Comparative study of optical analysis methods for thin films

Journal Pre-proof Comparative study of optical analysis methods for thin films Jaegang Jo, Eilho Jung, Jin Cheol Park, Jungseek Hwang PII:

S1567-1739(19)30294-9

DOI:

https://doi.org/10.1016/j.cap.2019.11.010

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CAP 5100

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Current Applied Physics

Received Date: 20 July 2019 Revised Date:

3 October 2019

Accepted Date: 12 November 2019

Please cite this article as: J. Jo, E. Jung, J.C. Park, J. Hwang, Comparative study of optical analysis methods for thin films, Current Applied Physics (2019), doi: https://doi.org/10.1016/j.cap.2019.11.010. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V. on behalf of Korean Physical Society.

Comparative Study of Optical Analysis Methods for Thin Films Jaegang Jo1, Eilho Jung1, Jin Cheol Park2, and Jungseek Hwang1,* 1

Department of Physics, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do, 16149, Republic of Korea 2

Department of Energy Science, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do, 16149, Republic of Korea

Abstract Three popular optical analysis methods (the transfer-matrix method, the Tinkham formula, and Beer’s law) have been used for analyzing the optical spectra of thin films. While the transfer-matrix method is an accurate method, the Tinkham formula and Beer’s law are approximate methods. Here we investigated the three methods using measured transmittance spectra of insulating transition-metal dichalcogenide (TMD) thin films on a quartz substrate. Three different semiconducting 2H-TMD systems (MoS2, MoSe2, and MoTe2) were measured and analyzed. The optical conductivities obtained from the measured transmittance spectra using the transfer-matrix method and Tinkham formula and the absorption coefficients obtained using the transfer-matrix method and Beer’s law were compared. The comparisons show some discrepancies. The reasons for the discrepancies between the results obtained via the two different methods were examined and the application limitations of the Tinkham formula and Beer’s law were discussed.

Corresponding Author *E-mail: [email protected]

1. Introduction The interactions between electromagnetic radiation and condensed matter have been extensively used for investigating the electronic and phononic properties of materials. Electromagnetic radiation in a wide frequency range (from microwave to ultraviolet (UV)) has been used for examining the optical characteristics of materials [1]. In spectroscopic studies of thin films, a substantially modified mathematical description is required compared with the spectroscopic studies of bulk material systems. Because the range of thickness for a thin film is usually a few nanometers to a few micrometers, a millimeter-size film can be considered to extend to almost infinity in the lateral dimensions [2]. These thin films are usually too thin to be freestanding; they should be placed on an optical transparent substrate in the frequency range of interest. In this study, we investigated three analysis methods for determining the optical properties of transition-metal dichalcogenide (TMD) materials in thin films in a wide spectral range of mid-infrared (MIR) to UV and compared the resulting optical properties. The three analysis methods were the Tinkham formula, Beer’s law, and the transfer-matrix method. While the Tinkham formula and Beer’s law are approximate methods, the transfer-matrix method is a rigorous method. Various optical constants, including the optical conductivity and the absorption coefficient, can be reliably obtained using the transfer-matrix method. However, using the Tinkham formula and Beer’s law, respectively, the optical conductivity and the absorption coefficient can be reliably obtained. Therefore, for comparison of the Tinkham formula (Beer’s law) and the transfer-matrix method, the optical conductivity spectra (absorption coefficient spectra) were used. Thin-film semiconducting 2H-MoS2, 2H-MoSe2, and 2H-MoTe2 were prepared on a quartz substrate. The thin films were 4-, 10-, and 16 nm-thick 2H-MoTe2, 4-nm-thick 2H-MoS2, and 4-nm-thick MoSe2. Transmittance spectra of these test samples (films on the substrate) and

(separately) the bare quartz substrate were measured in a wide spectral range of mid-infrared (MIR) to UV at room temperature. First, the measured transmittance spectra of the materials were analyzed using the transfer-matrix method, which has been known as an exact method. The resulting optical constants were comparable to the published values for similar materials [4–6], indicating that the transfer-matrix method is reliable. The optical conductivity spectra of the thin-film samples were obtained using the Tinkham formula and their absorption coefficient spectra were obtained using Beer’s law. The optical spectra obtained via the three methods (the transfer-matrix method, the Tinkham formula, and Beer’s law) were compared. The optical conductivity spectra obtained via the transfer-matrix method and Tinkham formula were compared and the absorption coefficient spectra via the transfer-matrix method and Beer’s law were compared. On the basis of these comparisons, the application limitations of the Tinkham formula and Beer’s law were discussed.

2. Experiments A. Sample preparation and optical measurement A non-crystallized Mo thin film was deposited on a 1-mm-thick double-polished quartz substrate using an electron-beam evaporation technique. To fabricate an MoTe2 film, Te powder and a Mo thin film were separately mounted in two-zone chemical vapor deposition equipment. The phase of MoTe2 (2H or 1T’) was determined by the ratio of Mo atoms to Te atoms. To control the Te ratio and obtain 2H-MoTe2, the temperatures of the two zones were separately controlled. Both zones were heated to 650

, and Ar and H were used as carrier

gases [7]. The thickness of the MoTe2 film was controlled by changing the thickness of the deposited Mo layer. Similarly, MoS2 and MoSe2 films [8] on quartz substrates were prepared. A total of five samples were prepared: a 4-nm-thick MoS2 sample (4-nm-thick MoS2 thin film on a quartz substrate); a 4-nm-thick MoSe2 sample; and 4-, 10-, and 16-nm-thick MoTe2

samples. The transmittance spectra of all the samples in a wide spectral range of MIR to UV (5000–50,000 cm-1) were measured by using a Lambda 950 UV/Vis spectrophotometer (Perkin-Elmer, USA). All measurements were performed at room temperature (300 K). The transmittance spectra of the quartz substrate were separately measured.

B. Three thin-film analysis methods 1) Tinkham formula The Tinkham formula [9] were used for obtaining the optical conductivity spectra of the thin films from the measured transmittance spectra. For a very thin film on a thick substrate ( ≪

and

≪ , where

,

, and

represent the thicknesses of the film and the

substrate and the wavelength, respectively), the Tinkham formula is a reasonable approximation method to determine the optical conductivity from the ratio of the transmittance spectrum of a thin film on a substrate to that of the substrate. It can be expressed as

=

/

=

!

