caching technique EFCT).With (10) formula substitution (13) formula,

By (14) formula as can be known, for the calculating target function gradient

We at first need to calculate

And Z.And in order to calculate Z, we also need at first to calculate Therefore calculating

Process in, we are only right

Once calculate, and its result of calculation reused, thus calculate Z and

Value.

Second method is Fast Fourier Transform (FFT) (fast Fourier transform FFT) technology.Because (13) formula has comprised a large amount of convolution algorithms, therefore calculate

Process have higher complexity.In order to reduce computation complexity, we replace convolution algorithm with the FFT computing, thereby (13) formula is deformed into:

Wherein, C is the scalar matrix of a N * N, and each element is:

$C(m,n)=\exp \left[\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000090475580000011.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}(\frac{m}{N}+\frac{n}{N})\right]$ m，n＝1，2，...，N

In addition, each calculating

The time, we all need calculate

By (10) formula as can be known,

Computation process also include convolution algorithm.Utilize the FFT computing to replace convolution algorithm, we can be deformed into (10) formula:

$E_P^{\mathrm{wafer}}({\mathrm{\&alpha;}}_s,{\mathrm{\&beta;}}_s)=F^{-1}\left\{\frac{2\mathrm{\&pi;}}{n_{w}R}{V_p}^{\&prime;}\mathrm{e\; F}\right\{\mathrm{Be\; M}\left\}\right\},$ p＝x，y，z。

Step 105, utilize steepest prompt drop method to upgrade matrix of variables Ω,

Wherein s is predefined optimization step-length.Further obtain the mask pattern of corresponding current Ω

In the PSM optimizing process,

Span be

(x, span y) is Ω (x, y) ∈ [∞ ,+∞] to Ω.

Step 106, calculate current mask Corresponding target function value D.When D reaches predetermined upper limit value less than predetermined threshold or the number of times that upgrades matrix of variables Ω, enter step 107.Otherwise return step 104.

Step 107 stops optimizing, and with current mask pattern Be defined as through the mask pattern after optimizing.

Embodiment of the present invention:

As shown in Figure 4,401 two pointolite A and B on surface of light source, being got.The x component of the 402 impulse Response Function H that emit beam for different pointolites for photoetching optical projection system on the y=0 position on the pupil.The y component of the 403 impulse Response Function H that emit beam for different pointolites for photoetching optical projection system on the y=0 position on the pupil.The z component of the 404 impulse Response Function H that emit beam for different pointolites for photoetching optical projection system on the y=0 position on the pupil.

As shown in Figure 5,501 is initial two-value mask synoptic diagram, and its critical size is 45nm, and it is 1 that white is represented transmission region, its rate of penetrating, black representative resistance light zone, and its rate of penetrating is 0.Mask graph is positioned at the XY plane, and lines are parallel with Y-axis.502 for turning to the surface of light source grid behind 31 * 31 pointolites under the resulting ring illumination imaging results in the binary mask air.503 for turning to the surface of light source grid behind 2 * 2 pointolites under the resulting ring illumination imaging results in the binary mask air.504 is the Y=0 place curve of light distribution contrast that two kinds of methods obtain.505 for turning to the surface of light source grid the resulting curve of light distribution behind 31 * 31 pointolites.506 for turning to the surface of light source grid the resulting curve of light distribution behind 2 * 2 pointolites.

402,403 and 404 can find from Fig. 4, for different pointolites, exist than big-difference between the impulse Response Function of lithographic projection system.This moment is if all adopt identical impulse Response Function bring error will inevitably for obtaining of aerial image to different electric light sources.505 and 506 can find in the comparison diagram 5, and to the rasterizing of surface of light source employing different densities, light distribution has than big-difference.This has proved that also the suitable method of employing is carried out the importance of rasterizing and the meaning that the present invention possessed to the partial coherence light source under super large NA optical patterning.

Be illustrated in figure 6 as imaging in initial phase-shift mask and the corresponding photoresist thereof, 601 is initial phase shift mask pattern, and its shape is consistent with targeted graphical, white is represented 0 ° of phase place opening portion, black is represented 180 ° of phase place opening portions, and grey is represented light-blocking part, and its critical size is 45nm.602 for adopting 601 as behind the mask, imaging in the photoresist of etching system, image error is 1526 (image error is defined as the value of objective function here), and the CD error is 20nm, and wherein the CD error is the critical size of imaging in the actual photoresist in Y=0 place and the difference of desirable critical size.

Be illustrated in figure 7 as based on imaging in the photoresist of scalar Model Optimization phase-shift mask and correspondence thereof.701 is the phase-shift mask figure based on the scalar model optimization.702 is to adopt 701 as behind the mask, imaging in the photoresist of etching system, and image error is 1447, the CD error is 15.

Be illustrated in figure 8 as based on imaging in the optimization phase-shift mask of the inventive method and the corresponding photoresist thereof.801 is the phase-shift mask figure of optimizing based on Abbe vector imaging model of the present invention.802 is to adopt 801 as behind the mask, imaging in the photoresist of etching system, and image error is 324, the CD error is 0.

Comparison diagram 6,7,8 because the scalar model can't be described the vector imaging characteristic of high NA etching system accurately, so can't effectively reduce image error and CD error based on the phase-shift mask optimization method of scalar model as can be known.On the other hand, because method proposed by the invention based on accurate Abbe vector imaging model, therefore can effectively reduce image error and CD error.

Only consider the situation of alternative expression PSM among the present invention, alternative expression PSM is: the opening portion transmissivity is 1 or-1, hinders the light zone simultaneously between different openings in addition; But on behalf of the present invention, this only be confined to the situation of alternative expression PSM, and the present invention also is applicable to various ways such as attenuation type PSM.

