An optical monitoring method for depositing dielectric layers of arbitrary thickness using reciprocal of transmittance

Qing-Yuan Cai; Yu-Xiang Zheng; Hai-Han Luo; Dong-Dong Zhao; Xiao-Feng Ma; Ding-Quan Liu

doi:10.1364/OE.23.004703

1. Introduction

The optical properties of thin films depend on the refractive index and thickness of each layer [1]. Monitoring, such as optical monitoring or quartz crystal monitoring, is very important in optical film deposition. The capability of getting the information of refractive index and thickness of the deposited layer is always the most important characteristic of monitoring method of optical film deposition. Optical monitoring is quite capable in this respect, and can provide error compensation for each layer in complex optical film deposition [2, 3]. The popular optical monitoring methods include turning point monitoring (TPM) [4] and level monitoring (LM) [5] of monochromatic light, broadband optical monitoring (BOM) [6, 7], ellipsometry monitoring (EM) [8]. Most of these methods are based on the measurement of transmittance or reflectance (absolute or relative value), which is hard to be converted to thickness of layer directly. Although several graphical technologies such as optical admittance loci technology, reflection coefficient diagram, etc., have been developed and used in optical monitoring process [9–11], optical monitoring is still one complex technology. These graphical technologies are not intuitionistic enough to estimate layer thickness from optical monitoring signal, compared to quartz crystal monitoring, in which, monitoring signal is in approximate linear relationship with layer thickness. In conventional reflectance or transmittance monitoring process, two turning points (two optical signal extrema) are necessary to calculate both practical refractive index and thickness of deposited layer, or one turning point to calculate thickness [12~14], and the calculation process is not simple. In this paper, we have derived the algorithm to find an effective and easy way to convert optical monitoring signal to phase thickness, and have given a trace diagram to help understanding the deposition process easily. On the basis of the proposed transformation formula and trace diagram, a realtime fitting or calculation could be performed to get correct refractive index and thickness of deposited layer.

2. Principle and method

The proposed method is used in monitoring the deposition process of non-absorbing film at normal incidence. The assumption is that a layer with refractive index n is about to be deposited on film with pre-deposited layers. The film can be considered as an equivalent surface with a starting input optical admittance Y_st = α + iβ, which can be explicated by imaging that a layer with refractive index n and phase thickness φ_st = 2πnd_st/λ (0 ≤ φ_st < π/2) deposited on infinitely thick substrate with refractive index n_s at reference wavelength λ, where, d_st denotes the starting geometrical thickness. The optical characteristic matrix of the film structure at reference wavelength is given by

[\begin{matrix} B^{'} \\ B^{'} (α + i β) \end{matrix}] = [\begin{matrix} \cos φ_{s t} & \frac{i}{n} \sin φ_{s t} \\ i n \sin φ_{s t} & \cos φ_{s t} \end{matrix}] [\begin{matrix} 1 \\ n_{s} \end{matrix}] .

Now, assuming that the refractive index of incident medium is also n, Fresnel reflective coefficients of the pre-deposited film

{r^{'}}_{s t}

and substrate

{r^{'}}_{s}

are connected by

{r^{'}}_{s t} = {r^{'}}_{s} \exp (- 2 i φ_{s t}),

where,

{r^{'}}_{s} = \frac{n - n_{s}}{n + n_{s}},

{r^{'}}_{s t} = \frac{n - Y_{s t}}{n + Y_{s t}} = \frac{(n^{2} - α^{2} - β^{2} - i 2 n β)}{{(n + α)}^{2} + β^{2}} .

From Eqs. (1)-(4), one gets

{r^{'}}_{s} = {\begin{cases} \frac{\sqrt{{(n^{2} - α^{2} - β^{2})}^{2} + 4 n^{2} β^{2}}}{{(n + α)}^{2} + β^{2}} (β > 0) \\ \frac{n - α}{n + α} (β = 0) \\ - \frac{\sqrt{{(n^{2} - α^{2} - β^{2})}^{2} + 4 n^{2} β^{2}}}{{(n + α)}^{2} + β^{2}} (β < 0) \end{cases},

φ_{s t} = \frac{1}{2} arc \cos \frac{n^{2} - α^{2} - β^{2}}{[{(n + α)}^{2} + β^{2}] {r^{'}}_{s}},

n_{s} = n \frac{1 - {r^{'}}_{s}}{1 + {r^{'}}_{s}} .

