CN102636951A - Computing method for diffraction field of double-absorbing-layer alternating phase shift contact hole mask - Google Patents

Computing method for diffraction field of double-absorbing-layer alternating phase shift contact hole mask Download PDF

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CN102636951A
CN102636951A CN2012101481993A CN201210148199A CN102636951A CN 102636951 A CN102636951 A CN 102636951A CN 2012101481993 A CN2012101481993 A CN 2012101481993A CN 201210148199 A CN201210148199 A CN 201210148199A CN 102636951 A CN102636951 A CN 102636951A
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CN102636951B (en
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李艳秋
杨亮
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Beijing Institute of Technology BIT
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Abstract

The invention provides a computing method for a diffraction field of a double-absorbing-layer alternating phase shift contact hole mask. According to the method, diffraction of the double-absorbing-layer alternating phase shift contact hole mask in photo-etching can be quickly computed. The computing method specifically comprises the following steps of: 1, setting harmonic number reserved in the x direction and setting harmonic number reserved in the y direction; 2, solving components of wave vector of each diffraction order in the tangential direction and the normal direction according to a Floquet condition; 3, performing Fourier series expansion on dielectric constant of a two-dimensional grating on each layer and a reciprocal of the dielectric constant; and 4, solving the diffraction field of an emission region by using an enhanced transmission matrix method. In two orthogonal directions, the Fourier series expansion is performed by selecting the minimum common multiple of cycles of three grating layers in the corresponding orthogonal direction, so that the diffraction of a plurality of layers of two-dimensional mask gratings with different cycles in the two orthogonal directions can be analyzed, and meanwhile, the diffraction field of the double-absorbing-layer alternating phase shift contact hole mask can be quickly solved.

Description

The computing method of biabsorption layer alternating phase-shift contact hole mask diffractional field
Technical field
The present invention relates to a kind of computing method of biabsorption layer alternating phase-shift contact hole 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 with figure transfer to wafer.
Structure more complicated on the mask, according to the periodicity on all directions, mask can be divided into one dimension, X-Y scheme.The one dimension figure only has periodically in one direction, and fairly simple, common lines/space (Line/Space) structure is exactly the one dimension figure.X-Y scheme all has on both direction periodically, is some complicated geometric figures, and is more approaching with the practical devices structure.Contact hole (Contact Hole), L figure, splicing figure and H figure all are two-dimensional structures.In addition, can be divided into three types of intensive figure, half intensive figure and isolation patterns again according to pattern density.
In order to understand the Physical Mechanism that said process takes place better, need set up model, and the propagation therein of analog simulation light.And lithography simulation has become development, has optimized the important tool of photoetching process.Here we introduce the computing method of mask diffraction.
Analog simulation mask diffraction mainly contains two kinds of methods: kirchhoff method (Kirchhoff approach) and strict electromagnetic method (Rgorous electromagnetic field).As what infinitely approach, the amplitude, the phase place that see through electric field are directly determined by mask layout (mask layout) the Kirchhoff method with mask.For example binary mask (binary masks, BIM) in, the light intensity of transmission region is 1, phase place is 0, light tight regional light intensity is 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 light intensity; Phase place is π; It is 1 that the non-etched area of transmission region sees through light intensity, and phase place is 0, light tight zone to see through light intensity all be 0.The principal feature of Kirchhoff method is that intensity, 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, adds and adopts large-numerical aperture (Numerical Aperture, liquid immersion lithography NA); The polarisation of light effect is fairly obvious, 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;
Figure BSA00000717055500021
), waveguide method (the waveguide method; ) and finite element method (finite element methods, FEM).Among the FDTD, Maxwell (Maxwell) equation is carried out discretize in the space, on the 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.
Figure BSA00000717055500023
and
Figure BSA00000717055500024
carries 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, 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 is no longer so steep through electric field magnitude, phase change through the zone with through the zone.
Prior art (J.Opt.Soc.Am.A, 1994,11,9:2494-2502; JOURNAL OF MUDANJIANG COLLEGE OF EDUCATION; 2009,6:57-59) diffraction characteristic of a kind of utilization
Figure BSA00000717055500031
analysis of two-dimensional sub-wave length grating is disclosed.But this method has following deficiency, and it can only the identical multilayer two-dimension grating of analytical cycle; This methods analyst be the dielectric diffraction properties, and convergence is relatively poor; This method has only been analyzed the diffraction of one deck two-dimensional grating simultaneously; And in the alternating phase-shift contact hole mask; Mask has three grating layers; Etch areas cycle of (x, y) on the direction of two quadratures is two times of corresponding cycle of mask absorption layer in the substrate of glass, and the cycle on two orthogonal directionss is all inequality, and substrate phase shift district presents the crossed grating characteristic.Therefore adopt said method can not calculate the diffraction of biabsorption layer alternating phase-shift contact hole mask.
Summary of the invention
The present invention provides a kind of computing method of biabsorption layer alternating phase-shift contact hole mask diffraction, and this method can be calculated the diffraction of biabsorption layer alternating phase-shift contact hole mask in the photoetching fast.
