Application of the discrete dipole approximation for dipoles embedded in film

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1728 J. Opt. Soc. Am. A/Vol. 25, No. 7 /July 2008 Bae et al.

Application of the discrete dipole approximationfor dipoles embedded in film

Euiwon Bae,1,* Haiping Zhang,2 and E. Daniel Hirleman1

1School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47906, USA2KLA-Tencor Corporation, One Technology Drive, Milpitas, California 95035, USA

*Corresponding author: [email protected]

Received February 22, 2008; revised April 25, 2008; accepted May 5, 2008;posted May 13, 2008 (Doc. ID 92998); published June 25, 2008

An implementation of the discrete dipole approximation for dipoles embedded in film on substrate is derived.It is capable of predicting the scattering response from various types of subsurface features such as trenchesand contact-vias. An arbitrarily shaped subsurface feature is modeled with dipoles inside the film material ontop of a substrate. Relative polarizability, direct interactions, and reflection interactions are derived andapplied to construct a system of equations that are solved with an iterative method. The far-field scatteringresponse is computed from the dipole moment solution of the system matrix with the help of the reciprocitytheorem. The validity of the proposed method is compared with that of other existing theories, and the effect offilm structure on far-field scattering is shown with high-aspect-ratio cylindrical contact-via models.© 2008 Optical Society of America

OCIS codes: 290.0290, 120.1880, 240.3695.

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. INTRODUCTIONnderstanding the light scattering from features on

tructures, both on and under the surface, is of greatnterest in a variety of industrial and research areas. Inhe semiconductor field, laser in-line scanning is widelyccepted as an inspection tool to provide a high yield inanufacturing products [1]. As the density of the memo-

ies and the circuits on the chips increases, the physicalize of the structures decrease. A modeling capability isritical to designing the next-generation inspectionystem using laser light scattering.

One of the many steps in integrated circuit fabricationnvolves a deposition of film onto a silicon substrate. Dur-ng this process, contamination can occur as a particle onhe surface or as defects inside the film, such as pits oroids. Another feature that is of great interest is aontact-via, which is a cylindrical or rectangular featurehat provides electrical connections in integrated circuits.ast and accurate detection of all these features presentsn inspection challenge for the semiconductor manufac-uring industry.

Extensive research has been performed in the geophys-cal community on the detection of subsurface objects.iegel [2] provided for scattering from a submergedntenna modeled by a dipole where the substrate was as-umed to be a half-space without a film. Kristensson [3]pplied expansion coefficients to solve scattering matri-es, while Hill [4] used the Rayleigh–Gauss approxima-ion to model the scattering response from objects of lowontrast, but the latter method had limitations on theypes of material to which it could be applied. Ermutlut al. [5] studied scattering from an object modeled as a

1084-7529/08/071728-9/$15.00 © 2

ingle electric dipole below the surface, and Eremin et al.6] and Orlov and Eremin [7] presented a numerical

odeling of detection of defects in silicon wafers usingxisymmetric models or applying the discrete sourceethod (DSM). Recently, Paulus and Martin [8,9] derivedGreen’s tensor method for stratified media that used a

iscrete dipole approximation (DDA) and showed variouspplications to the near-field computation.In this paper, we apply the DDA, which is also called

he coupled-dipole method, to an object embedded in alm on top of a substrate. We call our process DDEFILM.method of modeling the light scattering from a contami-

ant on a surface has been proposed by Schmehl et al.10] and Nebeker [11] that provided excellent agreementn predicting the scattering response. To extend the pre-ious results, Zhang [12] expanded the DDA to surfaceodeling of subsurface features, along with introduction

f the reciprocity theorem to efficiently compute thear-field scattering signature.

In DDEFILM, a small volume element defined as aipole is used to model an arbitrarily shaped featurembedded in film. For example, a cylindrical pit modeledith dipoles surrounded by a film material is shown inig. 1. The incident field induces a dipole moment thatepends on the material characteristics, shape, incidentolarization, and incidence angle. The incident light isombined with the transmitted incident beam and theeflected incident beam such that multiple interactions inhe film as used in thin-film measurement (ellipsometry)ay be applied. The total dipole moment at each dipole

ocation is deduced from the polarizability, direct interac-ion, and reflection interaction inside the film. In particu-

008 Optical Society of America

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Bae et al. Vol. 25, No. 7 /July 2008 /J. Opt. Soc. Am. A 1729

ar, the reflection interaction involves reflection from bothhe top interface (air–film) and the bottom interfacefilm–substrate).