From Eqs. (5) and (6), the starting phase thickness φ_st, which is a necessary parameter to calculate target ending phase thickness during one layer deposition, is determined. If a layer with refractive index n and phase thickness δ is deposited on the pre-deposited film, one has

[\begin{array}{l} B^{″} \\ C^{″} \end{array}] = [\begin{matrix} \cos δ & \frac{i}{n} \sin δ \\ i n \sin δ & \cos δ \end{matrix}] [\begin{matrix} 1 \\ α + i β \end{matrix}] .

Substituting Eq. (1) into Eq. (8) gives

[\begin{array}{l} B \\ C \end{array}] = [\begin{array}{l} B^{'} B^{″} \\ B^{'} C^{″} \end{array}] = [\begin{matrix} \cos φ & \frac{i}{n} \sin φ \\ i n \sin φ & \cos φ \end{matrix}] [\begin{matrix} 1 \\ n_{s} \end{matrix}],

where,

φ = φ_{s t} + δ,

B = \cos φ + \frac{i n_{s}}{n} \sin φ,

C = i n \sin φ + n_{s} \cos φ .

After deposition, the admittance of the film structure Y is given by

Y = \frac{C}{B} = \frac{i n \sin φ + n_{s} \cos φ}{\cos φ + \frac{i n_{s}}{n} \sin φ} .

When phase thickness φ = mπ/2 (m is an integer), Y is a real number, i. e.

Y = n_{s^{'}} = \frac{n^{2}}{n_{s}} or Y = n_{s} .

So far, admittances corresponding to two adjacent turning points are parameters n_s′ and n_s. The extrema of reflectance of deposited film with incident medium of air or vacuum are given by

R_{s^{'}} = {(\frac{1 - n_{s^{'}}}{1 + n_{s^{'}}})}^{2} and R_{s} = {(\frac{1 - n_{s}}{1 + n_{s}})}^{2} .

Ignoring absorption in film, the extrema of transmittance of deposited film are obtained as follows:

T_{s^{'}} = \frac{4 n_{s^{'}}}{{(1 + n_{s^{'}})}^{2}} and T_{s} = \frac{4 n_{s}}{{(1 + n_{s})}^{2}} .

As known, reflectance of any optical film in vacuum or air is given by

R = (\frac{B - C}{B + C}) {(\frac{B - C}{B + C})}^{*} .

Substituting Eqs. (11) and (12) into Eq. (17) gives

R = 1 - \frac{1}{\frac{{(1 + n_{s})}^{2}}{4 n_{s}} + \sin^{2} φ [\frac{{(n^{2} / n_{s} + 1)}^{2}}{4 n^{2} / n_{s}} - \frac{{(1 + n_{s})}^{2}}{4 n_{s}}]} .

Then, using Eq. (16), one has

R = 1 - \frac{1}{\frac{1}{T_{s}} + (\frac{1}{2} - \frac{1}{2} \cos (2 φ)) (\frac{1}{T_{s^{'}}} - \frac{1}{T_{s}})} .

A more concise formula to express the relation of optical signal (in this work, it is reciprocal of transmittance 1/T) and phase thickness is given by

\frac{1}{T} = \frac{1}{2} (\frac{1}{T_{s}} + \frac{1}{T_{s^{'}}}) + \frac{1}{2} \cos (2 φ) (\frac{1}{T_{s}} - \frac{1}{T_{s^{'}}}) .

From Eq. (20), a trace diagram is generated, as showed in Fig. 1. Each optical signal datum is represented by a trace point in the circle diagram. The angle between the 1/T axis and the radial direction to the trace point is twice the phase thickness. Before deposition, the starting phase thickness is φ_st, and during the deposition, the realtime phase thickness is φ. According to Eq. (10), it is easy to calculate the actually deposited phase thickness δ. The starting transmittance and ending transmittance during the deposition are represented by T_st and T_end.

