CN102540698A - Method for computing double absorption layer alternating phase shift mask diffraction field - Google Patents

Abstract

The invention provides a method for computing a double absorption layer alternating phase shift mask diffraction field. The method comprises the following specific steps; 1, setting the number n of spatial harmonics when an electromagnetic field is expanded; 2, performing Fourier series expansion on the dielectric constant of each grating layer, wherein, performing Fourier series expansion on TM polarized light by computing the reciprocal of the dielectric constant; and 3, for TE polarized light and the TM polarized light, solving the characteristic matrix of each grating layer, and obtaining a diffraction field corresponding to the TE polarized light by using continuous boundary conditions in the tangent direction of the electromagnetic field. Through the Fourier series expansion on the TM polarized light by computing the reciprocal of the dielectric constant, the convergence of the TM polarized light while entering a damaged mask grating can be improved, so that the calculated diffraction field has a higher accuracy.

Description

The computing method of biabsorption layer alternating phase-shift mask diffractional field

Technical field

The present invention relates to a kind of computing method of biabsorption layer alternating phase-shift mask diffractional field, belong to photoetching resolution enhancement techniques field.

Background technology

The develop rapidly of semiconductor industry mainly has benefited from the progress of the Micrometer-Nanometer Processing Technology of microelectric technique, and photoetching technique is one of manufacturing technology of most critical in the chip preparation.Because constantly bringing forth new ideas of optical lithography techniques, it breaks through the optical exposure limit of people's expection again and again, makes it to become the mainstream technology when prior exposure.

Etching system mainly is divided into: illuminator (light source), mask, optical projection system and wafer four parts.Light incides diffraction takes place on the mask, and diffraction light gets into after the optical projection system interference imaging on wafer, again through development and etch processes after, just mask graph is transferred on the wafer.

In order to understand some phenomenons that take place in the photoetching better, theoretical direction is carried out in practical operation, need the propagation of analog simulation light in total system.Lithography simulation has become the important tool of development, optimization photoetching process at present.Here our primary study mask diffractive effect.

Analog simulation mask diffraction mainly contains two kinds of methods: kirchhoff method (Kirchhoff approach) and strict electromagnetic method (Rigorous electromagnetic field).The Kirchhoff method is approximated to mask unlimited thin, and the amplitude, the phase place that see through electric field are directly determined by mask layout (mask layout).For example binary mask (binary masks, BIM) in, the electric field intensity that sees through of transmission region is 1, it is 0 that light tight zone sees through electric field intensity, the phase place of electric field that both see through is all 0.For example alternating phase-shift mask (alternating phase shift masks, Alt.PSM) in, it is 1 that the etched area of transmission region sees through intensity; Phase place is π, and it is 1 that the non-etched area of transmission region sees through intensity, and phase place is 0;, the intensity that sees through in light tight zone all is 0.The principal feature of Kirchhoff method is that the electric field intensity and the phase change of mask zones of different is very steep.

When mask feature size much larger than wavelength and thickness the time much smaller than wavelength, the polarisation of light characteristic is not obvious, this moment Kirchhoff approximate is very accurate.When developing into 45nm along with photoetching technique, the characteristic dimension of mask is near optical source wavelength (ArF), and mask thickness also reaches wavelength magnitude, and the polarization effect of light wave is fairly obvious.(Numerical Aperture, the polarization effect that liquid immersion lithography NA), mask cause is very remarkable, and then influences image quality to add the employing large-numerical aperture.At this moment must adopt strict electromagnetic field model to simulate the diffraction of mask.