Realize that technical scheme of the present invention is following:
A kind of computing method of biabsorption layer alternating phase-shift contact hole mask diffraction, concrete steps are:
The harmonic number that keeps on step 1, the setting x direction is L x, setting the harmonic number that keeps on the y direction is L y
Step 2, according to Bu Luokai (Floquet) condition, find the solution the (m, n) wave vector of the individual order of diffraction time along the tangential, the component of normal direction, wherein m is for getting time [D x, D x] between integer, n is for getting all over [D y, D y] between integer, L x=2D x+ 1, L y=2D y+ 1;
Wave vector is the component α of x, y axle along the tangential m, β nFor:
α m = α 0 - 2 πm / Λ x β n = β 0 - 2 πn / Λ y - - - ( 1 )
α 0=n Iksinθcosδ,β 0=n Iksinθsinδ (2)
Wherein k is an incident light wave wave vector in a vacuum, n IBe the refractive index of incidence zone, θ is the incident angle of light wave, and δ is the position angle of light wave, Λ xFor mask along the x direction lowest common multiple in three layers of grating cycle, Λ yFor mask along the y direction lowest common multiple in three layers of grating cycle;
Wave vector is the component r of z axle along the normal direction of grating planar Mn, t MnFor:
r mn = [ ( n I k ) 2 - α m 2 - β n 2 ] 1 / 2 α m 2 + β n 2 ≤ ( n I k ) 2 - j [ α m 2 + β n 2 - ( n I k ) 2 ] 1 / 2 α m 2 + β n 2 > ( n I k ) 2 - - - ( 3 )
t mn = [ ( n II k ) 2 - α m 2 - β n 2 ] 1 / 2 α m 2 + β n 2 ≤ ( n II k ) 2 - j [ α m 2 + β n 2 - ( n II k ) 2 ] 2 α m 2 + β n 2 > ( n II k ) 2 - - - ( 4 )
Wherein subscript I representes the incidence zone, n IThe refractive index of expression incidence zone, subscript II representes outgoing district, n IIThe refractive index in expression outgoing district, j representes imaginary unit;
Step 3, Fourier Fourier series expansion is carried out in the specific inductive capacity and the elastivity of each layer two-dimensional grating;
Specific inductive capacity can be expressed as as fourier expansion:
ϵ l ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ϵ l , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l = 1,2 ) - - - ( 5 )
ϵ l ′ ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ϵ l ′ , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l ′ = 3 ) - - - ( 6 )
ε wherein L, (p, q)Be l layer grating relative dielectric constant (p, q) individual Fourier component, ε L ', (p, q)Be l ' layer grating relative dielectric constant (p, q) individual Fourier component;
Elastivity can be expressed as as fourier expansion:
1 / ϵ l ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ξ l , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l = 1,2 ) - - - ( 7 )
1 / ϵ l ′ ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ξ l ′ , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l ′ = 3 ) - - - ( 8 )
ξ wherein L, (p, q)Be l layer grating relative dielectric constant (p, q) individual Fourier component, ξ reciprocal L ', pqBe l ' layer grating relative dielectric constant (p, q) individual Fourier component reciprocal;
Step 4, according to the α that calculates in the step 2 m, β n, r (m, n), t (m, n)And the ε that calculates in the step 3 L, (p, q), ε L ' (p, q), ξ L, (p, q)And ξ L ', (p, q)Find the solution the eigenmatrix of every layer of grating,, utilize to strengthen the diffractional field that transmission matrix method is found the solution the outgoing zone according to electromagnetic field tangential continuity boundary conditions.
Beneficial effect
Among the present invention at the direction (x of two quadratures; Y) on; Through choosing three grating layers at the make progress lowest common multiple in cycle of counterparty, carry out the Fourier series expansion, can analyze two orthogonal directions (x; Y) all different multilayer two-dimension mask grating diffration of last cycle, and diffraction that can as analysed basis base area crossed grating; The present invention is through adopting the situation that transmission matrix method is analyzed three layers of grating taper incident that strengthens, and the ability rapid solving obtains the diffractional field of biabsorption layer alternating phase-shift contact hole mask.
Description of drawings
Fig. 1 is a biabsorption layer alternating phase-shift contact hole mask synoptic diagram.
Fig. 2 is for finding the solution biabsorption layer alternating phase-shift contact hole mask diffraction process flow diagram.
Fig. 3 is matrix E of the present invention lSynoptic diagram.
Fig. 4 is matrix E of the present invention l' synoptic diagram.
Fig. 5 is a matrix A of the present invention lSynoptic diagram.
Fig. 6 is a matrix A of the present invention l' synoptic diagram.
Fig. 7 is a matrix K of the present invention xSynoptic diagram.
Fig. 8 is a matrix K of the present invention ySynoptic diagram.
Fig. 9 is the synoptic diagram of unit matrix I of the present invention.
Figure 10 is matrix Y of the present invention ISynoptic diagram.