Fig. 1 The proposed trace diagram to assist optical monitoring.

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When two turning points appear in the monitoring curve, it means that the extrema of optical signal 1/T_s and 1/T_s′ are obtained and the trace circle diagram can be drawn. At turning points, admittance values Y (n_s and n_s′) are real and can be calculated from extrema of optical signal, so one has

Y = {\begin{matrix} (1 + \sqrt{1 - T}) / (1 - \sqrt{1 - T}) (Y > 1) \\ (1 - \sqrt{1 - T}) / (1 + \sqrt{1 - T}) (Y < 1) \end{matrix} .

According to Eq. (14), the practical refractive index n of deposited layer can be calculated, and then the geometrical thickness d can be obtained, too.

During the monitoring process, quartz crystal monitoring is applied to keep steady rate of deposition, and the phase thickness is proportional to the deposition time during deposition approximately (maybe not at the beginning of deposition process). There must be a time period, in which the deposition rate is quite steady. During that time period, the experimental signal curve is an ideal cosine curve in the following format.

f (t) = a + b \cos (ω t + φ_{0}) .

The Eq. (22) can be fitted to get parameters a and b, which will be optimized in the next fitting process with more experimental signals involved. Values 1/T_s and 1/T_s′ can be calculated by solving the following equation set:

{\begin{matrix} a = \frac{1}{2} (\frac{1}{T_{s}} + \frac{1}{T_{s^{'}}}) \\ b = \frac{1}{2} (\frac{1}{T_{s}} - \frac{1}{T_{s^{'}}}) \end{matrix} .

Then parameters n and d of the layer can be obtained from Eqs. (10), (14), (20) and (21).

During the monitoring process, if no turning point appears, the fitting results of f(t) may be unstable and the parameters a, b, ω and φ₀ may not be precise enough. In this situation, the derivative function f′(t) should be used as an important monitoring signal, which is given by

f^{'} (t) = - b ω \sin (ω t + φ_{0}) .

When the optical thickness of deposited layer is bigger than 0.5 quarter-wavelength (qw), there must be an extreme point on the curve of f(t) or f′(t), which makes the fitting process easy. In the monitoring process, the value of f′(t) is given by

f^{'} (t_{j}) = \frac{\sum_{m = 1}^{N} [f (t_{j + m}) - f (t_{j - m})]}{N (N + 1) Δ t} .

Where, parameter Δt denotes the time interval of measurement. So, there is a lag time NΔt between the curve of f′(t) and f(t). It would not be a big problem, since the variation trend is predictable, as shown in Eq. (24). The deposition will stop at the time when the fitted phase thickness equals to the designed ending phase thickness φ_end, which is given by

φ_{end} = \frac{1}{2} [ω (t + N Δ t) + φ_{0}] .

In some extreme cases, for instance, an ultra thin layer in layer stacks, it is better to just apply quartz crystal monitoring or time monitoring, and it would not affect the final optical properties of optical films significantly due to the good thickness compensation mechanism of our method. The error compensation mechanism will be discussed in the following third paragraph containing Eq. (31).

Δ φ \approx | \frac{\frac{1}{T_{s}} \cot φ}{2 (\frac{1}{T_{s}} - \frac{1}{T_{s^{'}}})} + \frac{\frac{1}{T_{s^{'}}} \tan φ}{2 (\frac{1}{T_{s}} - \frac{1}{T_{s^{'}}})} | \frac{Δ T}{T} + | \frac{\frac{1}{T_{s}} \cot φ}{2 (\frac{1}{T_{s}} - \frac{1}{T_{s^{'}}})} | \frac{Δ T_{s}}{T_{s}} + | \frac{\frac{1}{T_{s^{'}}} \tan φ}{2 (\frac{1}{T_{s}} - \frac{1}{T_{s^{'}}})} | \frac{Δ T_{s^{'}}}{T_{s^{'}}} .

