Electromagnetic analysis for optical coherence tomography based through silicon vias metrology

W. A. Iff; J.-P. Hugonin; C. Sauvan; M. Besbes; P. Chavel; G. Vienne; L. Milord; D. Alliata; E. Herth; P. Coste; A. Bosseboeuf

doi:10.1364/AO.58.007472

Applied Optics
Vol. 58,
Issue 27,
pp. 7472-7488
(2019)
•https://doi.org/10.1364/AO.58.007472

Electromagnetic analysis for optical coherence tomography based through silicon vias metrology

W. A. Iff, J.-P. Hugonin, C. Sauvan, M. Besbes, P. Chavel, G. Vienne, L. Milord, D. Alliata, E. Herth, P. Coste, and A. Bosseboeuf

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W. A. Iff, J.-P. Hugonin, C. Sauvan, M. Besbes, P. Chavel, G. Vienne, L. Milord, D. Alliata, E. Herth, P. Coste, and A. Bosseboeuf, "Electromagnetic analysis for optical coherence tomography based through silicon vias metrology," Appl. Opt. 58, 7472-7488 (2019)

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Table of Contents Category
- Optical Devices, Sensors, and Detectors
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About this Article
History
- Original Manuscript: May 9, 2019
- Revised Manuscript: July 27, 2019
- Manuscript Accepted: August 7, 2019
- Published: September 16, 2019

Abstract

This paper reports on progress in the analysis of time-domain optical coherence tomography (OCT) applied to the dimensional metrology of through-silicon vias (TSVs), which are vertical interconnect accesses in silicon, enabling three-dimensional (3D) integration in microelectronics, and estimates the deviations from earlier, simpler models. The considered TSV structures are 1D trenches and circular holes etched into silicon with a large aspect ratio. As a prerequisite for a realistic modeling, we work with spectra obtained from reference interferograms measured at a planar substrate, which fully includes the dispersion of the OCT apparatus. Applying a rigorous modal approach, we estimate the differences to a pure ray tracing technique. Accelerating our computations, we focus on the relevant fundamental modes and apply a Fabry–Perot model as an efficient approximation. Exploiting our results, we construct and present an iterative procedure based on the minimization of a merit function, which concludes TSV heights reliably, accurately, and rapidly from measured interferograms.

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Supplementary Material (4)

Name	Description
Data File 1	Effective index n_eff(Delta_x) for propagation of the fundamental mode of an 1D trench at the center wavelength lambda_c = 1.329 µm of the applied light source spectrum; pure TE- and pure TM-polarization are considered.
Data File 2	Effective index n_eff(D) for propagation of the fundamental mode in radial direction (+/- 1st harmonic in azimuthal direction) at a circular hole of diameter D in silicon at the center wavelength lambda_c = 1.329 µm of the applied light source.
Visualization 1	Simulation of the OCT interferogram (intensity pattern I(z)−I_0) when varying the height H of a circular TSV (Through Silicon Via) of diameter D = 5 µm when illuminated by a Gaussian beam of the waist w = 2.7 µm. Dispersion is present.
Visualization 2	Simulation of the OCT interferogram (intensity pattern I(z)—I_0) when varying the height H of a circular TSV (Through Silicon Via) of diameter D = 5 µm when illuminated by a Gaussian beam of the waist w = 2.7 µm. Dispersion is absent.

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Trench width $Δ x / μm$	$H / μm$ from envelope detection plus Gaussian fits	$\bar{H}$ : weighted averaged $H / μm$ (15) from Gaussian fits	$H / μm$ from novel technique with correct $n_{eff}$	$\bar{H}$ : weighted averaged $H / μm$ (15) from our technique	$\| \frac{a_{b}}{a_{t}} \cdot \exp (- 2 k_{c} n_{eff} H) \|$ (values from electromagnetic analysis)
3.0	$18.9 \pm 0.1$	18.9	$17.1 \pm 0.1$	17.2	0.69
3.0	$18.9 \pm 0.1$	18.9	$17.2 \pm 0.1$	17.2	0.71
2.5	$14.6 \pm 0.1$	15.2	$16.8 \pm 0.2$	16.4	1.52
2.5	$15.7 \pm 0.1$	15.2	$16.0 \pm 0.2$	16.4	1.47
2.0	$16.6 \pm 0.1$	16.9	$15.8 \pm 0.2$	15.8	0.97
2.0	$17.1 \pm 0.1$	16.9	$15.9 \pm 0.2$	15.8	0.91

Trench width $Δ x / μm$	$H / μm$ from envelope detection plus Gaussian fits	$\bar{H}$ : weighted averaged $H / μm$ (15) from Gaussian fits	$H / μm$ from our technique with correct $n_{eff}$	$\bar{H}$ : weighted averaged $H / μm$ (15) from our technique	$\| \frac{a_{b}}{a_{t}} \cdot \exp (- 2 k_{c} n_{eff} H) \|$ (values from electromagnetic analysis)
3.0	$19.9 \pm 0.1$	20.0	$16.3 \pm 0.2$	17.4	0.48
3.0	$20.1 \pm 0.1$	20.0	$17.7 \pm 0.1$	17.4	0.45
2.5	$19.2 \pm 0.1$	19.1	$18.2 \pm 0.3$	17.8	1.26
2.5	$18.9 \pm 0.1$	19.1	$17.6 \pm 0.2$	17.8	1.34
2.0	$19.6 \pm 0.1$	19.7	$17.6 \pm 0.2$	18.1	0.50
2.0	$19.8 \pm 0.1$	19.7	$18.7 \pm 0.2$	18.1	0.50

Diameter $D / μm$	$H / μm$ from envelope detection plus Gaussian fits	$\bar{H}$ : weighted averaged $H / μm$ (15) from Gaussian fits	$H / μm$ from our technique with correct $n_{eff}$	$\bar{H}$ : weighted averaged $H / μm$ (15) from our technique
12.0	$20.7 \pm 0.1$	20.9	$19.7 \pm 0.1$	19.7
12.0	$21.0 \pm 0.1$	20.9	$19.9 \pm 0.2$	19.7
7.0	$16.4 \pm 0.1$	16.4	$17.0 \pm 0.3$	17.4
7.0	$16.4 \pm 0.1$	16.4	$17.6 \pm 0.2$	17.4
5.0	$19.7 \pm 0.2$	19.7	$15.9 \pm 0.1$	15.9
5.0	$19.7 \pm 0.3$	19.7	$15.9 \pm 0.1$	15.9
3.0	$6.5 \pm 0.2$	7.1	$13.8 \pm 0.3$	13.6
3.0	$9.3 \pm 0.4$	7.1	$13.5 \pm 0.2$	13.6
2.5	$1.4 \pm 0.5$	2.0	$10.7 \pm 0.5$	12.5
2.5	$2.4 \pm 0.4$	2.0	$15.2 \pm 0.6$	12.5
2.5	$16.7 \pm 0.1$	16.7	$14.2 \pm 0.3$	13.3
2.5	$16.8 \pm 0.1$	16.7	$12.3 \pm 0.3$	13.3
2.5	$17.1 \pm 0.2$	17.0	$12.9 \pm 0.3$	13.0
2.5	$17.0 \pm 0.2$	17.0	$13.5 \pm 0.6$	13.0

Trench width $Δ x / μm$	$H / μm$ from envelope detection plus Gaussian fits	$\bar{H}$ : weighted averaged $H / μm$ (15) from Gaussian fits	$H / μm$ from novel technique with correct $n_{eff}$	$\bar{H}$ : weighted averaged $H / μm$ (15) from our technique	$\| \frac{a_{b}}{a_{t}} \cdot \exp (- 2 k_{c} n_{eff} H) \|$ (values from electromagnetic analysis)
3.0	$18.9 \pm 0.1$	18.9	$17.1 \pm 0.1$	17.2	0.69
3.0	$18.9 \pm 0.1$	18.9	$17.2 \pm 0.1$	17.2	0.71
2.5	$14.6 \pm 0.1$	15.2	$16.8 \pm 0.2$	16.4	1.52
2.5	$15.7 \pm 0.1$	15.2	$16.0 \pm 0.2$	16.4	1.47
2.0	$16.6 \pm 0.1$	16.9	$15.8 \pm 0.2$	15.8	0.97
2.0	$17.1 \pm 0.1$	16.9	$15.9 \pm 0.2$	15.8	0.91

Trench width $Δ x / μm$	$H / μm$ from envelope detection plus Gaussian fits	$\bar{H}$ : weighted averaged $H / μm$ (15) from Gaussian fits	$H / μm$ from our technique with correct $n_{eff}$	$\bar{H}$ : weighted averaged $H / μm$ (15) from our technique	$\| \frac{a_{b}}{a_{t}} \cdot \exp (- 2 k_{c} n_{eff} H) \|$ (values from electromagnetic analysis)
3.0	$19.9 \pm 0.1$	20.0	$16.3 \pm 0.2$	17.4	0.48
3.0	$20.1 \pm 0.1$	20.0	$17.7 \pm 0.1$	17.4	0.45
2.5	$19.2 \pm 0.1$	19.1	$18.2 \pm 0.3$	17.8	1.26
2.5	$18.9 \pm 0.1$	19.1	$17.6 \pm 0.2$	17.8	1.34
2.0	$19.6 \pm 0.1$	19.7	$17.6 \pm 0.2$	18.1	0.50
2.0	$19.8 \pm 0.1$	19.7	$18.7 \pm 0.2$	18.1	0.50

Abstract

Supplementary Material (4)

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Figures (20)

Tables (3)

Equations (17)

Applied Optics