. METHOD. System Equationhen an electric field excites a dipole, a dipole moment is

nduced that is related to the total electric field as

Pi = �iEtot,i, �1�

here Pi is the dipole moment at the ith dipole location,i is the total incident field at the ith dipole location, andi is the dipole polarizability from the lattice dispersionelation (LDR) [9]. Since the total field Etot consists of thencident beam from the source Einc, the electric field fromther dipoles (direct interaction Edir), and the electriceld from other dipoles reflected from the interfacereflection interaction Eref), Eq. (1) can be modified as

1

�iPi − Edir,i − Eref,i = Einc,i. �2�

When we divide the object into an accumulation of Nubvolumes located on a cubical lattice, we can expressq. (2) with the dyadic Green’s function Gij and theommerfeld integral identity Sij as

1

�iPi −

k22

�2�j�i

N

Gij · Pj −k2

2

�2�j�i

N

Sij · Pj = Einc,i. �3�

The global matrix equation is constructed combiningll three interactions into a 3N�3N matrix as

�D + A + R�P = E, �4�

here D is the 3N diagonal matrix from polarizability, As the 3N�3N matrix from the second term of Eq. (3), Rs the 3N�3N matrix from the third term of Eq. (3), P ishe 3N�1 dipole moment matrix, and E is the 3N�1

ig. 1. Dipole configuration and coordinate definition for a sub-urface feature embedded in the film on a substrate. � is thengle an incident beam makes with the z axis, while � is itsngle with the plane of incidence (xz plane). The incident field is0 and the wave vector of the scattered field is denoted ksca.

otal incident beam matrix. The total matrix isonstructed via summation of submatrices as

D = diag� 1

�1x,

1

�1y,

1

�1iz, . . . ,

1

�Nx,

1

�Ny,

1

�Niz� , �5�

A = �A1N ¯ A1N

] ]

AN1 ¯ ANN�, R = �

R1N ¯ R1N

] ]

RN1 ¯ RNN� , �6�

P = �P1 ¯ PNT, E = �E1 ¯ ENT. �7�

The DDA method applied to dipoles embedded in a filmequires modifying the right-hand side of Eq. (2). Now thencident beam not only directly reaches the dipole but alsoeflects from the film–substrate and air–film boundariess shown in Fig. 2. We define the parameters for air, film,nd substrate, respectively, as �1, �1; �2, �2; and �3, �3.he total field that excites the ith dipole in the film has

he form [12]

ATE = A2TE�exp�− ik2zzi� + R23

TE exp�2ik2zdf + ik2zzi�,

ATM = A2TM�exp�− ik2zzi� + R23

TM exp�2ik2zdf + ik2zzi�,

�8�

here k2z is the z component of wave vector k2 in the film,23 is the reflection coefficient for the film–substrate

nterface, and df is the thickness of the film. Coefficient A2s the series sum of the multiply reflected beam inside thelm, which is represented as

A2TE =

T12TE

1 − R21TER23

TE exp�2ik2zdf,

A2TM =

T12TM

1 − R21TMR23

TM exp�2ik2zdf, �9�

ig. 2. Different types of interaction between ith and jthipoles. The incident beam impinges the dipole not only throughransmission, but also via multiple reflections from air–film andlm–substrate interfaces.

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here T12 is the transmission coefficient for the air–filmnterface, and R21 is the reflection coefficient for the film–ir interface. The generalized reflection and transmissionoefficients for the TM and TE waves are expressed as

RpqTE =

�qkpz − �pkqz

�qkpz + �pkqz, Rpq

TM =�qkpz − �pkqz

�qkpz + �pkqz, �10�

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TpqTE =

2�qkpz

�qkpz + �pkqz, Tpq

TM =2�qkpz

�qkpz + �pkqz, �11�

here the beam is propagating from medium p to medium. Therefore, the reflected incident field on the ith dipolembedded in the film for TE and TM waves is formulateds

