Determination the Optical Constants of Hafnium Oxide Film by

Spectroscopic Ellipsometry with Various Dispersion Models Weidong Gao∗, Yinhua Zhang, Hongxiang Liu

Institute of Optics and Electronics, Chinese Academy of Sciences

P.O. Box 350 (29), Shuang Liu，Cheng Du, Si Chuan, 610209, P. R. China

ABSTRACT

Optical constants of vacuum-deposited hafnium oxide film (HfO2) from infrared to ultraviolet spectral region (215nm-1700nm) have been determined by variable angle Spectroscopic ellipsometry with Cauchy dispersion model, Sellmeier dispersion model, Cauchy-Urbach dispersion model and Tauc-Lorentz dispersion model, respectively. The optical constants of the HfO2 film which were extracted with the four dispersion models have been compared. The surface roughness layer between HfO2 film and air and the interface layer between the film and the substrate have also been modeled with Bruggeman effective medium approximation (BEMA).

Key Words: HfO2, Film, Spectroscopic ellipsometry, Optical constants

1. Introduction

HfO2 is one of the most important oxide thin-film material for the manufacture of interference coatings due to its high

laser damage threshold and broad spectral range extending from 220 nm to 12μm[1,2]. However, it can also be used in

other optoelectronic devices, such as flexible thin film capacitors, fiber optic waveguides for communication networks, and computer-memory elements [3, 4]. HfO2 is also a promising dielectric material for future metal-oxide-semiconductor field effect transistor technologies due to its hardness, high melting point, thermal and chemical stability, and high dielectric constant [5,6]. Therefore, accurate knowledge of the optical constants of HfO2 film is indispensable for the design and analysis of various optoelectronic devices. In this paper, the optical constants of vacuum-deposited hafnium oxide film have been accurate determined by variable angle Spectroscopy ellipsometry with different dispersion models, the surface roughness layer between hafnium oxide film and air and the interface layer between the film and the substrate have also been modeled with BEMA [7].

2. Experimental

HfO2 film under study was evaporated with HfO2 onto silicon<111> wafer by an electron-gun in an APS1104 chamber, with a base pressure of 6.0×10-4Pa. To increase the packing density of the HfO2 film, ion beam assisted deposition technique has been used. During the deposition process, the temperature of substrate was controlled at 500K, the average

∗ wdgao3699@163.com

5th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Optical Test and Measurement Technology and Equipment, edited by Yudong Zhang, Jose M. Sasian, Libin Xiang, Sandy To,

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deposition rate was about 0.4nm/s, the argon flow rate was kept at 5 sccm and that of oxygen at 7 sccm. Spectroscopic Ellipsometry measurements of HfO2 film were carried out in air at room temperature by 2nm steps at three angles of incidence, φ=55°, 65° and 75° with the Variable Angle Spectroscopic Ellipsometry (VASE, J.A. Woollam Co., Inc.) for the spectral region from 215nm to 1700nm. Data acquisition and evaluation was made using WVASE32 software. The quality of the fit was assessed by the evaluation of the mean-squared error function (MSE):

∑= Δ ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−Δ+⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−

=N

i

icalii

cali

iiMN

MSE1

2

exp

exp2

exp

exp

21

σσψψ

ψ

(1)

Where N represents the number of (ψ, Δ) experimental pairs, M is the number of variable parameters in the model, and σ denotes the standard deviations on the experimental data points. Minimization of MSE was carried out using the Levenberg–Marquardt algorithm [8, 9]. In order to investigated the microstructure of the HfO2 film, conventional θ-2θ X-ray diffraction (XRD) was also carried out on the film surface from 20° to 80° in a Philips X’pert diffractometer, it was found that the vacuum-deposited HfO2 film was amorphous.