# " $

%&'

(

, where

)*+),

and

-./0/

)*+),

represent the transmittance spectra of the substrate and the film sample (film/substrate), respectively, 12 (377 Ω) represents the impedance of vacuum, n represents the reflective

index of the substrate, and 34 ≡ 3 + 73

represents the complex optical conductivity of

the film. A way how to derive the Tinkham formula was introduced in the Supplementary material. As mentioned previously, the Tinkham formula is valid when the phase difference through the film is very small; i.e., the thickness of the film is much smaller than the wavelength ( ≪ ). It is applicable for low-frequency spectra (e.g., microwave and infrared regions) [10, 11]. Because there is one formula but two unknowns (the real and imaginary parts of the optical conductivity), only when either the real or imaginary part is dominant the dominant quantity can be obtained by using the Tinkham formula. For example, for metallic

samples, because the real part of the optical conductivity is dominant in the low-frequency region, the real part can be obtained from the measured transmittance spectra using the Tinkham formula [12]. In a superconducting state, because the imaginary part of the optical conductivity in the low-frequency region is dominant, the imaginary part can be obtained from the measured transmittance spectra [13]. In this study, the applicability of the Tinkham formula for obtaining the real part of the optical conductivity of semiconducting film samples were investigated. The transmittance spectra of both the sample (film/substrate) and the substrate were measured and

ω were

obtained. The film thickness, d, was determined via atomic force microscopy, and the refractive index of the substrate, n, was taken from previous reports [14, 15]. The imaginary part of the optical conductivity was assumed small compared with the real part, as explained in the Discussion section. Under this assumption, the real part of the optical conductivity can be expressed as 3

:

≈ ;

$"

<=1/

− 1 @.

(1)

The real part of the optical conductivity of each thin film was obtained from the measured transmittance using the Tinkham formula. As discussed later in the paper, the validity of the aforementioned assumption was examined by comparing the optical conductivity spectra obtained using the assumption with those obtained using the transfer-matrix method. 2) Beer’s law Beer’s law, which is also known as the Beer–Lambert Law, implies that the intensity of transmittance decreases exponentially as the depth from the surface of an absorptive sample increases. This law is widely used to analyze the transmittance spectra of thin films [16, 17]. It can be used for obtaining the optical absorption coefficient (or the optical gap) from the measured transmittance of an insulating (or semiconducting) freestanding film with a known thickness [18]. It can also be used for thin-film samples on an optically transparent substrate.

The absorption coefficient, α, of a thin film with a thickness d can be obtained from the ratio of transmittance spectrum of the thin-film sample (film/substrate) to that of the substrate using Beer’s law [19], which can be described as A Here,

-./0/

)*+),

and

)*+),

=−

BCD

"

E

= − ln H "

I7JK/LMNLOPQOR

S.

represent the transmittance spectra of the thin-film

sample (film/substrate) and the substrate, respectively. In this formula, multiple reflections between two parallel surfaces of the thin film and the single bounce reflectance from the sample surface are not considered. However, this approximate Beer’s law has been commonly used to extract the optical gaps of thin films on optically transparent substrates. Because the absorption coefficient obtained using Beer’s law is roughly approximated, it may differ from that obtained via the transfer-matrix method. This difference will be discussed in further detail later in the paper by comparing the absorption coefficient spectra obtained using Beer’s law with those obtained using the transfer-matrix method. 3) Transfer-matrix method Transfer matrices [20] were used to model the optical properties of thin multilayered samples. Each transfer matrix contains the optical information of each layer. This model approach can be used to obtain optical constants from the measured optical spectra of thin multilayered samples; in this paper this approach will be called the “transfer-matrix method.” Because the transfer-matrix method includes all the multi-reflections at all the interfaces, in principle, it provides true and correct optical properties or optical constants, including the optical conductivity and the optical absorption coefficient. Let us assume that incident light is normal to a thin film with thickness

and a complex

index of refraction T4 sandwiched between two semi-infinite media with refraction indices

of T42 and T4 . By applying the boundary conditions of electric and magnetic fields at the 1

1

1

two interfaces, a matrix equation, UT4 V + U−T4 V P = W UT4 V O , can be obtained, where P 2 2

and O

represent the reflectance and transmittance coefficients of the single film,

respectively, and W represents the transfer matrix, which can be described as W ≡ X

cos \

−7T4 sin \ a b

δ = ` c T4

. :'

− 4 sin \ cos \

^ , where \

is the phase difference to proceed the film i.e.,

. After solving for P and O , the reflectance (d ) and transmittance (

film can be obtained as d ≡ P P

∗

and

≡

:4( O :4$

) of the

O ∗ . Here, the asterisk represents the

complex conjugate. This procedure can be extended to an N-layered multilayer system with a known thickness and the complex refraction index of each layer. The transfer matrix (W ) of the system is a product of the transfer matrices of all layers, i.e., W = W W ⋯ W. ⋯ Wg ,

where W. represents the transfer matrix of the ith layer. The reflectance and transmittance of

the multilayer system can be determined by using W . The complex refraction index of the ith layer, T4. , has the following relationship with the complex permittivity (h̃. ): T4. ≡ =h̃. .

The complex permittivity can be described by using the Drude-Lorentz model [21]. The Drude and Lorentz models can be used to describe unbound (or free charge carriers) and bound electrons, respectively. The complex permittivity of an insulting (or semiconducting) layer can be expressed with the Lorentz model alone as h̃

= hj + ∑p

( mn

( lm ( n.

om

, where

hj represents the high-frequency background permittivity. The second term is the Lorentz dielectric function, which can be used to describe contributions of bound carriers and phonons. Here,

p,

qp , and

rp

represent the resonant frequency, damping constant, and

plasma frequency (or oscillator strength), respectively, of the jth Lorentz absorption mode [21, 22]. The measured transmittance spectrum of our sample, which consisted of a two-layered (TMD/quartz) system, can be analyzed by using the transfer-matrix method. To determine the optical properties of each layer, the measured transmittance spectra of both the bare quartz

substrate and the TMD/quartz were needed. For obtaining good-quality fits to the transmittance spectra using the transfer-matrix method, a least-squares process [23–25] was used to adjust the fitting parameters (

p,

qp ,

rp ,

and sj ) in the Lorentz model. With the

resulting fitting parameters, the complex permittivity was obtained by using the Lorentz model. Other optical constants could be obtained by using the relationships between the optical constants [26, 27]. For example, to determine the optical conductivity 34

permittivity, a relation, 34

= 7 ta Dhj − h̃

from the

E, could be used.