Figure 11 is matrix Z of the present invention ISynoptic diagram.
Figure 12 is matrix Y of the present invention IISynoptic diagram.
Figure 13 is matrix Z of the present invention IISynoptic diagram.
Figure 14 is a matrix F of the present invention cSynoptic diagram.
Figure 15 is a matrix F of the present invention sSynoptic diagram.
Figure 16 is TE polarized light taper incident (θ=10 °; During biabsorption layer (CrO/Cr) the alternating phase-shift contact hole mask of
Figure BSA00000717055500061
λ=193nm); (0; 0), (0,2), (1,1), (2; 0) diffraction efficiency of the order of diffraction time is along with characteristic dimension (wafer yardstick, variation relation figure nm).
Embodiment
Below in conjunction with accompanying drawing the present invention is further elaborated.
Biabsorption layer alternating phase-shift contact hole mask synoptic diagram is as shown in Figure 1, below the mask that relates in the present embodiment is described.
The present invention is the z axle with the normal direction of mask plane (grating planar), according to the right-handed coordinate system principle, confirms x axle and y axle, set up coordinate system (x, y, z) as shown in Figure 1.
Biabsorption layer alternating phase-shift contact hole mask is divided into three layers along the z direction of principal axis, two-layer absorption layer and one deck phase shift layer; First absorption layer (the z 0<z<z 1) being generally CrO, thickness is d 1=z 1-z 0, the second absorption layer (z 1<z<z 2) being generally Cr, thickness is d 2=z 2-z 1, third phase moves layer, and its etching depth is d=λ/2 (n-1), to realize 180 ° phase shift.First absorption layer is periods lambda along the x axle 1xDistribute, dutycycle is f 1x, first absorption layer is periods lambda along the y axle 1yDistribute, dutycycle is f 1ySecond layer absorption layer is periods lambda along the x axle 2xDistribute, dutycycle is f 2x, second absorption layer is periods lambda along the y axle 2yDistribute, dutycycle is f 2yAnd the preceding two-layer cycle on x, y axle respectively is all identical with dutycycle, i.e. Λ 1x2x, f 1x=f 2x, Λ 1y2y, f 1y=f 2y, but f 1x≠ f 1y, f 2x≠ f 2yThe 3rd layer is dielectric, is periods lambda along the x axle 3xDistribute, dutycycle is f 3x, be periods lambda along the y axle 3yDistribute, dutycycle is f 3y, and Λ 3y=2 Λ 1y, Λ 3x=2 Λ 1xTop layer (L '=0), bottom (L '=4) are to be respectively incidence zone, outgoing district, and are infinite expanding along negative sense, the forward of z axle, and the refractive index of top layer is n I, the refractive index of bottom is n II
A branch of linearly polarized light incides diffraction takes place on the two-dimensional grating; Incident angle is θ, and position angle (plane of incidence and x axle clamp angle) is δ, and polarization angle (angle of incident electric field intensity and plane of incidence) is ψ; ψ=90 ° are when corresponding to the TE polarized light, and ψ=0 is ° corresponding to the TM polarized light.
As shown in Figure 2, the process flow diagram of biabsorption layer alternating phase-shift contact hole mask diffractional field computing method of the present invention; Concrete steps are:
The harmonic number that keeps on step 1, the setting x direction is L x, setting the harmonic number that keeps on the y direction is L yAbove-mentioned two harmonic numbers can be set as required, if hope fast speeds to be arranged finding the solution electric field energy, then can it be provided with lessly, if hope the electric field of being found the solution higher precision are arranged, then can its settings is bigger, and the while also can make L x=L y
Step 2, according to Bu Luokai (Floquet) condition, find the solution the (m, n) wave vector of the individual order of diffraction time along the tangential, the component of normal direction, wherein m is for getting time [D x, D x] between integer, n is for getting all over [D y, D y] between integer, L x=2D x+ 1, L y=2D y+ 1.
Wave vector is the component α of x, y axle along the tangential m, β nFor:
α m = α 0 - 2 πm / Λ x β n = β 0 - 2 πn / Λ y - - - ( 1 )
α 0=n Iksinθcosδ,β 0=n Iksinθsinδ (2)
Wherein k is an incident light wave wave vector in a vacuum, n IBe the refractive index of incidence zone, θ is the incident angle of light wave, and δ is the position angle of light wave, Λ xFor mask along the x direction lowest common multiple in three layers of grating cycle, because Λ 3y=2 Λ 1ySo, the Λ here y3y, Λ yFor mask along the y direction lowest common multiple in three layers of grating cycle, because Λ 3x=2 Λ 1xSo, the Λ here x3x
Wave vector is the component r of z axle along the normal direction of grating planar Mn, t MnFor:
r mn = [ ( n I k ) 2 - α m 2 - β n 2 ] 1 / 2 α m 2 + β n 2 ≤ ( n I k ) 2 - j [ α m 2 + β n 2 - ( n I k ) 2 ] 1 / 2 α m 2 + β n 2 > ( n I k ) 2 - - - ( 3 )
t mn = [ ( n II k ) 2 - α m 2 - β n 2 ] 1 / 2 α m 2 + β n 2 ≤ ( n II k ) 2 - j [ α m 2 + β n 2 - ( n II k ) 2 ] 2 α m 2 + β n 2 > ( n II k ) 2 - - - ( 4 )
Wherein subscript I representes the incidence zone, n IThe refractive index of expression incidence zone, subscript II representes outgoing district, n IIThe refractive index in expression outgoing district, j representes imaginary unit.