3. Results and Discussion

Measured SE data, ψ and Δ, of the HfO2 film evaporated on Si over the spectral range 215-1700nm is shown in Fig. 1. The solid, dotted and dashed lines represent the experimental data of ψ in Fig. 1(a), while the solid, dotted and dashed lines represent the experimental data of Δ in Fig. 1(b) for angle 55°, 65° and 75° respectively. It is noticed that the dispersion of the relative amplitude change, ψ (λ), and the relative phase change, Δ (λ), show angle dependence.

Experimental Data

Wavelength (nm)215 350 485 620 755 890 1025 1160 1295 1430 1565 1700

Ψ in

deg

rees

0

10

20

30

40

50

60

70

80

Exp E 55°Exp E 65°Exp E 75°

(a)

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Experimental Data

Wavelength (nm)215 350 485 620 755 890 1025 1160 1295 1430 1565 1700

Δ in

deg

rees

020406080

100120140160180200

Exp E 55°Exp E 65°Exp E 75°

(b)

Fig. 1. Measured ellipsometric data, ψ and Δ, of vacuum-deposited HfO2 film for the spectral region from 215nm to 1700nm.

To derive the optical constant through the fitting of the measured SE data (ψ and Δ), the HfO2 film deposited on Si might be modeled with two layers and a substrate: surface roughness/HfO2/interfaced layer/Si. The surface roughness (the top layer) was calculated using the BEMA consisting 50% air and 50% HfO2, while the middle layer (HfO2) can be modeled with the Sellmeier, Cauchy, Cauchy-urbach or Tauc-Lorentz model. The interfaced layer was again modeled using a 2-medium BEMA, mixing of 50% HfO2 and 50% silicon.

3.1 Determination of optical constants of HfO2 film with Cauchy and Cauchy-Urbach model

Cauchy dispersion model is most commonly used to describe the spectral dependence of refractive index (n) in transparent region. In this case, the index of refraction can be represented by a slowly varying function of wavelength:

( ) 0,//)( 42 =++= λλλλ kCBAn , (2)

Where A term describes the long-wavelength asymptotic index value, while B and C are the dispersion terms that add upward slope to the index curve as wavelength became shorter, is the wavelength in nanometers and k is extinction coefficient. First, the ψ and Δ spectra was fitted with Cauchy model, but the best fit only resulted in MSE=6.2 above 250nm, and the corresponding film thickness, interfaced layer and surface roughness are 175.89nm, 1.22nm and 1.98nm, respectively. When the spectral region moved down to shorter wavelength, a larger MSE was got and obvious discrepancy between the measured and fitted spectra appeared in the shorter wavelength, this indicated that absorption occurred and the Cauchy model could not fit the experimental data very well in the shorter wavelength . Cauchy-Urbach model is Cauchy term includes Urbach absorption [10], which is used to the exponential absorption tail that occurs below the band gap of many materials,

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=γλ

βαλ 1112400exp)(k (3)

whereα is the extinction coefficient amplitude, βis the exponent factor and γis the band edge.

The measured ψ and Δ spectra were then simulated with the Cauchy-Urbach model from 215nm to 1700nm, as shown in Fig. 2. The fit procedure resulted in the minimum MSE=7.1. The film thickness, interfaced layer and surface roughness

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deduced from the Cauchy-Urbach model were 175.88nm nm, 0.9nm and 2nm nm, respectively. While the corresponding

Urbach absorption parameters, α ,βare 0.044229 and 3.1239, γis set as 5.8eV.

Generated and Experimental

Wavelength (nm)215 350 485 620 755 890 1025 1160 1295 1430 1565 1700

Ψ in

deg

rees

0102030405060708090

100

Model Fit Exp E 55°Exp E 65°Exp E 75°

(a)

Generated and Experimental

Wavelength (nm)215 350 485 620 755 890 1025 1160 1295 1430 1565 1700

Δ in

deg

rees

0

20

40

60

80

100

120

140

160

180

Model Fit Exp E 55°Exp E 65°Exp E 75°

(b)

Fig.2. Measured and fitted (Cauchy-Urbach model) ellipsometric data, ψ and Δ, of vacuum-deposited HfO2 film for the spectral region

from 215nm to 1700nm.

3.2 Determination of optical constants of HO2 film with Sellmeier model

Sellmeier dispersion model is generally used to describe the spectral dependence of refractive index (n) in transparent region (below the band gap). In this model, the extinction coefficient (k) is assumed to be zero, and the refractive index (n) is given by [11],

( )20

222 /)( λλλλ −+= ss BAn (4)

Where λ is the wavelength in nanometer, λ0 is the wavelength of the oscillator, Bs is the oscillator strength and As is the contribution of the ultra-violet term. HfO2 films investigated by P. Torchio [12] were highly transparent above 250nm, though we modeled the HfO2 film with Sellmeier dispersion model above 250nm in our experiment. Through the fitting of SE data with Sellmeier dispersion model, the thickness of HfO2 film is 175.58nm, the Surface roughness layer and the interfaced layer is 2.08nm and 0.82nm, respectively. The best fitting resulted in MSE=3.6, as shown in Fig. 3.

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Generated and Experimental

Wavelength (nm)250 395 540 685 830 975 1120 1265 1410 1555 1700

Ψ in

deg

rees

0

20

40

60

80

100

Model Fit Exp E 55°Exp E 65°Exp E 75°

(a)

Generated and Experimental

Wavelength (nm)250 395 540 685 830 975 1120 1265 1410 1555 1700

Δ in

deg

rees

0

20

40

60

80

100

120

140

160

180

Model Fit Exp E 55°Exp E 65°Exp E 75°

(b)

Fig.3. Measured and fitted (Sellmeier model) ellipsometric data, Psi(ψ) and Delta(Δ), of vacuum-deposited HfO2 film for the spectral

region from 250nm to 1700nm .

3.3 Determination of optical constants of Ta2O5 film with Tauc-Lorentz model

For the HfO2 film evaporated on Si(111) substrate, as studied by XRD in Section 2 is amorphous, it’s optical functions can be described by the Tauc-Lorentz model[13], which recently derived by Jellison and Modine. This model uses a combination of Tauc band edge and the classical Lorentz oscillation function. The imaginary part of the complex dielectric function is given by

( )( )

( ) ( )

( )⎪⎩

⎪⎨

⎧

≤

>+−

−

=

g

gg

EE

EEEECEE

EECAEE

0

12222

02

20

2ε (5)

Where A is the amplitude (it includes the optical transition matrix elements), E0 is the peak transition energy, C is the broadening parameter, and Eg is the optical band gap. The real part of the dielectric function ε1 (E) is determined in a closed form from ε2 (E) using Kramers-kronig integration. The values of A, B, C, E0,, Eg and the thickness for deposited HfO2 layer as well as the thickness of the surface roughness layer and that of the interface layer are all consider as fit parameters. Eight parameters are varied to obtain the best fit as

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characterized by MSE. The fitting procedure resulted in MSE=5.1, indicating that the model fits the data. The dispersions of the refractive index and extinction coefficient can be extracted from the model fit. The optical constants exhibit a strong dispersion and decrease monotonically with increasing wavelength. Refractive indices of the HfO2 film in this study were found to be in the range of 2.486-1.969, while the extinction coefficients only exist below 233nm. This absorption tail could be due to the excitonic effect or electron transitions between the valence bands (and/or conduction bands).

Generated and Experimental

Wavelength (nm)215 463 710 958 1205 1453 1700

Ψ in

deg

rees

0

10

20

30

40

50

60

70

80

Model Fit Exp E 55°Exp E 65°Exp E 75°

(a)

Generated and Experimental

Wavelength (nm)215 463 710 958 1205 1453 1700

Δ in

deg

rees

0

20

40

60

80

100

120

140

160

180

Model Fit Exp E 55°Exp E 65°Exp E 75°

(b)

Fig.4. Measured and fitted (Tauc-Lorentz model) ellipsometric data, Psi(ψ) and Delta(Δ), of vacuum-deposited HfO2 film for the

spectral region from 215nm to 1700nm .