3. Results A. Transmittance spectra and analysis using transfer-matrix method The optical constants of a TMDC thin film were obtained from the measured spectra of the thin film on a quartz substrate by using the transfer-matrix method, as follows. First, a set of fitting parameters for the quartz substrate was obtained by fitting the measured transmittance of the bare quartz substrate. Then, another set of fitting parameters was obtained for the TMDC thin film separately by analyzing the transmittance of the TMDC/quartz sample. In this process, the set of substrate fitting parameters for the substrate layer was used. The thickness of the thin film was fixed as a known value. The high-frequency background permittivity, hj , of the film was set as a previously reported value [4]. The remainder of the fitting parameters were set as free ones for the least-squares process. The fits of all the samples, including the bare quartz substrate, are shown in Fig. 1 with the measured transmittance spectra. All six fits were similar and good-quality. From the fitting parameters of the Lorentz model, the optical conductivity of each TMD thin film was obtained by using the relationship between the optical conductivity and the permittivity [σ

=

/4w h

].

Fig. 1. Measured transmittance spectra [

] of all five samples (film/substrate) with

respect to the substrate, as well as the transmittance spectrum of the quartz substrate, are shown in solid black lines. The green dashed lines are the corresponding fits. The same quartz plate was used as the substrate for all the thin films.

B. Optical conductivity obtained using Tinkham formula As shown in Fig. 2, the optical conductivity spectra of all five TMD thin-film samples, which were obtained using the Tinkham formula of Eq. (1), were compared with those obtained using the transfer-matrix method. For the 4-nm-thick 2H-MoS2 sample [Fig. 2(a)], the two optical conductivity spectra agree well up to 29000 cm-1. Above 29000 cm-1, the optical conductivity obtained using the Tinkham formula was higher than that obtained using the transfer-matrix method. A similar result was observed for the 4-nm-thick 2H-MoSe2 sample [Fig. 2(b)], except that the characteristic frequency was 31000 cm-1 instead of 29000

cm-1. The optical conductivity spectra of MoS2 and MoSe2 obtained using the transfer-matrix method were compared with available optical conductivity data in literate [28]; the obtained results were reasonably consistent with the data in the literature. For all three 2H-MoTe2 samples [Figs. 2(c)–2(e)], the optical conductivity spectra obtained using the Tinkham formula were higher than those obtained using the transfer-matrix method in the lowfrequency region of approximately <20000 cm-1, depending on the thickness of the thin film. Around 20000 cm-1, the conductivity spectra obtained using the two methods were almost identical. Above roughly 25000 cm-1, the optical conductivity spectra obtained using the Tinkham formula were again higher than those obtained using the transfer-matrix method, again depending on the thickness of the thin film. The characteristic frequencies were dependent on the thickness of the thin film; as the film thickness decreased, the two optical conductivity spectra obtained using the two methods agreed in a wider spectral range.

Fig. 2. Optical conductivity spectra of all five thin-film samples. The dashed red and solid dark blue lines represent the optical conductivity spectra obtained using the Tinkham formula and the transfer-matrix method, respectively.

The optical conductivity spectra obtained using the Tinkham formula were higher than those obtained using the transfer-matrix method in frequency regions both lower and higher than 20000 cm-1. Additionally, as the film thickness increased, the two optical conductivity spectra deviated more significantly in both the low- and high-frequency regions. In the Tinkham formula, the phase difference of light to process the thin film is assumed to be approximately zero. The phase difference is the product of the thickness of the film d, the angular frequency of the light a:" b

, and the refractive index of the film material n (∆y =

: " z

=

). As the film thickness increases and the frequency (wavelength) becomes higher

(shorter), the phase difference becomes larger. Consequently, the optical conductivity obtained using the Tinkham formula was much higher than the correct conductivity obtained using the transfer-matrix method in the high-frequency region, as shown in Fig. 2. Additionally, as mentioned previously, the imaginary part of the optical conductivity was assumed to be negligible. However, when the imaginary part is significant comped with the real part, the optical conductivity obtained using the Tinkham formula of Eq. (1) is higher than the true value obtained using the transfer-matrix method. We discuss on this issue more in detail in the Discussion section.

C. Absorption coefficient obtained using Beer’s law The absorption coefficient spectra of all five thin films were obtained by using both Beer’s law and the transfer-matrix method. Fig. 3 shows the absorption coefficient spectra of all five thin films obtained using the two methods (Beer’s law and transfer-matrix method), for

comparison. For all five thin films, over the entire measured frequency range, the absorption coefficient spectra obtained using Beer’s law did not agree with those obtained using the transfer-matrix method. The absorption coefficient spectra of MoS2 and MoSe2 obtained using the transfer-matrix method were compared with available absorption coefficient spectra in literatures [6,29,30]; they reasonably agreed with the data in the literatures. The amplitude of the absorption coefficient obtained using Beer’s law was much higher than that obtained using the transfer-matrix method in the whole frequency region. However, interestingly, the peak positions in the two absorption spectra were located at very similar frequencies. For the 4-nm-thick 2H-MoS2 and 2H-MoSe2 thin-film samples, the absorption coefficient spectra obtained using Beer’s law were higher in the whole frequency region. Additionally, for the three 2H-MoTe2 thin-film samples, the absorption coefficient spectra obtained using Beer’s law were higher than those obtained using the transfer-matrix method, but the two sets of spectra were close to each other near 50000 cm-1. For the 10-nm-thick 2H-MoTe2 sample, the two sets of spectra come across near 50000 cm-1.

Fig. 3. Absorption coefficient spectra of all five thin-film samples. The dashed magenta and solid blue lines represent the absorption coefficients obtained using Beer’s law and the transfer-matrix method, respectively.