Eref,i = ��cos2��A2

TM − sin2��A2TE�E0,x + �A2

TM − A2TE�cos��sin��E0,y

�cos2��A2TE − sin2��A2

TM�E0,y + �A2TM − A2

TE�cos��sin��E0,x

A2TME0,z

� , �12�

here � is the azimuthal angle with the plane of inci-ence as shown in Fig. 1, and E0x, E0y, and E0z are the x,, and z components of the incident beam E0. Thus the to-al incident field at the ith dipole is formulated when weultiply the x and y components of the incident wave

ectors as

Einc,i = Eref,i exp�ik2xxi + ik2yyi. �13�

The dipole polarizability matrix D is formulated via theDR method as a function of material property andarameters such as incidence angle, polarizability,nd wavelength:

�i =�0i

1 + �0i

d3 ��b1 + mrel,i

2b2 + mrel,i

2b3S�k2d�

2−

2

3i�k2d

3�

,

�14�

�0i = 3�0�mrel,i2 − 1

mrel,i2 + 2��Vi. �15�

he parameter d is the dipole spacing, mi is the refractivendex, and k2 is the wave vector inside the film. The defi-itions of b1–b3 and the S parameters are shown in [13].n the case of a dipole embedded in film, we apply aelative refractive index mrel, which is the value of theefractive index of the particle relative to that of the film.

The direct interaction of dipoles i and j can be modeledsing the dyadic Green’s function [13,14]. If we consider a

attice of N dipoles, the direct field interaction from eachipole can be modeled with the relative permittivity of thelm medium, �2, and the corresponding wavenumber k2.etailed derivation of the elements of the submatrix isrovided by Schmehl [8].To compute the reflected dipole interaction embedded

n the film, the reflected electric field is decomposed intoertical electric dipole (VED) and horizontal electric di-ole (HED) radiation. Assuming cylindrical coordinates=�x2+y2, �=arctan�y /x�, and z and applying theommerfeld identity [14], the reflected field interaction is
odeled as
Rij = CijR�

cos2��E�H − sin2��E�

H sin��cos��E�H + E�

H� cos��EzV

sin��cos��E�H + E�

H� sin2��E�H − cos2��E�

H sin��EzV

cos��E�H sin��E�

H EzV � , �16�

here CijR= 1 / 4�2 . This method is derived by Kong [15]

nd Tang [16] for stratified media and the expressions for

�H, E�

H, EzH, E�

V, and EzV are derived as

E�V = i�

0

�− A0 exp�− ikzz� + B0 exp�ikzz�k�2J1�k��dk�,

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EzV = i�

0

�A0 exp�− ikzz� + B0 exp�ikzz�k�

3

kzJ0�k��dk�,

�18�

E�H = i�

0

�− A0 exp�− ikzz� + B0 exp�ikzz�k�kz

��J0�k�� −1

k��J1�k��dk� − i�

0

�C0 exp�− ikzz�

+ D0 exp�ikzz�k0

2

k��J1�k��dk�, �19�

E�H = − i�

0

�− A0 exp�− ikzz� + B0 exp�ikzz�kz

�J1�k��dk�

+ i�0

�C0 exp�− ikzz� + D0 exp�ikzz�k0

2k�

kz

��J0�k�� −1

k��J1�k��dk�, �20�

EzH =�

0

�A0 exp�− ikzz� + B0 exp�ikzz�k�2J1�k��dk�, �21�

here k� is the wave vector; J0 and J1 are Bessel functionf zeroth and first order; and A0, B0, C0, and D0 areeneralized reflection coefficients. Since there are nolosed-form solutions to these Sommerfeld integralsEqs. (17)–(21)] they must be calculated numerically.

To calculate these integral expressions correctly, singu-arities such as branch points and poles of the integrandshould be watched carefully. However, their existing do-ain can be ascertained on the complex plane [17,18].hus we can easily avoid handling the singularities by de-

orming the integration path from the real axis to someppropriate integration paths. Due to the oscillations ofhe integrands of these integrals, the variable-interval-idth Romberg method [19] is chosen to implement theumerical integration efficiently in DDEFILM.After constructing the matrices as Eq. (4), we perform a

D fast Fourier transform FFT to the direct and reflectionnteraction matrices to reduce the required computationrom O�N2� to O�N�Nz

� log�N /Nz�. The previous case re-uired storage of the six elements of the both submatrixij and Rij, while the dipole-embedded-in-film case re-uires storing nine elements of Rij due to the nonsymme-ry of their submatrices. Finally, the solution is providedy the quasi-minimal residue (QMR), technique whichhowed the best performance among the other iterativelgorithms [20].

