Dielectric interface effects in subsurface microscopy of integrated circuits

Optics Communications 285 (2012) 1675–1679

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Dielectric interface effects in subsurface microscopy of integrated circuits F. Hakan Köklü a,⁎, Bennett B. Goldberg b, M. Selim Ünlü a a b

Department of Electrical and Computer Engineering, Boston University, 8 Saint Mary's Street, Boston, MA 02215, United States Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, United States

a r t i c l e

i n f o

Article history: Received 29 September 2011 Received in revised form 6 December 2011 Accepted 9 December 2011 Available online 22 December 2011 Keywords: Solid immersion Numerical aperture Spatial resolution Diffraction limit Polarization

a b s t r a c t We investigate the defocus and image quality affected by a dielectric interface on high numerical aperture focusing of linearly polarized illumination in aplanatic mode. Theoretical and experimental demonstration is performed on subsurface backside microscopy of silicon integrated circuits, showing that the high longitudinal magnification provided by solid immersion lens microscopy allows the observation of significant astigmatism. It is shown that a 50 micron longitudinal displacement of the objective lens with respect to the sample is necessary to achieve maximum resolutions in two directions. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Nearly all biological and solid state imaging is performed near a dielectric interface. Tightly focused light near a dielectric interface exhibits aberrations when the light is incident from an optically denser medium due to total internal reflection (TIR) and leading to a Goos– Hänchen shift [1]. A reflection mode confocal microscope utilizing a solid immersion lens (SIL) displays similar aberrations, as the SIL provides a sufficiently high numerical aperture (NA) where a significant portion of the light is effected by TIR [2,3]. Industrial applications in backside microscopy of integrated circuits (IC) for fault isolation and failure analysis often employ a SIL and routinely image and interrogate structures located near a dielectric interface under tightly focused light. Also known as numerical aperture increasing lens (NAIL) microscopy, this technique takes advantage of the large refractive index of silicon as the immersion medium and the increased focusing angles to realize high resolution backside imaging through the substrate of an IC [4,5,6]. The effects of polarization and dielectric interfaces in the focal region can no longer be neglected under the high NA focusing conditions provided by the NAIL. Enhanced contrast and ellipticity of the focal spot induced by the linearly polarized illumination have been investigated thoroughly [2,7,8]. Yet there have not been any studies analyzing the effects of the dielectric interface on the image resolution in IC imaging, even though the most critical structures of an IC lie precisely at the transistor layer which is adjacent to the SiO2 insulation medium. IC microscopy with a NAIL is thus an ideal platform to study tightly focused light near a dielectric interface as it provides high longitudinal magnification in the aplanatic configuration ⁎ Corresponding author. E-mail address: [email protected] (F. Hakan Köklü). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.050

allowing detailed examination of defocus [9]. In prior work, we have examined interconnect layers located inside the insulation medium, beyond the Si–SiO2 interface [10]. In this article, we theoretically and experimentally demonstrate that the tightest spot on the interface occurs at two different focusing conditions for two orthogonal directions under linearly polarized illumination. TIR from the dielectric interface modifies the optical fields and results in astigmatism while imaging structures located adjacent to the interface. Both of the focusing conditions require a slight defocus from the interface although the structures of interest lie along the interface. Vectorial field calculations validate the amount of defocus needed for optimizing the imaging conditions. 2. Setup The confocal microscopy setup employed for imaging is a doublepath, reflection-mode fiber-optical scanning microscope. The details of this setup can be found in Ref. [10]. Excitation is done with a fiber-coupled laser diode at λ0 = 1.3 μm. The NAIL used in this work is an undoped silicon hemisphere with radius, R = 1.61 mm. The optimum substrate thickness (X) for aplanatic imaging with this NAIL is X = R/n = 460 μm where n is the refractive index at the operating wavelength [9]. The sample is a custom IC with 4 metal and 2 polysilicon layers fabricated by a 0.35 μm process. The substrate thickness was reduced to 458 ± 2 μm for aplanatic imaging. 3. Theory We use a ray optics approach to calculate the spherical aberrations which will then be incorporated into the vectorial field calculations.

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before reaching the interface but the θ angle ray reaches the interface before the optical axis. This happens when the light is focused between the central and aplanatic points and the interface is located in the focus area [9]. The third case is the opposite of the previous case. Reference ray reaches the interface but the θ angle ray does not. This happens when the light is focused beyond the aplanatic point. Lastly, in the fourth case all the rays are focused beyond the interface. In SiO2, 8 d2 ðθÞ > > > < sinσ plSiO2 ðθÞ ¼ sth −zl > > > d2 ðθÞ : −ðzl −sth Þ sinσ

if zh ð0Þb sth and zh ðθÞ > sth if zh ð0Þ > sth and zh ðθÞbsth

ð3Þ

if zh ð0Þ; zh ðθÞ > sth

The three cases in this equation correspond to the second, third and fourth cases in the previous equation. Finally, the full optical path length difference becomes ΦðθÞ ¼ plair ðθÞ þ n1 plSi ðθÞ þ n2 plSiO2 ðθÞ

Fig. 1. Focusing geometry for a hemispherical NAIL.