In Beer’s law, reflections from the interface between the film and the substrate and multiple reflections inside the film were not included. However, the total intensity of the incident beam should be the sum of the reflectance, transmittance, and absorbance of the film. To obtain a more accurate absorption coefficient α, the reflectance should be considered, as follows [18]: A

= −ln H

n|

S / , where T represents the transmittance, R represents the

reflectance, and d represents the thickness of the thin film. If one uses this formula instead of A

=−

BCD

"

E

as a modified Beer’s law, the resulting absorption coefficient spectra will be

lower. The other issue is related to the multiple internal reflections inside the film. In the transfer-matrix method, the multi-reflections inside both the film and the substrate were completely considered. If the multiple internal reflectance in the film in Beer’s law was included, the optical path through the film becomes longer. Consequently, the effective thickness of the film becomes larger than the actual thickness ( ), resulting in an absorption coefficient smaller than that obtained using Beer’s law. This may explain the difference between the absorption coefficient spectra obtained using the two methods (Beer’s law and transfer-matrix method).

4. Discussion For the Tinkham formula of Eq. (1), the imaginary part of the optical conductivity was ignored. However, if the imaginary part was included, the Tinkham formula is expressed as ω = 3 =

! :

;$ "

# " $

%&'

}~1/

(

=

H

#$ ' "%&'S

− H3

Because d is very small for thin-film samples, σ

(

;$

:

#

H (" $ S %&'

(

or

S − 1•.

12 and σ

(2) 12 can be smaller than 1.

When the contribution of σ is much larger than that of σ , Eq. (1) is reliable. However,

when the absolute value of σ is comparable to or larger than σ , the optical conductivity spectra (3 ) obtained using Eq. (1) are higher than those obtained using Eq. (2). One should know σ

to use Eq. (2). However, because both σ

and σ

are unknown, Eq. (2)

practically cannot be used to obtain σ from the measured transmittance spectrum alone.

Fig. 4 shows the real and imaginary parts (σ and σ ) of the optical conductivity of the

16-nm-thick 2H-MoTe2 film obtained using the transfer-matrix method. Around 25000 cm-1, σ had a zero-crossing, and as shown in Fig. 2, the optical conductivity spectra (3 ) obtained

using Eq. (1) and the transfer-matrix method agreed well each other near 25000 cm-1. In other frequency regions, because the absolute value of σ became larger farther away from 25000 cm-1, the optical conductivity spectra (3 ) obtained using Eq. (1) was higher than those obtained using Eq. (2), which were closer to those obtained using the transfer-matrix method. In the high-frequency region (40000–50000 cm-1), where σ

was larger than σ , the

discrepancy between the optical conductivity spectra (σ ) obtained using the two methods [the Tinkham formula (Eq. (1)) and the transfer-matrix method] increased. The results indicate that the Tinkham formula in Eq. (1) form can be applied only when the real part of the optical conductivity (σ ) is dominant, i.e., σ ≫ 3 . As shown in Figs. 2(c)–2(e), the Tinkham formula, Eq. (1), is more applicable for thinner films and in the low-frequency region.

Fig. 4. Complex optical conductivity spectra of the 16-nm-thick 2H-MoTe2 film obtained using the transfer-matrix method. Here, σ and σ represent the real and imaginary parts of the optical conductivity, respectively. Around 25000 cm-1, the imaginary part exhibits a zero-crossing.

Thin films of thickness 100 nm or thicker have been frequently used for many of studies [22,31,32]. Since these thin films are also placed on a substrate these samples can be analyzed by using one of the three methods. As we already mentioned previously, the transfer-matrix method will be the best rigorous one even though this method requires a lot of efforts. However, if someone is interested in estimating the bandgap of the thin film sample, then the Beer’s law can be a useful method even though the amplitude of the absorption coefficient is not completely reliable. If someone is interested in optical properties of a metallic thin film sample in a long-wavelength region, like in far-infrared region, then the Tinkham method can be a reasonable one. However, in general, the transmittance of a good metallic thin film with thickness 100 nm or thicker may be too weak to be measured. In this case reflectance may be more reliable quantity to be measured but the analysis of the measure reflectance may be not very trivial.

5. Conclusion The three popular thin-film analysis methods (the Tinkham formula, Beer’s law, and the transfer-matrix method) were introduced. We measured the transmittance spectra of TMD thin-film samples and obtained the optical constants of the thin films from the measured transmittance spectra using the three analysis methods. Since the optical constants obtained using the transfer-matrix method matched the reported reference spectra the results obtained using this method were considered as the correct spectra and were compared with the spectra obtained using the other two methods. The Tinkham formula and the transfer-matrix method were compared via the optical conductivity spectra [3

] and the Beer’s law and the

transfer-matrix method were compared via the absorption coefficient spectra [A

]. In the

Tinkham formula, Eq. (1), it was assumed that the phase change in the film was small and the

real part of the optical conductivity was dominant. We found that these assumptions caused errors and made the optical conductivity spectra obtained using the Tinkham formula [Eq. (1)] higher than those obtained using the transfer-matrix method. In Beer’s law, the reflections due to the interface between the film and the substrate and multiple-reflections inside the film were excluded. This exclusion caused errors and made the absorption coefficient spectra much higher than those obtained using the transfer-matrix method, as the thickness of the film was smaller than the effective thickness owing to the multiple internal reflections. Our results provide guidelines for appropriate applications of these popular thin-film analysis methods.

Acknowledgements We acknowledge financial support from the National Research Foundation of Korea (NRFK Grant No. 2017R1A2B4007387).

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Comparative Study of Optical Analysis Methods for Thin Films Jaegang Jo1, Eilho Jung1, Jin Cheol Park2, and Jungseek Hwang1,* 1

Department of Physics, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do, 16149, Republic of Korea 2

Department of Energy Science, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do, 16149, Republic of Korea

Abstract Three popular optical analysis methods (the transfer-matrix method, the Tinkham formula, and Beer’s law) have been used for analyzing the optical spectra of thin films. While the transfer-matrix method is an accurate method, the Tinkham formula and Beer’s law are approximate methods. Here we investigated the three methods using measured transmittance spectra of insulating transition-metal dichalcogenide (TMD) thin films on a quartz substrate. Three different semiconducting 2H-TMD systems (MoS2, MoSe2, and MoTe2) were measured and analyzed. The optical conductivities obtained from the measured transmittance spectra using the transfer-matrix method and Tinkham formula and the absorption coefficients obtained using the transfer-matrix method and Beer’s law were compared. The comparisons show some discrepancies. The reasons for the discrepancies between the results obtained via the two different methods were examined and the application limitations of the Tinkham formula and Beer’s law were discussed.