Noticing that T, T_s and T_s′ are directly measured values, and assuming that the relative errors ΔT/T, ΔT_s/T_s and ΔT_s′/T_s′ are same, when φ = mπ ± atan(T_s′/T_s)^0.5, minimal Δφ could be obtained. At turning points, Δφ is given by

Δ φ \approx \sqrt{\frac{1 / T}{| 1 / T_{s} - 1 / T_{s^{'}} |} | \frac{Δ T}{T} |} (T = T_{s} or T_{s^{'}}) .

Conventional optical monitoring signal is transmittance T or reflectance R. Using 1/T as monitoring signal, we have to reconsider the relative noise, which is written as

\frac{Δ (1 / T)}{1 / T} = - \frac{Δ T}{T} = \frac{R}{1 - R} \frac{Δ R}{R} .

During deposition process, substrate thickness is finite. The transmittances of front surface (deposition side) T and back surface T_b of the substrate both affect total transmittance T_total, and the relation of T_total with T and T_b is given by

\frac{1}{T_{t o t a l}} = \frac{1}{T} + \frac{1}{T_{b}} - 1.

Since T_b is known and keeps constant through the deposition process, we can use 1/T_total instead of 1/T as monitoring signal, and the relation of monitoring signal and phase thickness is still same, as given by Eq. (20).

The capability of compensation is a very important characteristic of optical monitoring. And it is why TPM method is widely used in the deposition of Narrow Band-pass Filters (NBF). The monitoring method using reciprocal of transmittance is also quite capable of deposition compensation, which is an intrinsic characteristic. As we have discussed above, our monitoring method is based on the measurement of phase thickness φ, which is calculated from cosine curve of monitoring signal, as given by

φ = m π \pm \frac{1}{2} arc \cos (\frac{\frac{2}{T} - \frac{1}{T_{s}} - \frac{1}{T_{s^{'}}}}{\frac{1}{T_{s}} - \frac{1}{T_{s^{'}}}}) m = 0, 1, 2, \dots .

The calculation process of φ, has no relationship with the thickness error of previous layers. However, according to Eq. (6), the initial phase thickness φ_st depends on the initial admittance Y_st, which has been determined by the previous layer. The monitoring error of previous layer Δφ_pre induces error Δφ_st of the initial phase thickness of the present layer. Since the target ending phase thickness φ during deposition is unchanged, the practically deposited thickness of layer δ will be different from originally designed one. It means that the thickness error Δφ_st of the present layer just compensates the thickness error Δφ_pre of the previous layer. The key of our monitoring method is to keep deposited phase thickness φ_act of each layer same with the theoretically designed one φ_des, and the compensation of layers will be completed, as shown in Fig. 2. Thickness errors in any individual layer are a combination of a compensation of the error in the previous layer together with the error committed in the layer itself, which will be compensated during the next layer deposition. In some sense, TPM method is just one special case of our method, since in TPM process, phase thickness φ of each layer is 0.5mπ (m is an integer). The monitoring curve using reciprocal of transmittance is a cosine curve, which makes arbitrary phase thickness monitoring feasible. So, in each monitoring case where TPM works, the method we proposed works.

Fig. 2 The proposed trace diagram for explanation of the error compensation mechanism. The dash line denotes monitoring curve of theoretically designed layers and the solid line denotes the one of actually deposited layers.

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3. Experiment and results

In this work, NBF with both quarter-wave and non-quarter-wave layers has been fabricated to prove the effectiveness of the monitoring method proposed in this paper. The designed film structure is Substrate | 2.295L1.227HL2HLHLHL2HLHLHL2H1.272L0.434H2L | Air, including 19 layers. Here, set reference wavelength as 2900nm, Ge as H, SiO as L, Si as substrate (n = 3.435 at room temperature, n ≈3.48 at 250°C [15], 3mm thickness). The material research of deposited layers has been done before NBF fabrication. All the films were deposited in an IR box-type coater(ZZS800-3/g, Chengdu Rankuum Machinery Ltd), which was equipped with E-Beam evaporator, electro-thermal evaporator, Hall ion source, crystal monitor(IC5, INFICON) and self-built optical monitoring system with InSb detector.