The geometry shown in Fig. 1 specifies all angles and distances and provides relevant equations for the focus coordinates. We assume we know the objective focus, zob, and the focusing angle, θ, for each ray when calculating the rest of the parameters. The geometry is drawn for n1 =nSi, n2 =nSiO2 when focusing beyond the Si–SiO2 interface. zh and zl depend on θ introducing the spherical aberrations due to the optical path length differences between rays. In this case, the light is focused around the Si–SiO2 interface making zh and zl very close to each other and to the thickness of the silicon substrate, sth. The ray traveling along the z-axis is called the reference ray and the ray intersecting the optical axis with the angle θ is called the θ angle ray. The distance from the optical axis to the points where the light rays intersect the medium interfaces are denoted as d1 and d2 which are given as d1(θ)=R sin(γ+α), and d2(θ)=(zh(θ)−sth)tan(γ). The path length difference between the reference ray and the θ angle ray in air is given as: plair ðθÞ ¼ ðR þ zob Þ−

d1 ðθÞ sinθ

ð1Þ

Likewise in silicon, 8 d1 ðθÞ > > ðR þ zh ð0ÞÞ > > sinγ > > > > d ð θ Þ−d > 1 2 ðθÞ > −ðR þ zh ð0ÞÞ < sinγ plSi ðθÞ ¼ d ð θ Þ > > > 1 −ðR þ sth Þ > > sinγ > > > > > : d1 ðθÞ−d2 ðθÞ −ðR þ sth Þ sinγ

if zh ð0Þ; zh ðθÞbsth if zh ð0Þbsth and zh ðθÞ > sth if zh ð0Þ > sth and zh ðθÞbsth

ð2Þ

if zh ð0Þ; zh ðθÞ > sth

The first case in this equation is valid when all the focused rays meet the optical axis before reaching the Si–SiO2 interface. The second case happens when the reference ray meets the optical axis

ð4Þ

in μm. The optical path length difference is also denoted as the spherical aberration function for each incoming ray. The complete focal field calculations are performed using angular spectrum representation (ASR) [11]. The computation is based on the exact treatment explained in Ref. [12] for focusing through a layered medium. The spherical aberration function calculated above is incorporated into the field integrals as additional phase terms in the same fashion as Ref. [13]. The field expressions and the open form of the integrals are listed in the Appendix. The resulting intensity of the focal fields with varying displacement of the focus from the interface is shown in y–z plane in Fig. 2. The light is focused from the bottom through the silicon into the SiO2 medium and the physical microscope focus displacement between consecutive pictures is 25 μm. This is the actual movement of the stage with respect to the microscope objective. This large displacement step is scaled down multiple times by the longitudinal magnification in the image space. The full span of displacement is from −100 μm to 100 μm and zero corresponds to focusing on the interface. Although the total displacement of the microscope is 200 μm, the change in the position of the optical focus is less than 1 μm due to very high longitudinal magnification provided by the NAIL in aplanatic configuration and the interface [14]. 4. Results The structures of interest in our sample are polysilicon lines that are fabricated on silicon substrate and covered with SiO2 as the insulation layer. Therefore, the best resolution is achieved when the field at the interface is tightest in the lateral direction. Fig. 3 demonstrates the change in the full-width at half maximum (FWHM) of the field at the interface in two directions. This data is extracted by taking a linecut through the field calculations shown in Fig. 2. The upper curve corresponds to the FWHMs of the linecuts in the direction parallel to the polarization direction. This means the linecut is taken in the x direction when the incoming light is linearly polarized also in the x direction. The lower curve corresponds to the FWHMs of the linecuts in the direction perpendicular to the polarization direction. These FWHM values are directly comparable to the FWHM values of experimental linecuts in the next section. As seen in the figure, the tightest spot on the interface does not occur when a linearly polarized light is focused on the interface but when it is focused behind or beyond the interface depending on the direction measured. The physical stage-microscope displacement between two cases is calculated to be 50 μm. This implies that two images taken at a certain focus with orthogonal polarizations are not enough to achieve the highest possible resolution in all directions.