Corresponding Author *E-mail: [email protected]

1. Introduction The interactions between electromagnetic radiation and condensed matter have been extensively used for investigating the electronic and phononic properties of materials. Electromagnetic radiation in a wide frequency range (from microwave to ultraviolet (UV)) has been used for examining the optical characteristics of materials [1]. In spectroscopic studies of thin films, a substantially modified mathematical description is required compared with the spectroscopic studies of bulk material systems. Because the range of thickness for a thin film is usually a few nanometers to a few micrometers, a millimeter-size film can be considered to extend to almost infinity in the lateral dimensions [2]. These thin films are usually too thin to be freestanding; they should be placed on an optical transparent substrate in the frequency range of interest. In this study, we investigated three analysis methods for determining the optical properties of transition-metal dichalcogenide (TMD) materials in thin films in a wide spectral range of mid-infrared (MIR) to UV and compared the resulting optical properties. The three analysis methods were the Tinkham formula, Beer’s law, and the transfer-matrix method. While the Tinkham formula and Beer’s law are approximate methods, the transfer-matrix method is a rigorous method. Various optical constants, including the optical conductivity and the absorption coefficient, can be reliably obtained using the transfer-matrix method. However, using the Tinkham formula and Beer’s law, respectively, the optical conductivity and the absorption coefficient can be reliably obtained. Therefore, for comparison of the Tinkham formula (Beer’s law) and the transfer-matrix method, the optical conductivity spectra (absorption coefficient spectra) were used. Thin-film semiconducting 2H-MoS2, 2H-MoSe2, and 2H-MoTe2 were prepared on a quartz substrate. The thin films were 4-, 10-, and 16 nm-thick 2H-MoTe2, 4-nm-thick 2H-MoS2, and 4-nm-thick MoSe2. Transmittance spectra of these test samples (films on the substrate) and

(separately) the bare quartz substrate were measured in a wide spectral range of mid-infrared (MIR) to UV at room temperature. First, the measured transmittance spectra of the materials were analyzed using the transfer-matrix method, which has been known as an exact method. The resulting optical constants were comparable to the published values for similar materials [4–6], indicating that the transfer-matrix method is reliable. The optical conductivity spectra of the thin-film samples were obtained using the Tinkham formula and their absorption coefficient spectra were obtained using Beer’s law. The optical spectra obtained via the three methods (the transfer-matrix method, the Tinkham formula, and Beer’s law) were compared. The optical conductivity spectra obtained via the transfer-matrix method and Tinkham formula were compared and the absorption coefficient spectra via the transfer-matrix method and Beer’s law were compared. On the basis of these comparisons, the application limitations of the Tinkham formula and Beer’s law were discussed.

2. Experiments A. Sample preparation and optical measurement A non-crystallized Mo thin film was deposited on a 1-mm-thick double-polished quartz substrate using an electron-beam evaporation technique. To fabricate an MoTe2 film, Te powder and a Mo thin film were separately mounted in two-zone chemical vapor deposition equipment. The phase of MoTe2 (2H or 1T’) was determined by the ratio of Mo atoms to Te atoms. To control the Te ratio and obtain 2H-MoTe2, the temperatures of the two zones were separately controlled. Both zones were heated to 650 ℃, and Ar and H2 were used as carrier gases [7]. The thickness of the MoTe2 film was controlled by changing the thickness of the deposited Mo layer. Similarly, MoS2 and MoSe2 films [8] on quartz substrates were prepared. A total of five samples were prepared: a 4-nm-thick MoS2 sample (4-nm-thick MoS2 thin film on a quartz substrate); a 4-nm-thick MoSe2 sample; and 4-, 10-, and 16-nm-thick MoTe2

samples. The transmittance spectra of all the samples in a wide spectral range of MIR to UV (5000–50,000 cm-1) were measured by using a Lambda 950 UV/Vis spectrophotometer (Perkin-Elmer, USA). All measurements were performed at room temperature (300 K). The transmittance spectra of the quartz substrate were separately measured.

B. Three thin-film analysis methods 1) Tinkham formula The Tinkham formula [9] were used for obtaining the optical conductivity spectra of the thin films from the measured transmittance spectra. For a very thin film on a thick substrate (𝑑 ≪ 𝑑𝑠𝑢𝑏 and 𝑑 ≪ 𝜆, where 𝑑, 𝑑𝑠𝑢𝑏 , and 𝜆 represent the thicknesses of the film and the substrate and the wavelength, respectively), the Tinkham formula is a reasonable approximation method to determine the optical conductivity from the ratio of the transmittance spectrum of a thin film on a substrate to that of the substrate. It can be expressed as 𝑇(𝜔) =

𝑇𝑓𝑖𝑙𝑚/𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 (𝜔) 𝑇𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 (𝜔)

=

1 𝑍 ̃ (𝜔)𝑑 0 | |1+𝜎

2

, where 𝑇𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 and 𝑇𝑓𝑖𝑙𝑚/𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒

𝑛+1

represent the transmittance spectra of the substrate and the film sample (film/substrate), respectively, 𝑍0 (377 Ω) represents the impedance of vacuum, n represents the reflective index of the substrate, and 𝜎̃ (≡ 𝜎1 + 𝑖𝜎2 ) represents the complex optical conductivity of the film. A way how to derive the Tinkham formula was introduced in the Supplementary material. As mentioned previously, the Tinkham formula is valid when the phase difference through the film is very small; i.e., the thickness of the film is much smaller than the wavelength (𝑑 ≪ 𝜆). It is applicable for low-frequency spectra (e.g., microwave and infrared regions) [10, 11]. Because there is one formula but two unknowns (the real and imaginary parts of the optical conductivity), only when either the real or imaginary part is dominant the dominant quantity can be obtained by using the Tinkham formula. For example, for metallic

samples, because the real part of the optical conductivity is dominant in the low-frequency region, the real part can be obtained from the measured transmittance spectra using the Tinkham formula [12]. In a superconducting state, because the imaginary part of the optical conductivity in the low-frequency region is dominant, the imaginary part can be obtained from the measured transmittance spectra [13]. In this study, the applicability of the Tinkham formula for obtaining the real part of the optical conductivity of semiconducting film samples were investigated. The transmittance spectra of both the sample (film/substrate) and the substrate were measured and 𝑇(ω) were obtained. The film thickness, d, was determined via atomic force microscopy, and the refractive index of the substrate, n, was taken from previous reports [14, 15]. The imaginary part of the optical conductivity was assumed small compared with the real part, as explained in the Discussion section. Under this assumption, the real part of the optical conductivity can be expressed as 𝜎1 (𝜔) ≈