Though combine accompanying drawing the specific embodiment of the present invention has been described; but to those skilled in the art; under the prerequisite that does not break away from the principle of the invention, can also make some distortion, replacement and improvement, these also should be considered as belonging to protection scope of the present invention.

Claims (5)

1. phase-shift mask optimization method based on Abbe vector imaging model is characterized in that concrete steps are:

Step 101, be the targeted graphical of N * N with size

As initial mask pattern M, and set the pairing phase place of each opening on the initial mask, the feasible phase differential that has 180 ° by the light of adjacent apertures;

Step 102, go up the out of phase corresponding opening at initial mask pattern M different transmissivity 1 or-1 are set, it is 0 that resistance light zone transmissivity is set; Set the matrix of variables Ω of N * N: when M (x, y)=1 o'clock,

When M (x, y)=-1 o'clock,

When M (x, y)=0 o'clock,

M (x, y) transmissivity of each pixel correspondence on the expression mask pattern wherein;

Step 103, with objective function D be configured to the Euler's distance between the imaging in the targeted graphical photoresist corresponding with current mask square, promptly

Wherein

Be the pixel value of targeted graphical, Z (x, y) pixel value of imaging in the photoresist of representing to utilize Abbe vector imaging model to calculate current mask correspondence;

Step 104, calculating target function D are for the gradient matrix of matrix of variables Ω

Step 105, utilize steepest prompt drop method to upgrade matrix of variables Ω,

Wherein s is predefined optimization step-length; Obtain the mask pattern of corresponding current Ω

Step 106, calculate current mask pattern

Corresponding target function value D; When D reaches predetermined upper limit value less than setting threshold or the number of times that upgrades matrix of variables Ω, enter step 107, otherwise return step 104;

Step 107 stops optimizing, with current mask pattern

Be defined as through the mask pattern after optimizing.

2. according to the described phase-shift mask optimization method of claim 1, it is characterized in that the concrete steps of utilizing Abbe vector imaging model to calculate imaging in the photoresist of current mask correspondence in the described step 103 are based on Abbe vector imaging model:

Step 201, mask graph M grid is turned to N * N sub regions;

Step 202, according to the shape of partial coherence light source surface of light source is tiled into a plurality of pointolites, with each grid region center point coordinate (x _s, y _s) represent the pairing pointolite coordinate of this grid region;

Step 203, at a single point light source, utilize its coordinate (x _s, y _s) imaging I (α in the air when obtaining this spot light on the corresponding wafer position _s, β _s);

Step 204, judge whether to calculate imaging in the air on the corresponding wafer positions of all pointolites, if then enter step 205, otherwise return step 203;

Step 205, according to Abbe Abbe method, to imaging I (α in the air of each pointolite correspondence _s, β _s) superpose, when obtaining the partial coherence light illumination, imaging I in the air on the wafer position;

Step 206, based on the photoresist approximate model, calculate the imaging in the photoresist of mask correspondence according to imaging I in the air.

3. according to the described phase-shift mask optimization method of claim 2, it is characterized in that, utilize its coordinate (x at a single point light source in the described step 203 based on Abbe vector imaging model _s, y _s) imaging I (α in the mask air on the corresponding wafer position when obtaining this spot light _s, β _s) detailed process be:

The direction of setting optical axis is the z axle, and according to the left-handed coordinate system principle with the z axle set up global coordinate system (x, y, z);

Step 301, according to pointolite coordinate (x _s, y _s), the near field distribution E of the light wave that the calculation level light source sends N * N sub regions on mask; Wherein, E is the vector matrix of N * N, and its each element is one 3 * 1 vector, 3 components of the diffraction near field distribution of mask in the expression global coordinate system;

Step 302, obtain the Electric Field Distribution of light wave at optical projection system entrance pupil rear according near field distribution E

Wherein, Be the vector matrix of N * N, its each element is one 3 * 1 vector, 3 components of the Electric Field Distribution at entrance pupil rear in the expression global coordinate system;

Step 303, to establish light wave direction of propagation in optical projection system approximate parallel with optical axis, further according to the Electric Field Distribution at entrance pupil rear

Obtain the Electric Field Distribution in optical projection system emergent pupil the place ahead

Wherein, the Electric Field Distribution in emergent pupil the place ahead

Be the vector matrix of N * N, its each element is one 3 * 1 vector, 3 components of the Electric Field Distribution in emergent pupil the place ahead in the expression global coordinate system;

Step 304, according to the Electric Field Distribution in optical projection system emergent pupil the place ahead

Obtain the Electric Field Distribution at optical projection system emergent pupil rear

Step 305, utilize Wolf Wolf optical imagery theory, according to the Electric Field Distribution at emergent pupil rear

Obtain the Electric Field Distribution E on the wafer ^Wafer, and according to E ^WaferImaging I (α in the mask air on the corresponding wafer position of acquisition point light source _s, β _s).

4. according to the described phase-shift mask optimization method of claim 2 based on Abbe vector imaging model, it is characterized in that, when described partial coherence light source is circle, described shape according to the partial coherence light source turns to the surface of light source grid: with central point on the surface of light source is the center of circle, k the concentric circless different with the radius of prior setting are divided into k+1 zone with the sphere shape light face, described k+1 zone begun to carry out from inside to outside 1～k+1 numbering from the center circle district, be a plurality of fan-shaped grid region with each area dividing that is numbered 2～k.

5. according to the described phase-shift mask optimization method of claim 4, it is characterized in that the number of the fan-shaped grid region that described each zone that is numbered 2～k is divided is identical based on Abbe vector imaging model.

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