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4

z0 = -75µm

z0 = -100µm

z0 = -50µm

3 2

z (µm)

1

SiO2

0

Si

-1 -2 -3 -4 4

z0 = -25µm

z0 = 0

z0 = 25µm

3 2

z (µm)

1 0

-1 -2 -3 -4 4

z0 = 50µm

z0 = 100µm

z0 = 75µm

3 2

z (µm)

1 0

-1 -2 -3 -4 -2

0 y (µm)

2 -2

0 y (µm)

2 -2

0 y (µm)

2

Fig. 2. Intensity distributions of the focal fields at different defocus values from the interface. Defocus changes from − 100 μm to 100 μm from left to right and top to bottom where consecutive pictures have 25 μm defocus difference. Center image corresponds to zero defocus. Dashed lines mark the interface. Light is focused from the bottom.

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FWHM (µm)

0.7

depth for the sharpest image of the lines in the polarization direction where the FWHM in the other direction is changing rapidly.

50 µm

0.6

5. Conclusions

0.5 0.4 0.3 0.2 -100

-50

0

50

100

defocus (µm) Fig. 3. Upper curve corresponds to the FWHM values of the focused field at the interface in the direction parallel to the polarization. Lower curve corresponds to the FWHM values in the direction perpendicular to the polarization.

Experimental verification is conducted on an L-shaped corner structure with a linewidth of 0.6 μm which allow for the characterization of the imaging performance in two lateral directions at the same time. Fig. 4 displays images taken at the two best foci for the horizontal and vertical lines, while the polarization direction is always along either one of the lines. The images on the left column are taken at −30 μm physical displacement from the case of focusing on the interface which results in the sharpest lines perpendicular to the polarization direction. The sharpest images of the lines in the polarization direction are taken at 50 μm physical displacement from the case of focusing which are shown on the right column in Fig. 4. 80 μm difference in total displacement is somewhat bigger than the expected value of 50 μm which can be attributed to the increased susceptibility to noise in the slowly changing FWHM function as seen in the lower curve in Fig. 3. At −30 μm displacement, resolutions calculated from the confocal line spread functions are 200 nm and 260 nm in the direction perpendicular and parallel to the polarization, respectively. The experimental values are found to be 250 nm and 260 nm, very close to the calculations. At 50 μm displacement, calculated resolutions are 195 nm and 290 nm in the direction perpendicular and parallel to the polarization, respectively, although the experimental values are 220 nm and 500 nm. This large error can again be attributed to the difficulty of pinpointing the probe

In this study, we show that best lateral resolution in backside IC microscopy is achieved at different probe depths due to the tightly focused linearly polarized light near a dielectric interface. Depending on the layout of the structures of interest with respect to the polarization direction, best edge responses are acquired at different probe depths displaying astigmatism. The dielectric interface redistributes the illumination so that the point spread function no longer has the narrowest width in all directions at one longitudinal focus as it is for a homogeneous medium. These findings could assist in the interpretation of images in layered samples such as integrated circuits. Considering defocus effects on theoretical point spread functions will also help improve deconvolution techniques. Appendix A. Focal field expressions The field before the dielectric interface is the sum of the original field and the reflected field from the interface. −ik f

ik f E e 1 E1 ðρ; ϕ; zÞ ¼ 1 0 0 1 2  I0 þ Ir0 þ I2 þ I r2 cos2ϕ  B C r @ I 2 þ I2 sin2ϕ A  r −2i I1 þ I1 cosϕ

ðA:1Þ

The field after the interface is the transmitted field and given as ik f E e−ik1 f E2 ðρ; ϕ; zÞ ¼ 1 0 21 0 t t I0 þ I2 cos2ϕ t @ I sin2ϕ A

ðA:2Þ

2

−2iI t1 cosϕ The integrals used in the focal field expressions are given as: pffiffiffiffiffiffiffiffiffiffiffi γ I 0 ¼ ∫0 max dγf w ðγÞ cosγ sinγð1 þ cosγÞ

ðA:3Þ

J 0 ðk1 ρ sinγ Þeik1 z cosγ eik0 ΦðθÞ

pffiffiffiffiffiffiffiffiffiffiffi γ I 1 ¼ ∫0 max dγf w ðγÞ cosγ sinγ sinγ

ðA:4Þ

J 1 ðk1 ρ sinγ Þeik1 z cosγ eik0 ΦðθÞ

pffiffiffiffiffiffiffiffiffiffiffi γ I 2 ¼ ∫0 max dγf w ðγÞ cosγ sinγð1− cosγ Þ

ðA:5Þ

ik1 z cosγ ik0 ΦðθÞ

J 2 ðk1 ρ sinγ Þe

e

  pffiffiffiffiffiffiffiffiffiffiffi γ I r0 ¼ ∫0 max dγf w ðγÞ cosγ sinγ rs ðγ Þ−r p ðγÞ cosγ −ik1 z cosγ 2ik1 z0 ik0 ΦðθÞ