𝑛+1 [√1/𝑇(𝜔) − 1 ]. 𝑍0 𝑑

(1)

The real part of the optical conductivity of each thin film was obtained from the measured transmittance using the Tinkham formula. As discussed later in the paper, the validity of the aforementioned assumption was examined by comparing the optical conductivity spectra obtained using the assumption with those obtained using the transfer-matrix method. 2) Beer’s law Beer’s law, which is also known as the Beer–Lambert Law, implies that the intensity of transmittance decreases exponentially as the depth from the surface of an absorptive sample increases. This law is widely used to analyze the transmittance spectra of thin films [16, 17]. It can be used for obtaining the optical absorption coefficient (or the optical gap) from the measured transmittance of an insulating (or semiconducting) freestanding film with a known thickness [18]. It can also be used for thin-film samples on an optically transparent substrate.

The absorption coefficient, α, of a thin film with a thickness d can be obtained from the ratio of transmittance spectrum of the thin-film sample (film/substrate) to that of the substrate using Beer’s law [19], which can be described as 𝛼(𝜔) = −

ln[𝑇(𝜔)] 𝑑

=−

𝑇𝑓𝑖𝑙𝑚/𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 (𝜔) 1 ln [ ]. 𝑑 𝑇𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 (𝜔)

Here, 𝑇𝑓𝑖𝑙𝑚/𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 and 𝑇𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 represent the transmittance spectra of the thin-film sample (film/substrate) and the substrate, respectively. In this formula, multiple reflections between two parallel surfaces of the thin film and the single bounce reflectance from the sample surface are not considered. However, this approximate Beer’s law has been commonly used to extract the optical gaps of thin films on optically transparent substrates. Because the absorption coefficient obtained using Beer’s law is roughly approximated, it may differ from that obtained via the transfer-matrix method. This difference will be discussed in further detail later in the paper by comparing the absorption coefficient spectra obtained using Beer’s law with those obtained using the transfer-matrix method. 3) Transfer-matrix method Transfer matrices [20] were used to model the optical properties of thin multilayered samples. Each transfer matrix contains the optical information of each layer. This model approach can be used to obtain optical constants from the measured optical spectra of thin multilayered samples; in this paper this approach will be called the “transfer-matrix method.” Because the transfer-matrix method includes all the multi-reflections at all the interfaces, in principle, it provides true and correct optical properties or optical constants, including the optical conductivity and the optical absorption coefficient. Let us assume that incident light is normal to a thin film with thickness 𝑑1 and a complex index of refraction 𝑛̃1 sandwiched between two semi-infinite media with refraction indices of 𝑛̃0 and 𝑛̃2 . By applying the boundary conditions of electric and magnetic fields at the 1

1

1

two interfaces, a matrix equation, (𝑛̃ ) + (−𝑛̃ ) 𝑟1 = 𝑀1 (𝑛̃ ) 𝑡1 , can be obtained, where 𝑟1 0 0 2

and 𝑡1 represent the reflectance and transmittance coefficients of the single film, respectively, and 𝑀1 represents the transfer matrix, which can be described as 𝑀1 ≡ (

𝑖 𝑛1

cos 𝛿1

− ̃ sin 𝛿1

−𝑖𝑛̃1 sin 𝛿1

cos 𝛿1

), where 𝛿1 is the phase difference to proceed the film i.e., δ1 =

2𝜋

( 𝜆 ) 𝑛̃1 𝑑1. After solving for 𝑟1 and 𝑡1 , the reflectance (𝑅1 ) and transmittance (𝑇1 ) of the film 𝑛̃

can be obtained as 𝑅1 ≡ 𝑟1 𝑟1 ∗ and 𝑇1 ≡ 𝑛̃2 𝑡1 𝑡1 ∗. Here, the asterisk represents the complex 0

conjugate. This procedure can be extended to an N-layered multilayer system with a known thickness and the complex refraction index of each layer. The transfer matrix (𝑀𝑇 ) of the system is a product of the transfer matrices of all layers, i.e., 𝑀𝑇 = 𝑀1 𝑀2 ⋯ 𝑀𝑖 ⋯ 𝑀𝑁 , where 𝑀𝑖 represents the transfer matrix of the ith layer. The reflectance and transmittance of the multilayer system can be determined by using 𝑀𝑇 . The complex refraction index of the ith layer, 𝑛̃𝑖 , has the following relationship with the complex permittivity (𝜀̃𝑖 ): 𝑛̃𝑖 ≡ √𝜀̃𝑖 . The complex permittivity can be described by using the Drude-Lorentz model [21]. The Drude and Lorentz models can be used to describe unbound (or free charge carriers) and bound electrons, respectively. The complex permittivity of an insulting (or semiconducting) 𝜔2

layer can be expressed with the Lorentz model alone as 𝜀̃(𝜔) = 𝜀𝐻 + ∑𝑗 𝜔2−𝜔2𝑝𝑗−𝑖𝜔𝛾 , where 𝑗

𝑗

𝜀𝐻 represents the high-frequency background permittivity. The second term is the Lorentz dielectric function, which can be used to describe contributions of bound carriers and phonons. Here, 𝜔𝑗 , 𝛾𝑗 , and 𝜔𝑝𝑗 represent the resonant frequency, damping constant, and plasma frequency (or oscillator strength), respectively, of the jth Lorentz absorption mode [21, 22]. The measured transmittance spectrum of our sample, which consisted of a two-layered (TMD/quartz) system, can be analyzed by using the transfer-matrix method. To determine the optical properties of each layer, the measured transmittance spectra of both the bare quartz

substrate and the TMD/quartz were needed. For obtaining good-quality fits to the transmittance spectra using the transfer-matrix method, a least-squares process [23–25] was used to adjust the fitting parameters (𝜔𝑗 , 𝛾𝑗 , 𝜔𝑝𝑗 , and 𝜖𝐻 ) in the Lorentz model. With the resulting fitting parameters, the complex permittivity was obtained by using the Lorentz model. Other optical constants could be obtained by using the relationships between the optical constants [26, 27]. For example, to determine the optical conductivity 𝜎̃(𝜔) from the permittivity, a relation, 𝜎̃(𝜔) = 𝑖

𝜔 4𝜋

[𝜀𝐻 − 𝜀̃(𝜔)], could be used.