Step 3, Fourier Fourier series expansion is carried out in the specific inductive capacity and the elastivity of each layer two-dimensional grating.Because three layers of two-dimensional grating cycles on x, y direction are different, should choose three layers of grating during series expansion at the make progress lowest common multiple in cycle of counterparty.When doing the Fourier series expansion, selected unit area is shown in dotted line among Fig. 1 (a).
Specific inductive capacity can be expressed as as fourier expansion:
ϵ l ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ϵ l , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l = 1,2 ) - - - ( 5 )
ϵ l ′ ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ϵ l ′ , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l ′ = 3 ) - - - ( 6 )
ε wherein L, (p, q)Be l layer grating relative dielectric constant (p, q) individual Fourier component, ε L ', (p, q)Be l ' layer grating relative dielectric constant (p, q) individual Fourier component.
Elastivity can be expressed as as fourier expansion:
1 / ϵ l ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ξ l , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l = 1,2 ) - - - ( 7 )
1 / ϵ l ′ ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ξ l ′ , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l ′ = 3 ) - - - ( 8 )
ξ wherein L, (p, q)Be l layer grating relative dielectric constant (p, q) individual Fourier component reciprocal.ξ wherein L ', pqBe l ' layer grating relative dielectric constant (p, q) individual Fourier component reciprocal.
Step 4, according to the α that calculates in the step 2 m, β n, r (m, n), t (m, n)And the ε that calculates in the step 3 L, (p, q), ε L ', (p, q), ξ L, (p, q)And ξ L ', (p, q)Find the solution the eigenmatrix of every layer of grating,, utilize to strengthen the diffractional field that transmission matrix method is found the solution the outgoing zone according to electromagnetic field tangential continuity boundary conditions.
Step 401, find the solution the eigenmatrix of each two-dimensional grating layer;
The eigenmatrix of two-dimensional grating is:
M l=F lG l(l=1,2) (9)
M l′=F l′G l′(l′=3) (10)
Wherein
F l = K y A l K x I - K y A l K y K x A l K x - I - K x A l K y ( l = 1,2 ) - - - ( 11 )
G l = K x K y αA l - 1 + ( 1 - α ) E l - K y 2 K x 2 - αE l - ( 1 - α ) A l - 1 - K x K y ( l = 1,2 ) - - - ( 12 )
α = f 1 y Λ 1 y f 1 x Λ 1 x + f 1 y Λ 1 y - - - ( 13 )
F l ′ = K y E l ′ - 1 K x I - K y E l ′ - 1 K y K x E l ′ - 1 K x - I - K x E l ′ - 1 K y ( l ′ = 3 ) - - - ( 14 )
G l ′ = K x K y A l ′ - 1 - K y 2 K x 2 - A l ′ - 1 - K x K y ( l ′ = 3 ) - - - ( 15 )
E wherein l, E l', A l, A l', K x, K y, I is (L t* L t) the rank matrix, L t=L x* L y, l and l ' all represent the number of plies.E lIn element be ε L, (p, q), E l' in element be ε L ', (p, q), A lIn element be ξ L, (p, q), A l' in element be ξ L ', (p, q)
For example the present invention sets L x=3, L y=3, because
Figure BSA00000717055500096
Figure BSA00000717055500097
Then P=[2 ,-1,0,1,2], q=[2 ,-1,0,1,2]; For l layer grating, according to the ε of step 3 generation L, (p, q)For
Figure BSA00000717055500099
Individual, be respectively: ε L, (2 ,-2), ε L, (2 ,-1)... ε L, (2,2)
The E that each layer grating pair answered lAllocation rule identical, below ignore consideration to the number of plies, be (L to size t* L t) i.e. 9 * 9 matrix E lThe distribution rule of last element describes:
With E lBe divided into L y* L y(promptly 9) individual L x* L xThe minor matrix of (promptly 3 * 3), and with each minor matrix e (i, j)Be used as an element, wherein i=[1,2,3], j=[1,2,3], e (1,1)For coordinate equals the minor matrix of (1,1), e (1,2)For coordinate equals the minor matrix of (1,2), and and the like.Be directed to each minor matrix e (i, j)Comprise 9 element e ' in it (i, j), (i ', j '), wherein i '=[1,2,3], j '=[1,2,3], e ' (i, j), (1,1)Be minor matrix e (i, j)Internal coordinate equals the minor matrix element of (1,1), e ' (i, j), (1,2)Be minor matrix e (i, j)Internal coordinate equals the minor matrix of (1,2), and and the like.