. Far-Field Computationnce the dipole moment of all the locations is computedy Eq. (4), the far-field radiation pattern can be computedy summation of the direct fields radiated from eachipole. If the Sommerfeld integrals, however, are usedithout any assumption at a point very far from the

adiating dipole, computation is very intensive. Variouspplications in antenna design had applied the reciprocityheorem [21–23] to simplify the computation of the fareld radiation pattern. When an incident field excites theipole and creates the dipole moment P, inside a body, thecattered far-field can be obtained by testing these dipoleoments with all possible far-field waves as shown inig. 3.When two current density sources I1 and I2 exist in

pace, the following integral equation satisfies theeciprocity theorem:

�V1

E1 · I2dv =�V2

E2 · I1dv, �22�

here I2 denotes a current density source and E2 is theeld radiated by I2. If we choose a unit volume and uniturrent density for test source I2, the scattered field E2an be obtained by the reciprocity theorem as

Esca�r� = − i��0

e−ik0r

4r �V2

E2 · I1dv. �23�

ere r denotes the distance from dipole to observationoint, k0 is the wave vector in vacuum, � is the angularrequency, and �0 is the vacuum permeability. Since cur-ent density is equal to −i�P for dipole radiation, and byiscretizing the volume integral V2 with dipoles, Eq. (23)s formulated as

Esca�ri� =k2

2

�0

e−ik2r

4r �j=1

N

��EjTM · Pj�e1 + �Ej

TE · Pj�e2, �24�

here indices i and j are used for the ith observation pointnd jth dipole center location, respectively. Ej

TM and EjTE

re the electric fields at dipole j due to the TM and TEncident plane wave with wave vector k=−ksca.

The scattered irradiance is computed by multiplyinghe complex conjugate of the scattered field. We define aifferential scattering cross section (DSC) as power dissi-ated per unit solid angle, and DSC is formulated as [24]

ig. 3. Formulation of the reciprocity theorem. When I2 is theurrent density source and E2 is the field radiated by I2, the scat-ered field E2 can be obtained from the reciprocity theorem whenunit current source I is assumed.
2

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dCsca

d�= lim

�→0�Csca

��

IscaA

Iinc�A/r2�=

r2Isca

Iinc, �25�

here Isca is the scattered irradiance, A is the detectionrea, r is the distance from the feature to the observationoint, Iinc is the incident irradiance, and � is the detec-ion solid angle. The total scattering cross section (TSC) isefined as the integral of the DSC over the hemisphericalistribution.

. NUMERICAL RESULTS. Comparison with Other Theorieso validate the proposed model, we have compared thear-field scattering from two different methods. Hill [4]rovided a theoretical development of scattering from auried object that has low contrast compared to the sur-oundings. He formulated the scattering from an object byonstructing a plane-wave scattering matrix and providedome analytical solutions for the simple shapes such asectangle, cylinder, and sphere. The refractive index ofhe scatterer is modified as the refractive index relativehat of the subsurface material and the Born approxima-ion is applied that is valid under the following condi-ions:

�n − 1� � 1 and a�k�n − 1�� 1, �26�

here n is the relative refractive index, a is the maximuminear dimension of the scatterer, and k is the wavenum-er.To compare our result with that from the Hill method, a

ipole model is constructed as shown in Fig. 4(a). Threeifferent shapes—cylinder (height and radius 0.1084�),ube (each side 0.2�), and sphere (radius 0.1241�)—areodeled with dipoles assuming a wavelength of

.6328 �m with TE �s� polarization. The depth from sur-ace is set to d/�=0.2, and normal incidence angle is as-umed. Since DDEFILM is formulated with the capabilityf filmed structure, we have set the refractive index of thelm equal to that the substrate such that an infinite half-pace is assumed. The refractive index of the scatterers iset to (1.703, 0.014) and that of the surrounding is1.7323, 0.0289); these satisfy the Born approximationondition of Eq. (26).

Comparison of far-field scattering patterns is shown inig. 4(b). The DSC output of Eq. (25) from DDEFILM isonverted to the absolute value of the scattered field di-ided by the wavelength for direct comparison. Solid,ashed, and dotted curves represent sphere, rectangle,nd cylinder, respectively. Heavier curves are results fromDEFILM, while the lighter curves are from Hill’sethod. Far-field scattering of the sphere at normal inci-

ence showed the maximum difference, about 6%, whilehe cylinder showed the minimum difference.