3. Results A. Transmittance spectra and analysis using transfer-matrix method The optical constants of a TMDC thin film were obtained from the measured spectra of the thin film on a quartz substrate by using the transfer-matrix method, as follows. First, a set of fitting parameters for the quartz substrate was obtained by fitting the measured transmittance of the bare quartz substrate. Then, another set of fitting parameters was obtained for the TMDC thin film separately by analyzing the transmittance of the TMDC/quartz sample. In this process, the set of substrate fitting parameters for the substrate layer was used. The thickness of the thin film was fixed as a known value. The high-frequency background permittivity, 𝜀𝐻 , of the film was set as a previously reported value [4]. The remainder of the fitting parameters were set as free ones for the least-squares process. The fits of all the samples, including the bare quartz substrate, are shown in Fig. 1 with the measured transmittance spectra. All six fits were similar and good-quality. From the fitting parameters of the Lorentz model, the optical conductivity of each TMD thin film was obtained by using the relationship between the optical conductivity and the permittivity [σ1 (𝜔) = 𝜔/4𝜋 𝜀2 (𝜔)].

Fig. 1. Measured transmittance spectra [𝑇(𝜔)] of all five samples (film/substrate) with respect to the substrate, as well as the transmittance spectrum of the quartz substrate, are shown in solid black lines. The green dashed lines are the corresponding fits. The same quartz plate was used as the substrate for all the thin films.

B. Optical conductivity obtained using Tinkham formula As shown in Fig. 2, the optical conductivity spectra of all five TMD thin-film samples, which were obtained using the Tinkham formula of Eq. (1), were compared with those obtained using the transfer-matrix method. For the 4-nm-thick 2H-MoS2 sample [Fig. 2(a)], the two optical conductivity spectra agree well up to 29000 cm-1. Above 29000 cm-1, the optical conductivity obtained using the Tinkham formula was higher than that obtained using the transfer-matrix method. A similar result was observed for the 4-nm-thick 2H-MoSe2 sample [Fig. 2(b)], except that the characteristic frequency was 31000 cm-1 instead of 29000

cm-1. The optical conductivity spectra of MoS2 and MoSe2 obtained using the transfer-matrix method were compared with available optical conductivity data in literate [28]; the obtained results were reasonably consistent with the data in the literature. For all three 2H-MoTe2 samples [Figs. 2(c)–2(e)], the optical conductivity spectra obtained using the Tinkham formula were higher than those obtained using the transfer-matrix method in the lowfrequency region of approximately <20000 cm-1, depending on the thickness of the thin film. Around 20000 cm-1, the conductivity spectra obtained using the two methods were almost identical. Above roughly 25000 cm-1, the optical conductivity spectra obtained using the Tinkham formula were again higher than those obtained using the transfer-matrix method, again depending on the thickness of the thin film. The characteristic frequencies were dependent on the thickness of the thin film; as the film thickness decreased, the two optical conductivity spectra obtained using the two methods agreed in a wider spectral range.

Fig. 2. Optical conductivity spectra of all five thin-film samples. The dashed red and solid dark blue lines represent the optical conductivity spectra obtained using the Tinkham formula and the transfer-matrix method, respectively. The optical conductivity spectra obtained using the Tinkham formula were higher than those obtained using the transfer-matrix method in frequency regions both lower and higher than 20000 cm-1. Additionally, as the film thickness increased, the two optical conductivity spectra deviated more significantly in both the low- and high-frequency regions. In the Tinkham formula, the phase difference of light to process the thin film is assumed to be approximately zero. The phase difference is the product of the thickness of the film d, the angular frequency of the light 𝜔, and the refractive index of the film material n (∆𝜙 = 2𝜋𝑛𝑑 𝜆

𝑛𝜔𝑑 𝑐

=

). As the film thickness increases and the frequency (wavelength) becomes higher

(shorter), the phase difference becomes larger. Consequently, the optical conductivity obtained using the Tinkham formula was much higher than the correct conductivity obtained using the transfer-matrix method in the high-frequency region, as shown in Fig. 2. Additionally, as mentioned previously, the imaginary part of the optical conductivity was assumed to be negligible. However, when the imaginary part is significant comped with the real part, the optical conductivity obtained using the Tinkham formula of Eq. (1) is higher than the true value obtained using the transfer-matrix method. We discuss on this issue more in detail in the Discussion section.

C. Absorption coefficient obtained using Beer’s law The absorption coefficient spectra of all five thin films were obtained by using both Beer’s law and the transfer-matrix method. Fig. 3 shows the absorption coefficient spectra of all five thin films obtained using the two methods (Beer’s law and transfer-matrix method), for

comparison. For all five thin films, over the entire measured frequency range, the absorption coefficient spectra obtained using Beer’s law did not agree with those obtained using the transfer-matrix method. The absorption coefficient spectra of MoS2 and MoSe2 obtained using the transfer-matrix method were compared with available absorption coefficient spectra in literatures [6,29,30]; they reasonably agreed with the data in the literatures. The amplitude of the absorption coefficient obtained using Beer’s law was much higher than that obtained using the transfer-matrix method in the whole frequency region. However, interestingly, the peak positions in the two absorption spectra were located at very similar frequencies. For the 4-nm-thick 2H-MoS2 and 2H-MoSe2 thin-film samples, the absorption coefficient spectra obtained using Beer’s law were higher in the whole frequency region. Additionally, for the three 2H-MoTe2 thin-film samples, the absorption coefficient spectra obtained using Beer’s law were higher than those obtained using the transfer-matrix method, but the two sets of spectra were close to each other near 50000 cm-1. For the 10-nm-thick 2H-MoTe2 sample, the two sets of spectra come across near 50000 cm-1.