Distribute rule to be: at minor matrix e (i, j)In, its (i ', j ') individual element e ' (i, j) (i ', j ')L, (i '-j ', i-j)
For example to minor matrix e (1,1)In (1,1) individual element e ' (1,1), (1,1), because i-j=0, so i '-j '=0 is e ' (1,1), (1,1)(just be equivalent to E lIn (1,1) individual element) equal ε L, (0,0)
For example to minor matrix e (1,1)In (1,2) individual element e ' (1,1), (1,2), because i-j=0, so i '-j '=-1 is e ' (1,1), (1,2)(just be equivalent to E lIn (1,2) individual element) equal ε L, (1,0)
For example to minor matrix e (2,1)In (1,2) individual element e ' (2,1), (1,2), because i-j=1, so i '-j '=-1 is e ' (2,1), (1,2)(just be equivalent to E lIn (4,2) individual element) equal ε L, (1,1)
For example to minor matrix e (2,1)In (3,3) individual element e ' (2,1), (3,3), because i-j=1, so i '-j '=0 is e ' (2,1), (3,3)(just be equivalent to E lIn (6,3) individual element) equal ε L, (0,1)
For example to minor matrix e (3,3)In (1,3) individual element e ' (3,3), (1,3),Because i-j=0, so i '-j '=-2 are e ' (3,3), (1,3)(just be equivalent to E lIn (7,9) individual element) equal ε L, (2,0)
For example to minor matrix e (3,3)In (3,3) individual element e ' (3,3), (3,3), because i-j=0, so i '-j '=0 is e ' (3,3), (3,3)Just be equivalent to E lIn (9,9) individual element equal ε L, (0,0)
According to the E that above-mentioned rule is obtained lAs shown in Figure 3.
E l', A lAnd A lThe distribution rule and the E of ' last element lIdentical, shown in Fig. 4-6.
K xBe diagonal matrix, its diagonal element is α m
For example the present invention sets L x=3, L y=3, because D x=(L x-1)/2, D then x=1, m=[1,0,1]; α according to the step 2 generation mBe 3, be respectively: α -1, α 0, α 1
Below be (L to size t* L t) i.e. 9 * 9 diagonal matrix K xThe distribution rule of diagonal element describes:
With K xBe divided into L y* L y(promptly 9) individual L x* L xThe minor matrix of (promptly 3 * 3), and with each minor matrix
Figure BSA00000717055500101
Be used as an element, wherein i=[1,2,3], j=[1,2,3],
Figure BSA00000717055500102
For coordinate equals the minor matrix of (1,1),
Figure BSA00000717055500103
For coordinate equals the minor matrix of (1,2), and and the like.Be directed to each minor matrix and comprise wherein i '=[1 of 9 elements
Figure BSA00000717055500105
in it; 2; 3], j '=[1; 2; 3];
Figure BSA00000717055500106
equals (1 for minor matrix internal coordinate; 1) minor matrix element; equals (1 for minor matrix
Figure BSA00000717055500109
internal coordinate; 2) minor matrix, and and the like.
Because K xBe diagonal matrix, then only exist
Figure BSA00000717055500111
And
Figure BSA00000717055500112
Diagonal position on have element value, the element value of all the other minor matrixs is all 0.
Distribute rule to be: in minor matrix
Figure BSA00000717055500113
(being j=i); Its (i '; J ') (being j '=i ') individual element
Figure BSA00000717055500114
is because this distribution rule and (i; J) irrelevant, so
Figure BSA00000717055500115
and
Figure BSA00000717055500116
is identical.
For example to minor matrix
Figure BSA00000717055500117
In (1,1) individual element
Figure BSA00000717055500118
Since i '=1, D x=1, i '-(D x+ 1)=-1, so
Figure BSA00000717055500119
(just be equivalent to K xIn (1,1) individual element) equal a -1
For example to minor matrix
Figure BSA000007170555001110
In (2,2) individual element
Figure BSA000007170555001111
Since i '=2, D x=1, i '-(D x+ 1)=0, so
Figure BSA000007170555001112
(just be equivalent to K xIn (2,2) individual element) equal a 0
According to the K that above-mentioned rule is obtained xAs shown in Figure 7.
K yBe diagonal matrix, its diagonal element is β n
For example the present invention sets L x=3, L y=3, because D y=(L y-1)/2, D then y=1, n=[1,0,1]; , according to the β of step 2 generation nBe 3, be respectively: β -1, β 0, β 1
Below be (L to size t* L t) i.e. 9 * 9 diagonal matrix K yThe distribution rule of diagonal element describes:
With K yBe divided into L y* L y(promptly 9) individual L x* L xThe minor matrix of (promptly 3 * 3), and with each minor matrix
Figure BSA000007170555001113
Be used as an element, wherein i=[1,2,3], j=[1,2,3],
Figure BSA000007170555001114
For coordinate equals the minor matrix of (1,1),
Figure BSA000007170555001115
For coordinate equals the minor matrix of (1,2), and and the like.Be directed to each minor matrix
Figure BSA000007170555001116
and comprise wherein i '=[1 of 9 elements
Figure BSA000007170555001117
in it; 2; 3], j '=[1; 2; 3];
Figure BSA000007170555001118
equals (1 for minor matrix internal coordinate; 1) minor matrix element;
Figure BSA000007170555001120
equals (1 for minor matrix
Figure BSA000007170555001121
internal coordinate; 2) minor matrix, and and the like.
Distribute rule to be: in minor matrix
Figure BSA000007170555001122
(being j=i); Its diagonal element
Figure BSA000007170555001123
since this distribute rule with (i '; J ') irrelevant, so the diagonal element of
Figure BSA000007170555001124
is identical in each minor matrix.
For example to minor matrix
Figure BSA000007170555001125
In (1,1) individual element
Figure BSA000007170555001126
Since i=1, D y=1, i-(D y+ 1)=-1, so
Figure BSA000007170555001127
(just be equivalent to K yIn (1,1) individual element) equal β -1
For example to minor matrix
Figure BSA00000717055500121
In (2,2) individual element
Figure BSA00000717055500122
Since i=2, D y=1, i-(D y+ 1)=0, so
Figure BSA00000717055500123
(just be equivalent to K yIn (5,5) individual element) equal β 0
According to the K that above-mentioned rule is obtained yAs shown in Figure 8.
I is a unit matrix, and is as shown in Figure 9.
Step 402, find the solution the matrix Y of incidence zone I, Z I, and transmission area matrix Y II, Z II
Y wherein I, Z IBe diagonal matrix, diagonal element is respectively
Figure BSA00000717055500124
Y II, Z IIAlso be diagonal matrix, diagonal element is respectively
Figure BSA00000717055500125
Matrix Y I, Z IY IIAnd Z IIThe distribution rule of last element is identical, below chooses Y IAs analytic target,, therefore ignore k the distribution rule of element on it is elaborated because k is constant.
For example the present invention sets L x=3, L y=3, because D x=(L x-1)/2, D then x=1, m=[1,0,1] is because D y=(L y-1)/2, D then y=1, n=[1,0,1] is according to the r of step 2 generation (m, n)Be 3 * 3=9, be respectively: r (1 ,-1), r (1,0), r (1,1)... R (1 ,-1), r (1,0), r (1,1)
Y IBe (L t* L t) i.e. 9 * 9 diagonal matrix, below to Y IThe distribution rule of diagonal element describes:
With Y IBe divided into L y* L y(promptly 9) individual L x* L xThe minor matrix of (promptly 3 * 3), and with each minor matrix y (i, j)Be used as an element, wherein i=[1,2,3], j=[1,2,3], y (1,1)For coordinate equals the minor matrix of (1,1), y (1,2)For coordinate equals the minor matrix of (1,2), and and the like.Be directed to each minor matrix y (i, j)Comprise 9 element y ' in it (i, j), (i ', j '), wherein i '=[1,2,3], j '=[1,2,3], y ' (i, j), (1,1)Be minor matrix y (i, j)Internal coordinate equals the minor matrix element of (1,1), y ' (i, j), (1,2)Be minor matrix y (i, j)Internal coordinate equals the minor matrix of (1,2), and and the like.
Because Y IBe diagonal matrix, then only at y (1,1), y (2,2)And y (3,3)Diagonal position on have element value, the element value of all the other minor matrixs is all 0.
Distribute rule to be: at minor matrix y (i, j)In (being j=i), its (i ', j ') (being j '=i ') individual element y ' (i, j), (i ', j ')=r ((i '-D x-1), (i-D y-1)).
For example to minor matrix y (1,1)In (1,1) individual element y ' (1,1), (1,1), since i '=1, D x=1, i=1, D y=1, i '-(D x+ 1)=-1, i-(D y+ 1)=-1, so y ' (1,1), (1,1)(just be equivalent to Y IIn (1,1) individual element) equal r (1 ,-1)
For example to minor matrix y (1,1)In (2,2) individual element y ' (1,1), (2,2), since i '=2, D x=1, i=1, D y=1, i '-(D x+ 1)=0, i-(D y+ 1)=-1, so y ' (1,1), (2,2)(just be equivalent to Y IIn (2,2) individual element) equal r (0 ,-1)
For example to minor matrix y (2,2)In (2,2) individual element y ' (2,2), (2,2), since i '=2, D x=1, i=2, D y=1, i '-(D x+ 1)=0, i-(D y+ 1)=0, so y ' (2,2), (2,2)(just be equivalent to Y IIn (5,5) individual element) equal r (0,0)
According to the Y that above-mentioned rule is obtained I, Z IY IIAnd Z IIShown in Figure 10-13.
Step 403, utilize the continuous boundary condition in electromagnetic field tangential, obtain the expression formula between incidence zone and the outgoing district electromagnetic field;
sin ψ δ m 0 δ n 0 j sin ψ n I cos θ δ m 0 δ n 0 - j n I cos ψ δ m 0 δ n 0 cos ψ cos θ δ m 0 δ n 0 + I 0 - j Y I 0 0 I 0 - j Z I R =
Π L = 1 N = 3 V L , 1 V L , 1 X L W L , 1 - W L , 1 X L w L , 2 - W L , 2 X L V L , 2 V L , 2 X L V L , 1 X L V L , 1 W L , 1 X L - W L , 1 W L , 2 X L - W L , 2 V L , 2 X L V L , 2 - 1 I 0 j Y II 0 0 I 0 j Z II T (L=1,2,3) (16)
Wherein
V L,1=F cW L,y-F sW L,x?V L,2=F cW L,x+F sW L,y
W L,1=F cV L,x+F sV L,y?W L,2=F cV L,y-F sV L,x
(17)
W L,x=[w L,x] W L,y=[w L,y]
V L,x=[v L,x] V L,y=[v L,y]
L representes L layer two-dimensional grating;
W L = w L , y w L , x Be L layer two-dimensional grating eigenmatrix M LThe eigenvector matrix;
q L, lBe L layer two-dimensional grating eigenmatrix M LThe eigenvalue matrix in (l, l) the positive square root l=of individual element [1,2,3 ..., 2L t];
X LRepresent the diagonal matrix in the L layer two-dimensional grating, (l l) is exp (kq to diagonal element L, ld L);
d LThe thickness of representing L layer two-dimensional grating;
V L = v L , y v L , x = F L - 1 Q L W L ;
Q LBe that (l l) is q to diagonal element L, lDiagonal matrix;
F cBe that diagonal element does
Figure BSA00000717055500143
Diagonal matrix;
F sBe that diagonal element does
Figure BSA00000717055500144
Diagonal matrix;
F cAnd F sThe allocation rule and the Y of diagonal element IIdentical, shown in Figure 14-15.
δ M0Be L x* 1 matrix, wherein when m=0, δ (m+D x+ 1,1)=1; When m ≠ 0, δ (m+D x+ 1,1)=0;
δ ' N0Be L y* 1 matrix, wherein when n=0, δ ' (n+D y+ 1,1)=1; When n ≠ 0, δ ' (n+D y+ 1,1)=0;
R is an intermediate variable;
T is the inferior amplitude of each order of diffraction of transmitted field to be found the solution;
Step 404, utilization strengthen transmission matrix method, find the solution the inferior amplitude T of each order of diffraction of transmitted field; Wherein T is 2L t* 1 matrix, each element among the T is the form of plural a+bj, wherein the amplitude of diffractional field does
Figure BSA00000717055500146
Promptly obtain the diffractional field in polarized light outgoing district.
Utilize to strengthen transmission matrix method, the expression formula between incidence zone and the outgoing district electromagnetic field is:
sin ψ δ m 0 δ n 0 j sin ψ n I cos θ δ m 0 δ n 0 - j n I cos ψ δ m 0 δ n 0 cos ψ cos θ δ m 0 δ n 0 + I 0 - j Y I 0 0 I 0 - j Z I R = f 1 g 1 T 1 - - - ( 18 )
Wherein
f L g L T L = V L , 1 V L , 1 X L W L , 1 - W L , 1 X L W L , 2 - W L , 2 X L V L , 2 V L , 2 X L I b L a L - 1 X L T L - - - ( 19 )
a L b L V L , 1 V L , 1 W L , 1 - W L , 1 W L , 2 - W L , 2 V L , 2 V L , 2 - 1 f L + 1 g L + 1 - - - ( 20 )
f 4 g 4 = I 0 j Y II 0 0 I 0 jZ II - - - ( 21 )
T = a 3 - 1 X 3 a 2 - 1 X 2 a 1 - 1 X 1 T 1 - - - ( 22 )
Further, the present invention also can solve the inferior diffraction efficiency of each order of diffraction;
The (m, n) the inferior diffraction efficiency of level is:
η ( m , n ) = | T s , ( m , n ) | 2 Re ( t ( m , n ) kn I cos θ ) + | T p , ( m , n ) | 2 Re ( t ( m , n ) / n II 2 kn I cos θ ) - - - ( 23 )
T wherein sBe the matrix that the first half element among the T is formed, T pMatrix for the composition of the latter half element among the T.T S, (m, n)Be T sIn ((m+D x+ 1)+(n+D y) L x) individual element, T P, (m, n)Be T pIn ((m+D x+ 1)+(n+D y) L x) individual element.
Further, the present invention can also find the solution each order of diffraction time degree of polarization (Degree of Polarization, DoP)
DoP ( m , n ) = η ( m , n ) TE - η ( m , n ) TM η ( m , n ) TE + η ( m , n ) TM · 100 % - - - ( 24 )
Wherein when incident light is the TE polarized light, with η (m, n)Be defined as
Figure BSA00000717055500158
When incident light is the TM polarized light, with η (m, n)Be defined as
Figure BSA00000717055500161
DoP representes the similar TE polaroid of mask for just, and DoP representes the similar TM polaroid of mask for negative.