Another comparison is attempted with the DSM fromremin [6]. DSM calculates the approximate solution

rom a finite linear combination of the fields of a discreteource. The DSM method models the discrete source fromipoles and multipoles, while the modeling capability isurrently limited to axially symmetric shapes. For com-arison, we construct a dipole model of a conical pit with

80 nm base diameter and 108° vertex angle as shown inig. 5(a). The incidence angle is set to 70° with s polariza-

ion, and a silicon substrate with refractive index (4.37,.08) is assumed. Since the Eremin case assumed a coni-al pit in the infinite half space, we set the refractive in-ex of the film the same as that of the substrate. Directomparison of the DSC is shown in Fig. 5(b) for both mod-ls. The incident beam impinges the conical pit at +70°,hile the forward scattering peak is shown around −70°.he maximum deviation is shown to be approximately0%.

. Numerical Computation of Film Structureo prove the effectiveness of the DDEFILM, we have mod-led and computed the scattering response from a typicaltructure embedded in the film. A contact-via is a common

ig. 4. Comparison of DDEFILM with Hill’s method [4]. (a)ipole model of three different shapes, cylindrical, rectangular,nd spherical. (b) Far-field scattering of the three shapes. Curves(solid), 2 (dashed), and 3 (dotted) denote sphere, rectangle, and

ylinder, respectively. Heavy curves represent the result ofDEFILM, while light curves result from Hill’s method. Theepth from the surface was set to d /�=0.2.

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tructure applied in integrated circuits to provide electri-al channels for multilayer circuits. During the fabrica-ion process, an underetched via could cause malfunctionnd thus decrease manufacturing yields. In this paper, weodeled a high-aspect-radio (HAR) cylindrical via witheight–diameter ratio of 8:1 in a SiO2 film on a Si surface.igure 6(a) shows the dipole model of a cylindrical via of80 nm diameter and 1440 nm height. Total number ofipoles is 3380 with dipole spacing of 0.022 �m, which ispproximately � /15 and which is commensurate withipole spacing rules from previous research [11,25]. Thencidence angle is set to 70° and s polarization is as-umed. The refractive indices of SiO2 and Si are set to1.46, 0.0) and (4.3, 0.08), respectively. Computation iserformed in a PC with a 1.7 GHz CPU and 2 gigabytes ofAM, which takes 85 s in the fully etched case and 163 s

or the d=1.22D case with target residue of 0.001.

ig. 5. Comparison of DDEFILM with DSM. (a) Dipole model ofconical pit on Si as rendered by Eremin et al. [6]. Diameter of

he top is 180 nm and the vertex angle is 108°. (b) Comparison ofar-field scattering pattern. The incidence angle is set to 70° withavelength of 488 nm and s polarization.

Figure 6(b) shows the DSC comparison of underetchedias. We define the ideal depth as H �1440 nm�. The solidurve �h=H� represents a fully etched via, while dashednd dotted curves represent h=0.75H and h=0.5H, of thenderetched cases. The forward and backward scatteringhowed the maximum value of DSC, and the fully etchedylindrical via showed the maximum among three casest ±70°. In the center region around 0° of scatteringngle, the h=0.75H and h=H cases showed similar mag-itude, while in the ±20° – ±40° region, the h=0.5H and=H cases showed similar magnitude. The result clearlyhows that the etching process could be monitored andifferentiated with a light-scattering method in a filmtructure.

Figure 6(c) shows the DSC comparison of etched viasith differing radial dimension. d=D (dashed curve)

epresents a 180 nm designed via, while d=1.22D and=0.77D designate when the actual diameter of theylindrical via is 220 nm and 132 nm, respectively. In theadial variation cases, the magnitude of the DSC is pro-ortional to the total volume of the scatterers over allcattering angles, while the overall location of peaks andalleys are similar.

Figure 6(d) shows the DSC pattern versus film thick-ess variations. The solid curve �f=H� represents filmhickness equal to the depth of the fully etched via; theashed curve �f=1.5H�, a film thickness of 2.16 �m; andhe dotted curve �f=2H�, a film thickness of 2.88 �m. Thelm clearly creates more peaks and valleys depending on

ts thickness; the scattering pattern in the 0– ±90° regionhows different numbers of fluctuations.

. DISCUSSIONhe results in Fig. 6 for the HAR contact-via shows that

he number and location of peaks and valleys varyepending on the physical parameters of the scatterers.he comparison of the fully etched and underetched casesrovides fast identification of types of defect and the sta-us of the contact-via during the fabrication process. Theesults of Fig. 6(b) and Fig. 6(c) can be cross referenced tondicate whether an etched via is fabricated as designedr is defective. In addition, the types of defect could be in-icated, e.g., whether they are underetched or the radialimension is changed.Compared to the infinite half-space, the presence of

lm shows a mechanism of scattering amplification.hen the same dipole model was computed in the

bsence of film, i.e., in the infinite half-space, the maxi-um DSC value was around 10−5, which is four orders ofagnitude smaller than that of the cylindrical vias

mbedded in film on top of the substrate, as shown inig. 6(b). The effect might be attributed to the additional

nteraction of dipole radiation with the substrate–film in-erface along with the air–film interface. In addition, ashown in Fig. 6(d), the film thickness contributes to theifferent DSC pattern in the far field. The maxima andinima that are the result of constructive and destructive

nterference are influenced by the thickness of the filmhen the other parameters are the same.In Mie scattering, the scattering efficiency showed a

uctuating pattern depending on the size parameter [24].

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ompared with the Mie scattering, which is in free space,urface and subsurface scattering provide differentcattering patterns depending on the incidence angle andolarization. The presence of the interface provides aariable that could be applied to provide another differen-iating mechanism. Figure 7(a) shows the hemisphericalistribution of the DSC above the fully etched HARontact-via with varying incidence angle and polarization.he results show the TSC for both polarization and se-

ected incidence angles. In the case of normal incidence,oth polarizations show similar values of TSC. For the5° incidence angle, the p polarization showed a widerorward lobe of high intensity than the s polarization,hile the s polarization showed a stronger backscattering

obe. This accounts for the higher TSC for s polarizationt 45°, which could be applied in the design of in-line in-

ig. 6. Comparison of DDEFILM results for 108 nm �D� �1440olarization is set to s, incidence angle is set to 70°, and wavelen

ull �h=H� and underetched (h=0.75H, h=0.5H) cases and the fuhe via varies (d=1.22D, d=D, and d=0.77D). (d) Results whenully etched via.

pection tools. Similar trends are observed when the inci-ence angle is increased to 70°. However, the s polariza-ion showed both stronger forward and backward lobesompared with the p polarization case.

Figure 7(b) shows the TSC versus incidence angle in 5°teps. For p and s polarization, TSC showed its peak at5°, while s showed larger values than p polarization.owever, at 75° incidence angle, s polarization showed.5 times higher value of TSC compared with p polariza-ion. This incidence angle dependency could also be ap-lied to the instrument design to increase the sensitivityf the scattering pattern discrimination.

. CONCLUSIONpractical implementation of the discrete dipole approxi-ation method for a dipole embedded inside a film on top

� cylindrical contact-via modeled with SiO2 film on Si surface.0.488 �m. (a) Dipole model of cylindrical via. (b) Results for theH� case without film structure. (c) Results when the diameter of

thickness f is equal to, 1.5 times, and 2 times the depth of the

nm �Hgth isll �h=

the film

Fi


ig. 7. (a) Hemispherical distribution of DSC above surface. Incident light impinges from the +xy plane. All cases show the highestntensity in the forward scattering direction. (b) Total scattering cross section (TSC) versus the incidence angle for both polarizations.

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f a surface (DDEFILM) has been derived. The modelingonsiders the polarizability, direct interaction, and reflec-ion interactions among the dipoles, which is turn also in-ludes the multiple reflections due to the film structurend the far-field scattering pattern. The DDEFILM results compared to two other published methods for infinitealf-space cases and shows good agreement. The proposedethod is applied to the scattering of HAR contact-viasith different parameters and shows that the far-field

cattering pattern can be applied to pinpoint the scatter-ng from a defective contact-via (such as underetchedias). The effects of film thickness, polarization, andncidence angle are shown to provide more distinctivecattering patterns.

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integrity using light scattering,” M.S. thesis (Arizona StateUniversity, Tempe, 1993).

2. M. Siegel, “Radiation from linear antennas in a dissipativehalf-space,” IEEE Trans. Antennas Propag. AP-19,477–485 (1971).

3. G. Kristensson, “Electromagnetic scattering from buriedinhomogeneities—A general three-dimensional formalism,”J. Appl. Phys. 51, 3486–3500 (1980).

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Documents

Application of the discrete dipole approximation for dipoles embedded in film