Fig. 3. Absorption coefficient spectra of all five thin-film samples. The dashed magenta and solid blue lines represent the absorption coefficients obtained using Beer’s law and the transfer-matrix method, respectively. In Beer’s law, reflections from the interface between the film and the substrate and multiple reflections inside the film were not included. However, the total intensity of the incident beam should be the sum of the reflectance, transmittance, and absorbance of the film. To obtain a more accurate absorption coefficient α, the reflectance should be considered, as 𝑇(𝜔)

follows [18]: 𝛼(𝜔) = −ln [1−𝑅(𝜔)] /𝑑, where T represents the transmittance, R represents the reflectance, and d represents the thickness of the thin film. If one uses this formula instead of 𝛼(𝜔) = −

ln[𝑇(𝜔)] 𝑑

as a modified Beer’s law, the resulting absorption coefficient spectra will be

lower. The other issue is related to the multiple internal reflections inside the film. In the transfer-matrix method, the multi-reflections inside both the film and the substrate were completely considered. If the multiple internal reflectance in the film in Beer’s law was included, the optical path through the film becomes longer. Consequently, the effective thickness of the film becomes larger than the actual thickness (𝑑), resulting in an absorption coefficient smaller than that obtained using Beer’s law. This may explain the difference between the absorption coefficient spectra obtained using the two methods (Beer’s law and transfer-matrix method).

4. Discussion For the Tinkham formula of Eq. (1), the imaginary part of the optical conductivity was ignored. However, if the imaginary part was included, the Tinkham formula is expressed as 𝑇(ω) =

𝜎1 =

1 𝑍 2 ̃ (𝜔)𝑑 0 | |1+𝜎 𝑛+1

𝑛+1 𝑍0 𝑑

=

1 𝑍 2 𝑍 2 [1+𝜎1 𝑑 0 ] +[𝜎2 𝑑 0 ] 𝑛+1 𝑛+1

𝑍

2

0 {√1/𝑇(𝜔) − [𝜎2 𝑑 𝑛+1 ] − 1}.

or

(2)

Because d is very small for thin-film samples, σ1 𝑑𝑍0 and σ2 𝑑𝑍0 can be smaller than 1. When the contribution of σ1 is much larger than that of σ2 , Eq. (1) is reliable. However, when the absolute value of σ2 is comparable to or larger than σ1 , the optical conductivity spectra (𝜎1 ) obtained using Eq. (1) are higher than those obtained using Eq. (2). One should know σ2 to use Eq. (2). However, because both σ1 and σ2 are unknown, Eq. (2) practically cannot be used to obtain σ1 from the measured transmittance spectrum alone. Fig. 4 shows the real and imaginary parts (σ1 and σ2 ) of the optical conductivity of the 16-nm-thick 2H-MoTe2 film obtained using the transfer-matrix method. Around 25000 cm-1, σ2 had a zero-crossing, and as shown in Fig. 2, the optical conductivity spectra (𝜎1 ) obtained

using Eq. (1) and the transfer-matrix method agreed well each other near 25000 cm-1. In other frequency regions, because the absolute value of σ2 became larger farther away from 25000 cm-1, the optical conductivity spectra (𝜎1 ) obtained using Eq. (1) was higher than those obtained using Eq. (2), which were closer to those obtained using the transfer-matrix method. In the high-frequency region (40000–50000 cm-1), where σ2 was larger than σ1 , the discrepancy between the optical conductivity spectra (σ1 ) obtained using the two methods [the Tinkham formula (Eq. (1)) and the transfer-matrix method] increased. The results indicate that the Tinkham formula in Eq. (1) form can be applied only when the real part of the optical conductivity (σ1 ) is dominant, i.e., σ1 ≫ 𝜎2. As shown in Figs. 2(c)–2(e), the Tinkham formula, Eq. (1), is more applicable for thinner films and in the low-frequency region.

Fig. 4. Complex optical conductivity spectra of the 16-nm-thick 2H-MoTe2 film obtained using the transfer-matrix method. Here, σ1 and σ2 represent the real and imaginary parts of the optical conductivity, respectively. Around 25000 cm-1, the imaginary part exhibits a zero-crossing.

Thin films of thickness 100 nm or thicker have been frequently used for many of studies [22,31,32]. Since these thin films are also placed on a substrate these samples can be analyzed by using one of the three methods. As we already mentioned previously, the transfer-matrix method will be the best rigorous one even though this method requires a lot of efforts. However, if someone is interested in estimating the bandgap of the thin film sample, then the Beer’s law can be a useful method even though the amplitude of the absorption coefficient is not completely reliable. If someone is interested in optical properties of a metallic thin film sample in a long-wavelength region, like in far-infrared region, then the Tinkham method can be a reasonable one. However, in general, the transmittance of a good metallic thin film with thickness 100 nm or thicker may be too weak to be measured. In this case reflectance may be more reliable quantity to be measured but the analysis of the measure reflectance may be not very trivial.

5. Conclusion The three popular thin-film analysis methods (the Tinkham formula, Beer’s law, and the transfer-matrix method) were introduced. We measured the transmittance spectra of TMD thin-film samples and obtained the optical constants of the thin films from the measured transmittance spectra using the three analysis methods. Since the optical constants obtained using the transfer-matrix method matched the reported reference spectra the results obtained using this method were considered as the correct spectra and were compared with the spectra obtained using the other two methods. The Tinkham formula and the transfer-matrix method were compared via the optical conductivity spectra [𝜎1 (𝜔)] and the Beer’s law and the transfer-matrix method were compared via the absorption coefficient spectra [𝛼(𝜔)]. In the Tinkham formula, Eq. (1), it was assumed that the phase change in the film was small and the

real part of the optical conductivity was dominant. We found that these assumptions caused errors and made the optical conductivity spectra obtained using the Tinkham formula [Eq. (1)] higher than those obtained using the transfer-matrix method. In Beer’s law, the reflections due to the interface between the film and the substrate and multiple-reflections inside the film were excluded. This exclusion caused errors and made the absorption coefficient spectra much higher than those obtained using the transfer-matrix method, as the thickness of the film was smaller than the effective thickness owing to the multiple internal reflections. Our results provide guidelines for appropriate applications of these popular thin-film analysis methods.

Acknowledgements We acknowledge financial support from the National Research Foundation of Korea (NRFK Grant No. 2017R1A2B4007387).

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: