Decoupling substrate thickness and refractive index ... - OSA Publishing

Vol. 26, No. 2 | 22 Jan 2018 | OPTICS EXPRESS 1697

Decoupling substrate thickness and refractive index measurement in THz time-domain spectroscopy FARAH VANDREVALA 1 1 Department 2 Department

AND

E RIK E INARSSON 1,2,*

of Electrical Engineering, University at Buffalo, New York 14260, USA of Materials Design and Innovation, University at Buffalo, New York 14260, USA

* [email protected]

Abstract: Terahertz time-domain spectroscopy (THz–TDS) relies heavily on knowing precisely the thickness or refractive index of a material. In practice, one of these values is assumed to be known, or their product is numerically optimized to converge on suitable values. Both approaches are prone to errors and may mask some real features or properties of the material being studied. To eliminate these errors, we use THz–TDS in reflection geometry to accurately and independently determine both thickness and refractive index by illuminating the step-edge of a substrate atop a metal stage. This method relies solely on the relative time delay among three reflected pulses, and therefore forgoes the need for optimization or assumption of substrate parameters. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (300.6495) Spectroscopy, terahertz; (080.2720) Mathematical methods (general); (120.3940) Metrology.

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#315301 Journal © 2018

https://doi.org/10.1364/OE.26.001697 Received 8 Dec 2017; revised 5 Jan 2018; accepted 7 Jan 2018; published 16 Jan 2018


1.

Introduction

The non-contact and nondestructive nature of terahertz time-domain spectroscopy (THz–TDS) is making it a popular choice for characterizing the electrical properties of materials ranging from two-dimensional crystals [1, 2] to biological samples [3, 4]. Although transmission geometry is more commonly used, reflection geometry is needed for the study of highly absorbing or highly reflective materials. In such applications, samples are typically placed atop substrates such as quartz or high-resistivity silicon, and their characteristics are determined by studying light–matter interactions at the substrate–sample interface [5]. One can then ascertain the complex optical properties of the sample material through equations that depend on the substrate thickness and its complex refractive index. Consequently, accurate knowledge of both these values is necessary to correctly isolate the sample properties from those of the underlying substrate. Since THz–TDS measures the electric field in the time domain, an imprecise value of the substrate thickness can cause errors to cascade into sample property calculations related to the detected phase [6]. Numerical algorithms have traditionally been used to optimize the values of substrate thickness Lsub , refractive index nsub , and extinction coefficient κsub [7–9]. Alternatively, Lsub can be held constant while nsub is calculated [10], or nsub can be kept fixed while computing Lsub [11]. More recent methods, proposed to minimize errors in parameter extraction resulting from substrate thickness variations, take one of the following approaches: assume that the substrate dielectric properties are known and calculate a calibration factor to account for beam displacement [12], perform a baseline calibration using a reference sample and substrate [13], or use substrates of different thicknesses to account for errors [14]. These techniques rely on assumptions of nsub and/or Lsub , or require multiple iterations to arrive at values that numerically minimize a chosen error function. In this work, we demonstrate an experimental method whereby we independently determine the substrate thickness and refractive index from a single time-domain measurement. We do this by illuminating the step-edge of a substrate—which sits atop a metal stage—to obtain three reflections. One reflection comes from the stage, while the other two come from the top and bottom surfaces of the substrate itself. The time delay between the first two reflections yields the substrate thickness completely independent of the refractive index. Using this value of Lsub and the time delay between the first and third reflections, we can then determine a value of nsub that is decoupled from the substrate thickness measurement. 2.

Fractional reflection measurement method

Our THz–TDS system (Advantest TAS7500TS) uses 50 fs light pulses from a 1550 nm infrared laser at a repetition rate of 50 MHz to excite a photoconductive antenna. This generates a THz beam consisting of pulses having a spectral range of 0.1–4 THz. Parabolic mirrors guide the p-polarized beam such that it is made incident onto a horizontal sample stage at an angle of θ i = 10°. We adjust the vertical position of the stage so that the beam is focused to a spot size of approximately 3 mm on the sample or substrate surface. Figure 1(a) shows a ray diagram for these pulses corresponding to fractional reflection, i.e., when the THz spot is focused on the substrate step-edge such that the portions of the incident THz beam reflected from the top surface of the substrate and from the stage are nearly equal in amplitude, as seen in the time-domain measurement in Fig. 1(b). The left side of Fig. 1(a) shows the part of the beam that travels outside the substrate, whereas the right side of Fig. 1(a) shows the part of the beam that propagates through the interior of the substrate. The first detected pulse (main) is the reflection from the top surface of the substrate, whereas the second pulse (stage) is the reflection from the metal stage. Since neither the main nor stage pulse travels through the substrate, the time delay between their arrival at the detector is completely


(a) Ray diagrams showing the THz beam paths outside the substrate (left), and inside the substrate (right). echo main

stage

(b) THz time-domain measurement showing the main, stage, and echo reflections.

Fig. 1. Fractional reflection at substrate step-edge.

independent of the substrate refractive index. Therefore, we can calculate the substrate thickness as follows: ∆t = tstage − tmain 2Lsub 2Lsub − sin θ i sin θ i c∆t = cos θ i cos θ i

(1) (2)

= 2Lsub cos θ i

(3)

c∆t 2 cos θ i

(4)

This leads to Lsub =

The third pulse (echo) is due to the portion of the incident THz beam that propagates inside the substrate, eventually being reflected from the substrate–stage interface. Given that the THz beam will bend toward the surface normal when it enters the substrate, θ t is always less than θ i . For a small θ i (10° in this case), θ t will have a negligible effect on path length. Moreover, any light incident on the substrate edge would be internally reflected, preventing light from escaping through the substrate edge. Consequently, the delayed arrival of the echo pulse at the detector depends on the substrate refractive index, which we can accurately determine from: ∆t 0 = techo − tmain c∆t 0 =

(5)

2Lsub 2Lsub nsub − sin θ t sin θ i cos θ t cos θ t

(6)


Using Snell’s Law, nsub sin θ t = nair sin θ i nsub cos θ t =

2 nsub

(7)

− sin θ i 2

1/2

(8)

we get 2Lsub 2Lsub nsub − nsub sin2 θ t cos θ t cos θ t = 2Lsub nsub cos θ t

c∆t 0 =

(9) (10)

Combining Eq. (8) and Eq. (10), the expression for substrate refractive index is nsub =

c∆t 0 2Lsub

2

+ sin2 θ i

1/2 (11)

In the above calculations, we have considered the optical path difference between the stage and main pulses in Eq. (2), as well as the echo and main pulses in Eq. (6), to be from the point where the THz beam reflects from the top of the substrate to the point where both beams have the same wavefront [12]. Since reflection from the stage is a crucial component of this scheme, it is advantageous for adequate reflectivity to have a metallic stage that is smooth enough for the substrate to lie flat on it with no air gaps. Having a large spot size, small angle of incidence, and sharp step edge will facilitate the measurement by ensuring that differences in path length inside the substrate are negligible. 3. 3.1.

Results and discussion Time-domain measurements

We use the fractional reflection setup described above to measure the thickness and refractive index around the perimeter of a quadrant of a 100 mm-diameter intrinsic Si wafer. Figure 2(a) shows a series of measurements taken in increments of approximately 2 mm along the two straight edges of the wafer section (inset). Based on these measurements, we find that the refractive index changes little with position, but the thickness decreases as we move to the center of the wafer. Using these edge measurements to establish boundary conditions and an accurate value for nsub , we then use the same measurement geometry to scan and obtain a thickness map of the entire wafer section. The result is shown in Fig. 2(b). It is known that variations in resistivity should alter a substrate’s refractive index [15]. To illustrate this effect, we repeated the fractional-reflection measurements using silicon wafers of different thicknesses and resistivities. The results are summarized in Table 1. We also measured the cross sections of two of the wafers using scanning electron microscopy (SEM) to independently verify their thickness. Despite differing significantly from the manufacturer’s specified values (Lmfr ), the thickness determined via fractional reflection differs from the wafer Table 1. Thicknesses L and refractive indices nsub for Si wafers with different resistivities ρ. ρSi [W cm] >1000 10–30 1–10

Lmfr [µm] 500 280 280

Lsub [µm] 485.2 ± 3.9 294.9 ± 2.4 305.2 ± 4.7

LSEM [µm] 487.7 – 305.1

nsub 3.409 ± 0.009 3.393 ± 0.007 3.363 ± 0.024


(GJH>PP@

(GJH>PP@ (a) Lsub and nsub measured every 2 mm along edges.