3.4 Comparison of the optical constants of HfO2 film extracted from above dispersion model

Refractive index and the extinction of the HfO2 film obtained from the Cauchy fitting, Sellmeier fitting, Cauchy-Urbach fitting and Tauc-Lorentz fitting in the spectral ranges from 250nm to 1700nm are shown in Fig. 5a, while the refractive index and extinction coefficient of the HfO2 film extracted from the Cauchy-Urbach fitting and Tauc-Lorentz fitting in the spectral ranges from 215nm to 250nm are shown in Fig. 5b. The spectral dependence of the refractive index extracted through the four dispersion model in the spectral region from 250nm to 1700nm is very similar, although there is some difference in the infrared wavelengths. From Fig. 5a, when the spectral range extends from the visible into near infrared, the Cauchy model, which flattens out at long wavelength, while the Sellmeier model pulls the index downwards at longer wavelength, which is more consistent with the refractive index variation tendency of many transparent materials. The

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Tauc-Lorentz results lie between of them and the extinction appears almost all the measured spectral range and increases abrupt below 300nm for the Cauchy-Urbach model fit. Fig. 5b shows the refractive index and the extinction of the HfO2 film from 215nm to 250nm obtained with Cauchy-Urbach fitting and Tauc-Lorentz fitting, clear discrepancy exists in this spectral region. The refractive index obtained from Cauchy-Urbach model is a little bigger than that obtained from Tauc-Lorentz model. The absorption appears only below 233nm in the Tauc-Lorentz model, while the extinction exists almost all the measured spectral range in the Cauchy-Urbach model.

Optical Constants

Wavelength (nm)250 395 540 685 830 975 1120 1265 1410 1555 1700

Inde

x of

refra

ctio

n 'n'

Extinction C

oefficient 'k'

1.96

2.02

2.08

2.14

2.20

2.26

0.0000

0.0003

0.0006

0.0009

0.0012

0.0015

cauchy, nsellmeir, ntauc-lorentz, ncauchy-urbach, ncauchy, ksellmeir, ktauc-lorentz, kcauchy-urbach, k

(a)s Optical Constants

Wavelength (nm)215 220 225 230 235 240 245 250

Inde

x of

refra

ctio

n 'n'

Extinction C

oefficient 'k'

1.90

2.00

2.10

2.20

2.30

2.40

2.50

0.000

0.010

0.020

0.030

0.040cauchy-urbach, ntauc-lorentz, ncauchy-urbach, ktauc-lorentz, k

(b)

Fig. 5 the refractive index and extinction index of HfO2 extracted from the above four dispersion model, (a) from 250nm to 1700nm;

(b) from 215nm to 250nm

4. Conclusions

Optical constants of vacuum-deposited hafnium oxide film have been determined by variable angle Spectroscopy ellipsometry with Cauchy model, Cauchy-Urbach model, Sellmeir model and Tauc-Lorentz model, the surface roughness layer and the interface layer between the film and the substrate have also been modeled with BEMA. It is found that a good consistency of film thickness, surface roughness layer and the interface layer thickness obtained with the four dispersion model, but some discrepancy exist in the optical constants, therefore, to accurately determine the optical

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constants of the HfO2 film with SE，a befitting model is needed in the corresponding spectral range, furthermore, the

optical characteristics of the coating such as transmittance, reflectance and absorption can also be helpful.

References

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York, 1999, P.93 9. K. Levenberg, Q. Appl. Math. 2 (1944) 164. 10. D. Marquardt, SIAM J. Appl. Math. 11 (1963) 431. 11. H.G. Tompkins, W.A. McGahan, Spectroscopic Ellipsometry and Reflectometry, John Wiley & Sons, Inc., New

York, 1999, p. 93. 12. P. Torchio, A. Gatto, M. Alvisi, G. Albrand, N. Kaiser, and C. Amra, Appl. Opt. 41, 3256 (2002). 13. G. E. Jellison, Jr. and F. A. Modine, Appl. Phys. Lett. 69 (1996), 371

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