Strict electromagnetic field model has been considered 3D (Three Dimensional) effect of mask and the influence of material fully.The numerical method that adopts mainly comprises: Finite-Difference Time-Domain Method (finite-difference time domain method; FDTD), rigorous coupled wave method (rigorous coupled wave analysis; RCWA), waveguide method (the waveguide method; WG) and finite element method (finite element methods, FEM).Among the FDTD, Maxwell Maxwell equation is carried out discretize on room and time, the equation of these discretizes carries out integration to the time and has just obtained the mask diffractional field, the size of step-length when the precision of separating depends on discretize.RCWA and WG carry out Fourier Fourier series expansion with mask electromagnetic field, specific inductive capacity to obtain the eigenwert equation, obtain separating of problem through finding the solution the eigenwert equation again, and the precision of separating depends on the exponent number when Fourier launches.The FEM more complicated is understood the very difficulty of getting up also, and is not all the fashion.Through these strict electromagnetic field models, or obtain amplitude, the phase place in mask near field, or directly obtain amplitude, the phase place of far field construction light.Strict electromagnetic field model shows that mask sees through zone, no longer so steep through electric field magnitude, phase change through the zone.

Prior art (J.Opt.Soc.Am.A, 1995,12:1077-1086; Laser Journal, 2007,28:26-27) a kind of diffraction characteristic that utilizes the approximate method simulation arbitrary face shape dielectric grating of multilayer is disclosed.But this method has the deficiency of following two aspects.The first, the multi-layer grating structure that this method analytical cycle is identical.The second, this methods analyst be the dielectric diffraction properties, when analyzing the incident of TM polarized light and diminish the mask grating, the convergence variation.

Prior art (J.Opt.Soc.Am.A, 1996,13:779-784) disclose the constringent method of a kind of TM of improvement polarization, but it analyzes the individual layer grating diffration.And in alternating phase-shift mask, in the substrate of glass cycle of etch areas different with the cycle of mask absorption layer, simultaneously alternating phase-shift mask has two absorption layers, so said method is not suitable for finding the solution the alternating phase-shift mask diffractional field.

Summary of the invention

The present invention provides a kind of computing method of biabsorption layer alternating phase-shift mask diffractional field, and this method can quick and precisely be calculated the formed diffractional field of TM polarized light incident mask, and can analyze the tri-layer mask grating diffration with different cycles.

Realize that technical scheme of the present invention is following:

A kind of computing method of biabsorption layer alternating phase-shift mask diffractional field, concrete steps are:

Step 1, space harmonics number (the number of space harmonics) n when setting the electromagnetic field expansion;

Step 2, the specific inductive capacity of each layer grating is carried out Fourier Fourier series expansion;

For the TE polarized light, then be:

${\mathrm{\&epsiv;}}_l\left(x\right)=\underset{h=-D}{\overset{D}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l,h}\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000676237800021.png,HSA00000676237800022.png"\; state="\{\{state\}\}">j</figure-callout>}\frac{2\mathrm{\&pi;hx}}{\mathrm{\&Lambda;}}\right)$

For the TM polarized light, then be:

$\frac{1}{{\mathrm{\&epsiv;}}_l\left(x\right)}=\underset{h=-D}{\overset{D}{\mathrm{\&Sigma;}}}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_{l,h}\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000676237800021.png,HSA00000676237800022.png"\; state="\{\{state\}\}">j</figure-callout>}\frac{2\mathrm{\&pi;hx}}{\mathrm{\&Lambda;}}\right)$

Wherein, the x direction is the grating vector direction, and Λ is the alternating phase-shift mask lowest common multiple in each grating cycle, l=[1,2,3], D=n-1, ε _l(x) be the specific inductive capacity of l layer grating, ε _{L, h}Be h Fourier Fourier of l layer grating relative dielectric constant component,

Be l layer grating relative dielectric constant h Fourier component reciprocal;

Step 3, to the TE polarized light, utilize the ε in the step 2 _{L, h}, find the solution the eigenmatrix of every layer of grating, utilize the continuous boundary condition in electromagnetic field tangential again, obtain the pairing diffractional field of TE polarized light;

To the TM polarized light; Utilize

in the step 2 to find the solution the eigenmatrix of every layer of grating; Utilize the continuous boundary condition in electromagnetic field tangential again, obtain the pairing diffractional field of TM polarized light.

Beneficial effect

The first, the present invention is directed to the TM polarized light, through the Fourier series expansion is carried out in elastivity, convergence when having improved the incident of TM polarized light and diminishing the mask grating makes the diffractional field that calculates have higher accuracy.

The second, the present invention carries out the Fourier series expansion through choosing the lowest common multiple in each grating cycle, can analyze the multilayer mask grating diffration with different cycles.

Description of drawings

Fig. 1 is biabsorption layer alternating phase-shift mask and diffraction synoptic diagram thereof.

Fig. 2 is the computing method process flow diagram of biabsorption layer alternating phase-shift mask diffractional field.

Fig. 3 is K _xThe synoptic diagram of matrix.

Fig. 4 is E _lThe synoptic diagram of matrix.

Fig. 5 is Y _I, Z _I, Y _IIAnd Z _IIThe synoptic diagram of matrix.

When Fig. 6 was TE, TM polarized light normal incidence Alt.PSM, 0,1 grade time diffraction efficiency was with the variation of live width.

When Fig. 7 was TE, TM polarized light normal incidence Alt.PSM, 0,1 grade time degree of polarization was with the variation of live width.

Embodiment

Below in conjunction with accompanying drawing the present invention is further elaborated.

Fig. 1 is biabsorption layer alternating phase-shift mask and diffraction synoptic diagram thereof, below biabsorption layer alternating phase-shift mask related in the present embodiment is described.

The present invention is the z direction with mask border normal direction (the normal to the boundary), and (the gratingvector) is the x direction with the grating vector direction, is the y direction with grating grizzly bar direction pointed; Set up coordinate system; (x, y z) meet right-hand rule to the coordinate system of wherein being set up.

Alternating phase-shift mask is divided into three layers along the z direction, and wherein first and second layer is an absorption layer, and the 3rd layer is phase shift layer.Each layer is periodic alternately to arrange along the x direction, wherein first and second layer cycle identical, for diminishing medium, the 3rd layer is dielectric, its cycle be before two times of two-layer cycle.In Fig. 1, ground floor (z ₀＜z＜z ₁) be generally CrO, the second layer (z ₁＜z＜z ₂) being generally Cr, the 3rd layer etching depth is d=λ/2 (n-1), and to realize 180 ° phase shift, wherein λ is the incident light wavelength, and n is the 3rd layer a refractive index.One line polarisation TE (electric field is perpendicular to plane of incidence) or TM (magnetic field is perpendicular to plane of incidence) are with angle θ; Be incident on the alternating phase-shift mask from the incidence zone I of mask top; Diffraction takes place then, and from the outgoing district II outgoing of mask below, the refractive index that wherein defines the incidence zone is n _I, the refractive index in outgoing district is n _IIThe object of the invention is: calculate incident light in the outgoing district the formed diffractional field of II.

As shown in Figure 2, the computing method of biabsorption layer alternating phase-shift mask diffractional field of the present invention, concrete steps are:

Step 1, space harmonics number (the number of space harmonics) n when setting the electromagnetic field expansion; N can carry out suitable choosing as required in this step, for example when required degree of precision result of calculation, n is chosen big, when needs faster during computing velocity, n is chosen more less.

Step 2, the specific inductive capacity of each layer grating is carried out Fourier Fourier series expansion.

For the TE polarized light, then be:

${\mathrm{\&epsiv;}}_l\left(x\right)=\underset{h=-D}{\overset{D}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l,h}\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000676237800021.png,HSA00000676237800022.png"\; state="\{\{state\}\}">j</figure-callout>}\frac{2\mathrm{\&pi;hx}}{\mathrm{\&Lambda;}}\right)$

For the TM polarized light, then be:

$\frac{1}{{\mathrm{\&epsiv;}}_l\left(x\right)}=\underset{h=-D}{\overset{D}{\mathrm{\&Sigma;}}}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_{l,h}\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000676237800021.png,HSA00000676237800022.png"\; state="\{\{state\}\}">j</figure-callout>}\frac{2\mathrm{\&pi;hx}}{\mathrm{\&Lambda;}}\right)$

Wherein, Λ is the alternating phase-shift mask lowest common multiple in each grating cycle, l=[1,2,3], D=n-1, ε _l(x) be the specific inductive capacity of l layer grating, ε _{L, h}Be h Fourier Fourier of l layer grating relative dielectric constant component,

Be l layer grating relative dielectric constant h Fourier component reciprocal.