We deposited a layer of Ge on the sapphire substrate with Physical Vapor Deposition (PVD) method of E-Beam Evaporation (EBE) in the vacuum environment (background pressure < 1.0 × 10⁻³ Pa) at temperature of 250°C. The monitoring wavelength was 2900nm, at which Ge layer is approximately transparent. The monitoring curve is showed in Fig. 3. The thermal-optical coefficient of sapphire substrate is very small (~10 × 10⁻⁶ K⁻¹) [16], and the refractive index could be set as 1.71 at wavelength of 2900nm from room temperature to 250°C. Using the signal data from in situ measurement as showed in Fig. 3, the total optical thickness of Ge layer is easily determined as 7.12 qw at 2900nm. The ex-situ transmittance measurement of deposited Ge layer was performed in Fourier Transform Infrared (FTIR) spectrometer (PerkinElmer, Spectrum GX) from 6000cm⁻¹ to 2200cm⁻¹ per 8 cm⁻¹ at room temperature. The optical thickness of Ge layer at room temperature was 6.89 qw at 2900nm, smaller than in situ measurement results, for its large thermal-optical coefficient (4.5~5.7 × 10⁻⁴ ± 0.5 × 10⁻⁴ K⁻¹, room temperature to 250°C) [15]. It means that the refractive index of Ge has changed −3.23% from deposition temperature to room temperature. The fitting of ex-situ transmittance measurement shows that the complex refractive index of Ge at room temperature is 4.189-i0.001552, and the geometrical thickness is 1192nm. The fitting result of the ex-situ measurement was showed in Fig. 4, with root-mean-square error (RMSE) as 0.189. Then, with geometrical thickness unchanged and ignoring extinction coefficient, the calculated refractive index of Ge at 250°C during deposition process is 4.331. The calculated average thermal-optical coefficient of Ge layer is 6.16 × 10⁻⁴ K⁻¹, larger than that of reference [15]. The error may originate from ignoring temperature up of sample during the deposition and ignoring the inhomogeneity of refractive index in Ge layer.

Fig. 3 The proposed trace diagram of Ge layer deposited on sapphire substrate. The black line denotes reciprocal of transmittance and the red line denotes derivative curve derived from Eq. (25) with (N + 1)Δt ≈3.6 s.

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Fig. 4 Fitting result of ex-situ FTIR measurement of Ge layer deposited on sapphire substrate using Cauchy dispersion model.

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Material research of SiO has also been done in both the ways of in situ monitoring and ex-situ FTIR measurement, and little change in optical thickness has been observed. The layer of SiO was deposited on silicon substrate in the method of thermal evaporation of bulk SiO in the vacuum environment (background pressure < 1.0 × 10⁻³ Pa) at temperature of 250°C. The monitoring curve of SiO layer deposition is showed in Fig. 5. Obviously, the SiO layer is homogeneous except the first quarter-wave thickness, and that is why we chose 2.295L not 0.295L as first layer. The measured refractive index of SiO is about 1.76 and be set constant from room temperature to 250 °C.

Fig. 5 The proposed monitoring curve of SiO layer deposited on silicon substrate.

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As seen in the trace diagrams of Ge and SiO layers deposition, as showed in Fig. 3 and Fig. 5, the proposed monitoring method is to draw the circle, and the trace points representing different thicknesses would be distributed on the circle symmetrically. It means that the values of reciprocal of transmittance at two turning points need to be obtained, whether from measured data or fitting results. In the other way, once two turning values of reciprocal of transmittance is obtained, the phase thickness can be easily calculated using Eq. (31), instead of drawing a circle diagram. Signal-Noise-Ratio (SNR) is a very key characteristic for an optical monitoring system, and high SNR ensures the measured signal 1/T and its 1st derivative value versus time d(1/T)/dt reliable. For layer thicker than 0.5 qw, at least one feature point on the curve of d(1/T)/dt exists, which could be zero point or extreme point, and higher SNR makes the fitting process of cosine function easier.