(b) THz thickness map for entire wafer section.

Fig. 2. Time-domain measurement results. thickness determined by SEM (LSEM ) by less than 1%. The minimum measurable thickness and the precision with which it can be measured are limited by the THz pulse width and the time resolution of the spectrometer. In our case, this corresponds to a minimum measurable substrate thickness of approximately 120 µm with a resolution of 0.3 µm. 3.2.

Frequency-domain analysis

To illustrate and corroborate the results from our method with those in literature, and to confirm whether the properties extracted from the above time-domain measurements correctly describe the substrate behavior in the frequency domain, we performed self-referenced measurements on the three Si substrates listed in Table 1. The Fresnel coefficients are defined at the air–substrate and substrate–air interfaces as described in [12]. We determine the complex refractive index (n˜ sub = nsub + j κsub ) using the following transfer function equation, in which | E˜ | represents the amplitude of the THz electric field and φ represents its phase. Tair−sub Rsub−airTsub−air | E˜echo | 2ω exp − j[φecho − φmain ] = exp − j n˜ sub Lsub cos θ t (12) Rair−sub c | E˜main | Since time delay and phase difference are correlated by Fourier transform, the refractive index equation shown below has the same form as its time-domain analogue in Eq. (11), nsub =

c[φecho − φmain ] 2ωLsub

2

+ sin2 θ i

1/2 (13)

and the extinction coefficient is given by the amplitude attenuation ratio, κsub =

−c | E˜echo | log − log[Tair−subTsub−air ] 2ωLsub cos θ t | E˜main |

(14)

In Fig. 3 we plot the real part nsub (upper panel) and the imaginary part κsub (lower panel) of the complex refractive index for the three Si substrates. The value of nsub matches the one calculated from the time-domain measurements, and we observe that κsub gets stronger as resistivity decreases, which is characteristic of increased absorption with doping level. Furthermore, the directly proportional relationship between resistivity and refractive index signifies a reduction

Lsub> m@


nsub

:DYHOHQJWK > m@

6L cm 6L cm 6L cm

sub

)UHTXHQF\f>7+]@

Fig. 3. Complex refractive index of Si wafers with different resistivities.

in phase shift between the electric field and the current density induced in the substrate [16]. Although this correlation between complex refractive index and resistivity is in agreement with earlier findings [15, 16], the accuracy with which we can calculate n˜ sub is limited if the thickness is not precisely known. This is because wafer specifications provided by a manufacturer include a significant uncertainty in thickness (typically ±25 µm) and, in case of moderately doped wafers, a broad range of resistivity values. In other words, by eliminating uncertainty in the thickness we can attribute any differences in the complex refractive index (Fig. 3) to variations in the doping level of the Si substrate. With the substrate properties well-characterized, it will be easier to accurately subtract the substrate contribution when characterizing a substrate-supported sample. 4.

Conclusion

We have demonstrated a fractional-reflection measurement, taken at the step-edge of a substrate, that allows us to decouple thickness and refractive index measurements using THz–TDS. Aside from not requiring any numerical optimization routines, the advantage of our approach is that all the necessary information is obtained in a single time-domain measurement. Moreover, it is easily applied to a wide range of substrates and will yield accurate results despite unknown or uncertain doping levels. Using these independently determined values, the optical properties of the substrate itself can be calculated with high confidence. This is critical when the effects of the supporting substrate must be carefully accounted for, such as when studying coatings, thin-films, two-dimensional materials, or biological samples. Funding Air Force Office of Scientific Research (AFOSR) (FA9550-16-1-0188).