In three grating layers of the mask of choosing in the present embodiment, the cycle of first and second layer equates, the 3rd layer the cycle be before the twice in two-layer cycle, so Λ is the 3rd layer cycle.

Step 3, to the TE polarized light, utilize the ε in the step 2 _{L, h}, find the solution the eigenmatrix of every layer of grating, utilize the continuous boundary condition in electromagnetic field tangential again, obtain the pairing diffractional field of TE polarized light; To the TM polarized light, utilize in the step 2

Find the solution the eigenmatrix of every layer of grating, utilize the continuous boundary condition in electromagnetic field tangential again, obtain the pairing diffractional field of TM polarized light.

The detailed process of this step is:

Step 301, open the Floquet condition according to the cloth Lip river, find the solution the component of the wave vector of i the order of diffraction time along tangential and normal direction respectively, wherein i gets the integer in time [m, m], 2m+1=n, and promptly the number of i institute value is n;

The component of (being the x direction) is wave vector along the tangential:

k _xi＝k _o[n _Isinθ-i(λ ₀/Λ)]

Wherein, k _oBe incident light wave vector in a vacuum, λ ₀Be incident light wavelength in a vacuum, n _IBe the refractive index of incidence zone, θ is an angle of incidence of light.

Wave vector along the component of normal direction (being the z direction) is:

$k_{l^{\&prime;},\mathrm{zi}}=\left\{\begin{array}{cc}+{[{\left(k_o{n}_{l^{\&prime;}}\right)}^2-k_{\mathrm{xi}}^2]}^{1/2}& k_{\mathrm{xi}}^2<{\left(k_o{n}_{l^{\&prime;}}\right)}^2\\ -j{[k_{\mathrm{xi}}^2-{\left(k_o{n}_{l^{\&prime;}}\right)}^2]}^{1/2}& k_{\mathrm{xi}}^2>{\left(k_o{n}_{l^{\&prime;}}\right)}^2\end{array}\right.$

l′＝I，II

Wherein, subscript I representes the incidence zone, and subscript II representes the outgoing district; When l '=I, n _{L '}The refractive index of expression incidence zone, when l '=II, n _{L '}The refractive index in expression outgoing district, j representes imaginary unit.

Step 302, find the solution the eigenmatrix of every layer of grating.

For the TE polarized light, then

$A_l^{\mathrm{TE}}=K_x^2-E_l$

Wherein,

K _x, E _lAll be (matrix of n * n),

For required that find the solution, for the eigenmatrix of TE polarized light.

K _xBe diagonal matrix, and diagonal element

Be k _Xi/ k _o,

As shown in Figure 3, for example, n=3, because 2m+1=n, m=1 then, i=[1,0,1] generates k through step 301 _XiComprise k _X-1, k _X0And k _X1K _xBe the matrix of (3 * 3), when i=-1, promptly coordinate be (1+2 ,-1+2), promptly coordinate is that the element of (1,1) is k _X-1, when i=0, promptly coordinate is that (0+2,0+2), promptly coordinate is that the element of (2,2) is k _X0, when i=1, promptly coordinate is that (1+2,1+2), promptly coordinate is that the element of (3,3) is k _X1

E _lBe the matrix of harmonic component (the permittivity harmonic components) composition of l layer specific inductive capacity, (p q) equals ε to its element _{L, p-q}, p=[1,2 ..., n], q=[1,2 ..., n], as shown in Figure 4, for example, and n=3, because D=n-1, D=2 then, h=[2 ,-1,0,1,2], the ε on the step 2 _{L, h}Comprise ε _{L ,-2}, ε _{L ,-1}, ε _{L, 0}, ε _{L, 1}And ε _{L, 2}E _lBe the matrix of (3 * 3),

Work as p=1, during q=1, promptly coordinate is that the element of (1,1) is ε _{L, p-q}=ε _{L, 0},

Work as p=1, during q=2, promptly coordinate is that the element of (1,2) is ε _{L, p-q}=ε _{L ,-1},

Work as p=1, during q=3, promptly coordinate is that the element of (1,3) is ε _{L, p-q}=ε _{L ,-2},

Work as p=2, during q=1, promptly coordinate is that the element of (2,1) is ε _{L, p-q}=ε _{L, 1},

Work as p=2, during q=2, promptly coordinate is that the element of (2,2) is ε _{L, p-q}=ε _{L, 0},

Work as p=2, during q=3, promptly coordinate is that the element of (2,3) is ε _{L, p-q}=ε _{L ,-1},

Work as p=3, during q=1, promptly coordinate is that the element of (3,1) is ε _{L, p-q}=ε _{L, 2},

Work as p=3, during q=2, promptly coordinate is that the element of (3,2) is ε _{L, p-q}=ε _{L, 1},

Work as p=3, during q=3, promptly coordinate is that the element of (3,3) is ε _{L, p-q}=ε _{L, 0}

For the TM polarized light, then

$A_l^{\mathrm{TM}}={\stackrel{\&OverBar;}{E}}_l^{-1}(K_x{E}_l^{-1}{K}_x-I)$

Wherein, For required that find the solution, for the eigenmatrix of TM polarized light,

For (matrix of n * n), it representes the matrix that the Fourier component of l layer elastivity is formed, (p q) does its element

Be E _lInverse matrix,

For

Inverse matrix, I representation unit matrix.

Step 303, find the solution incidence zone matrix Y _I, Z _I, find the solution outgoing district matrix Y _II, Z _II

Wherein, Y _I, Y _IIBe all (diagonal matrix of n * n), Y _IThe diagonal element of matrix

Be (k _{I, zi}/ k _o), Y _IIThe diagonal element of matrix Be (k _{II, zi}/ k _o); Z _I, Z _IIBe all (diagonal matrix of n * n), Z _IThe diagonal element of matrix

For The diagonal element of ZII matrix

For

N=3 for example, because 2m+1=n, m=1 then, i=[1,0,1] generates k through step 2 _{I, zi}Comprise k _{I, z-1}, k _{I, z0}And k _{I, z1}, generate k _{II, zi}Comprise k _{II, z-1}, k _{II, z0}And k _{II, z1}, the matrix Y that then generates _I, Z _IY _IIAnd Z _II, as shown in Figure 5.

Step 304, utilize electromagnetic field tangential continuity boundary conditions, obtain the expression formula between incidence zone and the outgoing district electromagnetic field.

For the TE polarized light, then

$\left[\begin{array}{c}\mathrm{\&delta;}\\ {\mathrm{jn}}_l\cos \mathrm{\&theta;\&delta;}\end{array}\right]+\left[\begin{array}{c}I\\ -{\mathrm{jY}}_I\end{array}\right]R=\underset{l=1}{\overset{3}{\mathrm{\&Sigma;}}}\left[\begin{array}{cc}{W}_l^{\mathrm{TE}}& W_l^{\mathrm{TE}}{X}_l^{\mathrm{TE}}\\ V_l^{\mathrm{TE}}& -V_l^{\mathrm{TE}}{X}_l^{\mathrm{TE}}\end{array}\right]{\left[\begin{array}{cc}{W}_l^{\mathrm{TE}}{X}_l^{\mathrm{TE}}& W_l^{\mathrm{TE}}\\ V_l^{\mathrm{TE}}{X}_l^{\mathrm{TE}}& -V_l^{\mathrm{TE}}\end{array}\right]}^{-1}\left[\begin{array}{c}I\\ {\mathrm{jY}}_{\mathrm{II}}\end{array}\right]T^{\mathrm{TE}}$

Wherein, δ is the matrix of n * 1; Work as i=0;

i ≠ 0;

is the feature matrix of eigenmatrix

; is diagonal matrix; Diagonal element is the positive square root of eigenmatrix

eigenwert, and R is an intermediate variable. is diagonal matrix; Diagonal element is that

is the positive square root of eigenmatrix

eigenwert; M '=[1; 2; N]; When m '=1; Then

is the positive square root of the pairing eigenwert of eigenmatrix

first row, and the rest may be inferred.

For the TM polarized light, then

$\left[\begin{array}{c}\mathrm{\&delta;}\\ \mathrm{j\&delta;}\cos \left(\mathrm{\&theta;}\right)/n_I\end{array}\right]+\left[\begin{array}{c}I\\ -{\mathrm{jZ}}_I\end{array}\right]R=\underset{l=1}{\overset{3}{\mathrm{\&Sigma;}}}\left[\begin{array}{cc}{W}_l^{\mathrm{TM}}& W_l^{\mathrm{TM}}{X}_l^{\mathrm{TM}}\\ V_l^{\mathrm{TM}}& -V_l^{\mathrm{TM}}{X}_l^{\mathrm{TM}}\end{array}\right]{\left[\begin{array}{cc}{W}_l^{\mathrm{TM}}{X}_l^{\mathrm{TM}}& W_l^{\mathrm{TM}}\\ V_l^{\mathrm{TM}}{X}_l^{\mathrm{TM}}& -V_l^{\mathrm{TM}}\end{array}\right]}^{-1}\left[\begin{array}{c}I\\ {\mathrm{jZ}}_{\mathrm{II}}\end{array}\right]T^{\mathrm{TM}}$

Wherein,

is the feature matrix of eigenmatrix

; is diagonal matrix, and diagonal element is the positive square root of eigenmatrix

eigenwert.

is diagonal matrix, and diagonal element is that

is the positive square root of eigenmatrix

eigenwert.

Step 305, utilization strengthen transmission matrix method, obtain the matrix T that the inferior amplitude of each order of diffraction of TE polarized light is formed ^TE, T wherein ^TEBe the matrix of n * 1, T ^TEIn each element be the form of plural a+bj, wherein the amplitude of diffractional field does

Promptly obtain the corresponding diffractional field of TE polarized light.

Utilize to strengthen transmission matrix method, obtain the matrix T that the inferior amplitude of each order of diffraction of TM polarized light is formed ^TM, T wherein ^TMBe the matrix of n * 1, T ^TMIn each element be the form of plural a+bj, wherein the amplitude of diffractional field does Promptly obtain the corresponding diffractional field of TM polarized light.

Wherein strengthening transmission matrix method is prior art, and the present invention simply provides it, and to find the solution formula following:

For the TE polarized light, then

$\left[\begin{array}{c}\mathrm{\&delta;}\\ {\mathrm{jn}}_I\cos \mathrm{\&theta;\&delta;}\end{array}\right]+\left[\begin{array}{c}I\\ -{\mathrm{jY}}_I\end{array}\right]R=\left[\begin{array}{c}{f}_1\\ g_1\end{array}\right]{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000676237800013.png,HSA00000676237800021.png"\; state="\{\{state\}\}">T</figure-callout>}}_1^{\mathrm{TE}}$

For the TM polarized light, then

$\left[\begin{array}{c}\mathrm{\&delta;}\\ \mathrm{j\&delta;}\cos \left(\mathrm{\&theta;}\right)/n_I\end{array}\right]+\left[\begin{array}{c}I\\ -{\mathrm{jZ}}_I\end{array}\right]R=\left[\begin{array}{c}{f}_1\\ g_1\end{array}\right]{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000676237800013.png,HSA00000676237800021.png"\; state="\{\{state\}\}">T</figure-callout>}}_1^{\mathrm{TM}}$

Wherein,

$\left[\begin{array}{c}{a}_N\\ b_N\end{array}\right]={\left[\begin{array}{cc}{W}_N^{\mathrm{\&Gamma;}}& W_N^{\mathrm{\&Gamma;}}\\ V_N^{\mathrm{\&Gamma;}}& -V_N^{\mathrm{\&Gamma;}}\end{array}\right]}^{-1}\left[\begin{array}{c}{f}_{N+1}\\ g_{N+1}\end{array}\right]$ N＝[1，2，3]，Г＝TE，EM

$\left[\begin{array}{c}{f}_N\\ g_N\end{array}\right]T_N^{\mathrm{\&Gamma;}}=\left[\begin{array}{cc}{W}_N^{\mathrm{\&Gamma;}}& W_N^{\mathrm{\&Gamma;}}{X}_N^{\mathrm{\&Gamma;}}\\ V_N^{\mathrm{\&Gamma;}}& -V_N^{\mathrm{\&Gamma;}}{X}_N^{\mathrm{\&Gamma;}}\end{array}\right]\left[\begin{array}{c}I\\ b_N{a}_N^{-1}{X}_N^{\mathrm{\&Gamma;}}\end{array}\right]T_N^{\mathrm{\&Gamma;}}=\left[\begin{array}{c}{W}_N^{\mathrm{\&Gamma;}}(I+X_N^{\mathrm{\&Gamma;}}{b}_N{a}_N^{-1}{X}_N^{\mathrm{\&Gamma;}})\\ V_N^{\mathrm{\&Gamma;}}(I-X_N^{\mathrm{\&Gamma;}}{b}_N{a}_N^{-1}{X}_N^{\mathrm{\&Gamma;}})\end{array}\right]T_N^{\mathrm{\&Gamma;}}$

$T^{\mathrm{\&Gamma;}}=a_3^{-1}{X}_3^{\mathrm{\&Gamma;}}{a}_2^{-1}{\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="HSA00000676237800021.png,HSA00000676237800022.png"\; state="\{\{state\}\}">X</figure-callout>}}_2^{\mathrm{\&Gamma;}}{a}_1^{-1}{\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HSA00000676237800013.png,HSA00000676237800021.png"\; state="\{\{state\}\}">X</figure-callout>}}_1^{\mathrm{\&Gamma;}}{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HSA00000676237800013.png,HSA00000676237800021.png"\; state="\{\{state\}\}">T</figure-callout>}}_1^{\mathrm{\&Gamma;}}$

When incident light is the TE polarized light,

f ₄＝I，g ₄＝jY _II

When incident light is the TM polarized light,

f ₄＝I，g ₄＝jZ _II

Further, the present invention can also find the solution the inferior diffraction efficiency of each order of diffraction.

For the TE polarized light, then:

${\mathrm{\&eta;}}_i^{\mathrm{TE}}=T_i^{\mathrm{TE}}{\left(T_i^{\mathrm{TE}}\right)}^{*}\mathrm{Re}\left(\frac{k_{\mathrm{II},\mathrm{zi}}}{k_o{n}_I\cos \mathrm{\&theta;}}\right)$

Wherein,

Be T ^TEIn

Individual element,

For Conjugation.

For the TM polarized light, then:

${\mathrm{\&eta;}}_i^{\mathrm{TM}}=T_i^{\mathrm{TM}}{\left(T_i^{\mathrm{TM}}\right)}^{*}\mathrm{Re}\left(\frac{k_{\mathrm{II},\mathrm{zi}}}{n_{\mathrm{II}}^2}\right)/\left(\frac{k_o\cos \mathrm{\&theta;}}{n_I}\right)$

Wherein,

Be T ^TMIn Individual element,

For

Conjugation.

A nearlyer step ground, the present invention can find the solution the inferior degree of polarization of each order of diffraction, and (Degree of Polarization DoP), judges the type of mask, judges that promptly mask is similar TE polaroid or TM polaroid.