Before NBF deposition, the sensitivity of each layer was analyzed, and it shows that the especially sensitive layers include 5th and 11th layers, but not include 17th layer. According to Eqs. (27) and (28), the monitoring error around the turning point is much greater than that at other point. For the 17th layer with non-quarter wave thickness, the sensitivity of the layer has been reduced. In order to demonstrate how the random thickness errors affect the spectral performance of the designed NBF, we ran simulation by introducing thickness error to each layer. Simulation was performed with an optical film designing software Essential Macleod (Version 9.5.390, Thin Film Center, Inc., Tucson, USA). Figure 6 shows the effect of random error in layer thickness with standard deviation 0.1%, 0.3%, 0.5% and 1%, respectively. According to Eq. (28), the monitoring thickness error of each individual layer is greater than 1% at the turning point. Obviously, without error compensation made in certain layers, it is difficult to successfully fabricate the 19 layers NBF. The successful experiment for fabrication of NBF shows the error compensation of the proposed method is powerful.

Fig. 6 The effect of random error in layer thickness of standard deviation: (a) 0.1%, (b) 0.3%, (c) 0.5% and (d) 1%. Ten lines of different colors represent ten sets of independent random thickness errors.

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The NBF was deposited on intrinsic silicon substrate in the vacuum environment (background pressure < 1.0 × 10⁻³ Pa) at deposition temperature of 250°C. The monitoring wavelength was set as 2900nm with resolution of ~10nm. In designing process, the refractive index of Ge at 2900nm wavelength was set as 4.19 at room temperature and 4.33 at deposition temperature. The refractive index of SiO was set as 1.76 for most SiO layers except the first quarter-wave layer (n = 1.73) adjacent to substrate. The monitoring curve is showed in Fig. 7, using reciprocal of transmittance as monitoring signal. Since the film structure includes quarter-wave and non-quarter-wave layers, the traditional TPM method is not suitable in the monitoring process with only the monochromatic light at the reference wavelength. In this work, the monitoring signal can be easily transformed to phase thickness φ with Eq. (31), which makes the monitoring of quarter-wave and non-quarter-wave layers both very easy. For layers with phase thickness φ = 0.5mπ (m is an integer), the derivative curve obtained from Eq. (25) was used to determine turning point. For layers with phase thickness φ ≠ 0.5mπ, such as the 1st and 17th layers, designed phase thickness φ was used to calculate termination signal value. Quartz crystal monitoring was used for deposition rate control in the deposition process of all layers, and thickness control for the 7th and 13th layers where the optical monitoring signal changed slightly with thickness.

Fig. 7 Monitoring curve of optimally designed NBF.

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After deposition, the in-situ transmittance spectra of NBF in the vacuum chamber were obtained in the range of 2600-3300 nm per 10 nm at temperature of 250°C immediately and at ~20 °C on the next day, as showed in Fig. 8. A blue shift of the spectrum is observed, and it is due to the change of the refractive index of Ge layer from 250°C to 20 °C. The SEM photograph of the deposited NBF sample is showed in Fig. 9. The thickness of quarter-wave layers of same material (H or L layer) in the picture is almost same, which indicates that the deposited thickness is in accordance with the designed one. The SEM photograph also indicates that the microstructure of NBF is compact.

Fig. 8 In situ measurement of transmittance spectra of NBF after deposition at different temperatures.

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Fig. 9 The SEM photograph of cross-section structure of deposited NBF.

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4. Conclusion

A concise formula has been derived to express the relation of optical monitoring signal and deposited phase thickness of layer based on the theory of optical film characteristic matrix. A trace diagram was proposed to assist optical monitoring. During deposition, the refractive index and thickness of each deposited layer can be calculated by performing cosine curve fitting on the experimental signal curve. The effectiveness of the proposed method has been-proved by monitoring the deposition of Ge layer, SiO layer and NBF. This method is intuitionistic and easy for operators to know detailed information during deposition and to have better control for deposition. It is suitable for monitoring deposition of both quarter-wave and non-quarter-wave layers, and may be applied in precise deposition control.

Acknowledgments

This work has been financially supported by the National Natural Science Foundation of China (No. 61275160), the STCSM project of China with the Grant No. 13ZR1463700, 12XD1420600.

References and links

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An optical monitoring method for depositing dielectric layers of arbitrary thickness using reciprocal of transmittance

Abstract

1. Introduction

2. Principle and method

3. Experiment and results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Equations (31)

Optics Express