Invention instance one:
Here calculated in biabsorption layer (CrO/Cr) the alternating phase-shift contact hole mask, (θ=10 ° are during
Figure BSA00000717055500162
λ=193nm) in TE taper incident; During different mask linewidths (wafer yardstick); (0,0), (0,2), (1; 1), the inferior diffraction efficiency of (2,0) level.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 the dutycycle of 55nm. mask on the x axle be 0.5, the dutycycle on the y axle is 0.6.
Figure 16 is TE polarized light taper incident (θ=10 °; During biabsorption layer (CrO/Cr) the alternating phase-shift contact hole mask of
Figure BSA00000717055500163
λ=193nm); (0; 0), (0,2), (1,1), (2; 0) diffraction efficiency of the order of diffraction time is along with characteristic dimension (wafer yardstick, variation relation figure nm).(a) (0,0) level diffraction of light efficient is with the graph of a relation of line width variation, (b) (0; 2) level diffraction of light efficient is with the graph of a relation of line width variation, and (c) (1,1) level diffraction of light efficient is with the graph of a relation of line width variation; (d) (2,0) level diffraction of light efficient is with the graph of a relation of line width variation.
In sum, more than being merely preferred embodiment of the present invention, is not to be used to limit protection scope of the present invention.All within spirit of the present invention and principle, any modification of being done, be equal to replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (1)

1. the computing method of a biabsorption layer alternating phase-shift contact hole mask diffractional field is characterized in that concrete steps are:
The harmonic number that keeps on step 1, the setting x direction is L x, setting the harmonic number that keeps on the y direction is L y
Step 2, according to Bu Luokai (Floquet) condition, find the solution the (m, n) wave vector of the individual order of diffraction time along the tangential, the component of normal direction, wherein m is for getting time [D x, D x] between integer, n is for getting all over [D y, D y] between integer, L x=2D x+ 1, L y=2D y+ 1;
Wave vector is the component α of x, y axle along the tangential m, β nFor:
α m = α 0 - 2 πm / Λ x β n = β 0 - 2 πn / Λ y - - - ( 1 )
α 0=n Iksinθcosδ,β 0=n Iksinθsinδ (2)
Wherein k is an incident light wave wave vector in a vacuum, n IBe the refractive index of incidence zone, θ is the incident angle of light wave, and δ is the position angle of light wave, Λ xFor mask along the x direction lowest common multiple in three layers of grating cycle, Λ yFor mask along the y direction lowest common multiple in three layers of grating cycle;
Wave vector is the component r of z axle along the normal direction of grating planar Mn, t MnFor:
r mn = [ ( n I k ) 2 - α m 2 - β n 2 ] 1 / 2 α m 2 + β n 2 ≤ ( n I k ) 2 - j [ α m 2 + β n 2 - ( n I k ) 2 ] 1 / 2 α m 2 + β n 2 > ( n I k ) 2 - - - ( 3 )
t mn = [ ( n II k ) 2 - α m 2 - β n 2 ] 1 / 2 α m 2 + β n 2 ≤ ( n II k ) 2 - j [ α m 2 + β n 2 - ( n II k ) 2 ] 2 α m 2 + β n 2 > ( n II k ) 2 - - - ( 4 )
Wherein subscript I representes the incidence zone, n IThe refractive index of expression incidence zone, subscript II representes outgoing district, n IIThe refractive index in expression outgoing district, j representes imaginary unit;
Step 3, Fourier Fourier series expansion is carried out in the specific inductive capacity and the elastivity of each layer two-dimensional grating;
Specific inductive capacity can be expressed as as fourier expansion:
ϵ l ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ϵ l , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l = 1,2 ) - - - ( 5 )
ϵ l ′ ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ϵ l ′ , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l ′ = 3 ) - - - ( 6 )
ε wherein L, (p, q)Be l layer grating relative dielectric constant (p, q) individual Fourier component, ε L ' (p, q)Be l ' layer grating relative dielectric constant (p, q) individual Fourier component,
Figure FSA00000717055400023
Elastivity is expressed as as fourier expansion:
1 / ϵ l ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ξ l , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l = 1,2 ) - - - ( 7 )
1 / ϵ l ′ ( x , y ) = Σ p = - ∂ x ∂ x Σ q = - ∂ y ∂ y ξ l ′ , ( p , q ) exp [ j 2 π ( px / Λ x + qy / Λ y ) ] ( l ′ = 3 ) - - - ( 8 )
ξ wherein L, (p, q)Be l layer grating relative dielectric constant (p, q) individual Fourier component, ξ reciprocal L ', pqBe l ' layer grating relative dielectric constant (p, q) individual Fourier component reciprocal;
Step 4, according to the α that calculates in the step 2 m, β n, r (m, n), t (m, n)And the ε that calculates in the step 3 L, (p, q), ε L ', (p, q), ξ L, (p, q)And ξ L ', (p, q)Find the solution the eigenmatrix of every layer of grating,, utilize to strengthen the diffractional field that transmission matrix method is found the solution the outgoing zone according to electromagnetic field tangential continuity boundary conditions.
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