${\mathrm{DoP}}_i=\frac{{\mathrm{\&eta;}}_i^{\mathrm{TE}}-{\mathrm{\&eta;}}_i^{\mathrm{TM}}}{{\mathrm{\&eta;}}_i^{\mathrm{TE}}+{\mathrm{\&eta;}}_i^{\mathrm{TM}}}\&CenterDot;100\%$

η representes diffraction efficiency, and the η subscript is represented transverse electric field TE (transverse electric), transverse magnetic TM (transverse magnetic), and subscript representes that the order of diffraction is inferior.DoP representes the similar TE polaroid of mask for just, and DoP representes the similar TM polaroid of mask for negative.

Embodiment of the present invention

Here calculated among the CrO/Cr Alt.PSM, when TE, TM normal incidence (193nm), 0,1 grade time diffraction efficiency and degree of polarization during different mask linewidths.Wherein CrO refractive index, extinction coefficient and thickness be respectively 1.965,1.201 and 18nm.Cr refractive index, extinction coefficient and thickness be respectively 1.477,1.762 and 55nm. analyzes is 1: 1 intensive lines here, dutycycle is 0.5.

When Fig. 6 was TE, TM polarized light normal incidence Alt.PSM, 0,1 grade time diffraction efficiency was with the variation of live width.In the Kirchhoff method, 0 grade time diffraction efficiency is 0.And strict electromagnetic field model is when showing little live width, 0 grade diffraction efficiency of TM polarized light is not 0, and it is far longer than the TE polarization.0 order of diffraction of this non-zero time has caused intensity energy imbalance (intensity imbalancing phenomena) just.

When Fig. 7 was TE, TM polarized light normal incidence Alt.PSM, 0,1 grade time degree of polarization was with the variation of live width.In the time of can seeing little live width, all become a TM polarizer (TM polarizer) 0,1 grade time, this can be lowered into image contrast.

Though described embodiment of the present invention in conjunction with accompanying drawing, for the technician in present technique field,, can also do some distortion, replacement and improvement not breaking away under the prerequisite of the present invention, these also are regarded as belonging to protection scope of the present invention.

Claims (1)

1. the computing method of a biabsorption layer alternating phase-shift mask diffractional field is characterized in that concrete steps are:

Step 1, the space harmonics when setting the electromagnetic field expansion are counted n;

Step 2, the specific inductive capacity of each layer grating is carried out Fourier Fourier series expansion;

For the TE polarized light, then be:

${\mathrm{\&epsiv;}}_l\left(x\right)=\underset{h=-D}{\overset{D}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l,h}\exp \left(j\frac{2\mathrm{\&pi;hx}}{\mathrm{\&Lambda;}}\right)$

For the TM polarized light, then be:

$\frac{1}{{\mathrm{\&epsiv;}}_l\left(x\right)}=\underset{h=-D}{\overset{D}{\mathrm{\&Sigma;}}}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_{l,h}\exp \left(j\frac{2\mathrm{\&pi;hx}}{\mathrm{\&Lambda;}}\right)$

Wherein, the x direction is the grating vector direction, and Λ is the alternating phase-shift mask lowest common multiple in each grating cycle, l=[1,2,3], D=n-1, ε _l(x) be the specific inductive capacity of l layer grating, ε _{L, h}Be h Fourier Fourier of l layer grating relative dielectric constant component, Be l layer grating relative dielectric constant h Fourier component reciprocal;

Step 3, to the TE polarized light, utilize the ε in the step 2 _{L, h}, find the solution the eigenmatrix of every layer of grating, utilize the continuous boundary condition in electromagnetic field tangential again, obtain the pairing diffractional field of TE polarized light;

To the TM polarized light; Utilize

in the step 2 to find the solution the eigenmatrix of every layer of grating; Utilize the continuous boundary condition in electromagnetic field tangential again, obtain the pairing diffractional field of TM polarized light.

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