J 0 ðk1 ρ sinγÞe

e

e

pffiffiffiffiffiffiffiffiffiffiffi γ I r1 ¼ ∫0 max dγf w ðγÞ cosγ sinγr p ðγ Þ sinγ

ðA:7Þ

−ik1 z cosγ 2ik1 z0 ik0 ΦðθÞ

J 1 ðk1 ρ sinγ Þe

e

e

  pffiffiffiffiffiffiffiffiffiffiffi γ I r2 ¼ ∫0 max dγf w ðγÞ cosγ sinγ r s ðγ Þ þ r p ðγ Þ cosγ −ik1 z cosγ 2ik1 z0 2ik1 z0 ik0 ΦðθÞ

J 2 ðk1 ρ sinγ Þe

e

e

e

  pffiffiffiffiffiffiffiffiffiffiffi γ I t0 ¼ ∫0 max dγf w ðγÞ cosγ sinγ t s ðγÞ þ t p ðγÞ cosγ ik2 z cosσ iz0 ðk1 cosγ−k2 cosσ Þ ik0 ΦðθÞ

J 0 ðk1 ρ sinγ Þe Fig. 4. Images recorded for the two longitudinal focus values of − 30 μm and 50 μm which give the best edge responses in directions parallel and perpendicular to the polarization direction, respectively. Arrows indicate the direction of polarization.

e

e

pffiffiffiffiffiffiffiffiffiffiffi γ I t1 ¼ ∫0 max dγf w ðγÞ cosγ sinγt p ðγÞnr sinγ J 1 ðk1 ρ sinγ Þeik2 z cosσ eiz0

ðA:6Þ

ðk1 cosγ−k2 cosσ Þ ik0 ΦðθÞ

e

ðA:8Þ

ðA:9Þ

ðA:10Þ

F. Hakan Köklü et al. / Optics Communications 285 (2012) 1675–1679

  pffiffiffiffiffiffiffiffiffiffiffi γ I t2 ¼ ∫0 max dγf w ðγ Þ cosγ sinγ t s ðγÞ−t p ðγÞ cosγ ik2 z cosσ iz0 ðk1 cosγ−k2 cosσ Þ ik0 ΦðθÞ

J 2 ðk1 ρ sinγ Þe

e

e

ðA:11Þ

where nr is n2/n1, Φ is the spherical aberration value calculated before, z0 is the physical defocus of the microscope and rs, rp, ts, and tp are the reflection and transmission coefficients. Although integrating the result of the ray optics spherical aberration calculation into electromagnetic optics is not a conventional approach, it is necessary and sufficiently accurate for modeling aplanatic focusing with a NAIL. There are at least two cases of this approach documented in the literature [9,13]. The refractions at the spherical interface introduced by the NAIL cannot be dealt with directly inside the scope of angular spectrum representation in a straightforward way. References [1] L. Novotny, G. R. D., K. Karrai, Optics Letters 26 (2001) 789. [2] K. Karrai, L. Xaver, L. Novotny, Applied Physics Letters 77 (2000) 3459. [3] S.M. Mansfield, G.S. Kino, Applied Physics Letters 57 (1990) 2615.

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[4] S.B. Ippolito, B.B. Goldberg, M.S. Ünlü, Applied Physics Letters 78 (2001) 4071. [5] F.H. Köklü, J.I. Quesnel, A.N. Vamivakas, S.B. Ippolito, B.B. Goldberg, M.S. Ünlü, Optics Express 16 (2008) 9501. [6] E. Ramsay, K.A. Serrels, M.J. Thomson, A.J. Waddie, M.R. Taghizadeh, R.J. Warburton, D.T. Reid, Applied Physics Letters 90 (2007) 131101. [7] K.A. Serrels, E. Ramsay, R.J. Warburton, D.T. Reid, Nature Photonics 2 (2008) 311. [8] F.H. Köklü, S.B. Ippolito, B.B. Goldberg, M.S. Ünlü, Optics Letters 34 (2009) 1261. [9] S.B. Ippolito, B.B. Goldberg, M.S. Ünlü, Journal of Applied Physics 97 (2005) 053105. [10] F.H. Köklü, M.S. Ünlü, Optics Letters 35 (2010) 184. [11] B. Richards, E. Wolf, Proceedings of the Royal Society of London Series A 253 (1959) 358. [12] A.N. Vamivakas, R.D. Younger, S.B. Ippolito, B.B. Goldberg, A. Swan, M.S. Unlu, E.R. Behringer, American Journal of Physics 76 (2008) 758. [13] S.H. Goh, C.J.R. Sheppard, Optical Communication 282 (2009) 1036. [14] K.A. Serrels, E. Ramsay, P.A. Dalgarno, B.D. Gerardot, J.A. O'Connor, R.H. Hadfield, R.J. Warburton, D.T. Reid, Journal of Nanophotonics 2 (2008) 021854.