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Limitations and possible improvements of DLC dielectric response model based on parameterization of density of states
Diamond & Related Materials 18 (2009) 413–418
Contents lists available at ScienceDirect
Diamond & Related Materials j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / d i a m o n d
Limitations and possible improvements of DLC dielectric response model based on parameterization of density of states Daniel Franta ⁎, David Nečas, Lenka Zajíčková, Vilma Buršíková Department of Physical Electronics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
a r t i c l e
i n f o
a b s t r a c t
Available online 5 November 2008 PACS: 71.23.Cq 71.55.Jv 78.20.Ci 78.66.Jg Keywords: Diamond-like carbon Amorphous hydrogenated carbon Optical properties Band structure
Several extensions of previously published PDOS model of DLC dielectric response are tested on a typical DLC film on Si. The extended models are applied to the processing of optical measurements, namely UV/VIS/NIR ellipsometry and spectrophotometry as well as FTIR transmittance spectroscopy. The tested extensions of PDOS approach, that originally took into account only π → π⁎ and σ → σ⁎ interband transitions, are (i) the inclusion of π → σ⁎ and σ → π⁎ transitions, (ii) the addition of transitions from σ to bands above σ⁎, (iii) the removal of the assumption that the bands are symmetrical with respect to the Fermi level. However, it is found that these extensions are not needed for the processing of optical data in the measured range, although they could become important in the VUV and XUV regions. It is shown that the PDOS models can be successfully combined with models of phonon absorption in IR region. The application of the combined model in the wide spectral range allows correct calculation of absorption peaks because the interference and dispersion effects are taken into account. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Diamond-like carbon (DLC) is composed of carbon in two different bonding configurations, sp3 and sp2, and various amount of hydrogen [1–3]. The sp3 bonded carbon has four σ electrons, whereas sp2 bonded carbon has three σ and one π electrons. The hydrogen forms bonds only with σ electrons. Therefore, π and σ bands are distinguished in the electronic structure of DLC films. Dielectric response of DLC material above 1 eV, i.e., in UV, visible and near-IR (UV/VIS/NIR) spectral region is caused by excitations of electrons from occupied to empty states, namely from valence to conduction states. The complete electronic structure of DLC is schematically plotted in Fig. 1. The contribution of the j → k⁎ interband transitions to the imaginary part of the dielectric function εi is given by the following convolution [4–6]: ei;jYk4 ðEÞ =
eh mE
2 jp
jYk4
j2
4πe0 B0
∞
× ∫ N j ðSÞN k4 ðS + EÞdS; −∞
ð1Þ
where E, e, h, m, ε0, B0 and |pj → k⁎|2 are photon energy, electron charge, Planck's constant, electron mass, permittivity of vacuum, volume of certain part of Brillouin zone of corresponding crystalline material and squared momentum-matrix element, respectively. The
⁎ Corresponding author. Tel.: +420 54949 3836; fax: +420 541211214. E-mail address: [email protected] (D. Franta). 0925-9635/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2008.10.037
functions N j(S) and N k (S + E) represent energy distributions of the density of states (DOS) in j occupied and k⁎ empty bands, respectively, where S is the energy of the electronic states with respect to the Fermi level. Not all the transitions have a significant influence on the dielectric function in the UV/VIS/NIR region due to very different transition probabilities and the factor 1/E2 in Eq. (1). In the previously published model of DLC [7–10] based on parameterization of density of states (PDOS) the σ → π⁎ and π → σ⁎ interband transitions were disregarded because of their low probability due to the high localization of π electrons [3]. This model, here referred as the base PDOS model, used also some additional simplifications as listed below. The transitions from the core level and to the excited states above σ⁎ were omitted because their energies lie in VUV and XUV spectral range, far from the measured spectral range. The transitions from the valence bands to higher excited states can be expected for energies above 25 eV while the transitions from the core level to the conduction bands start at 285 eV [11]. Therefore, only π → π⁎ and σ → σ⁎ transitions were considered in the base PDOS model. In addition, the symmetry of valence and conduction bands with respect to the Fermi level was assumed, i.e., the π⁎ and σ⁎ conduction bands are mirror images of the valence π and σ bands, respectively. This paper will investigate the influence of above mentioned simplifications on the interpretation of optical measurements. Furthermore, the PDOS model will be extended with Gaussian absorption peaks describing the phonon absorption in order to interpret transmittance data in IR region below 0.5 eV (4000 cm− 1).
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refractive index n and extinction coefficient k, are calculated from the complex dielectric function as
2. Model of DLC dielectric function 2.1. Base PDOS model The imaginary part of DLC dielectric function is composed of only two terms in case of the base model [7–10]: ei ðEÞ = ei;πYπ4 ðEÞ + ei;σYσ 4 ðEÞ;
ð2Þ
where εi,π → π⁎ and εi,σ → σ⁎ denote the contributions of π → π⁎ and σ → σ⁎ transitions, respectively. The individual contributions are calculated using following convolution ei;jYk4 ðEÞ =
1 ∞ ∫ Nj ðSÞNk4 ðS + EÞdS; E2 −∞
ð3Þ
where j = k = π,σ. Here, the functions N represent the unnormalized density of states [10] that differ from the density of states N because they include the constant factors in front of the integral in Eq. (1). From similarity with DOS distribution in crystals in vicinity of energy extrema of the bands the unnormalized density of valence states N j (S) can be modeled as a parabolic function [10]: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 32Q j −S−Eg j =2 Ehj =2 + S for 2 Nj ðSÞ = π Ehj −Egj > : 0 otherwise:
−Ehj =2bS b−Eg j =2
ð4Þ
The parameters E gj, E hj and Q j are the minimum energy limit of transitions (band gap), maximum energy limit of transitions and parameters proportional to the valence electron densities of the material, respectively. Assuming the symmetry with respect to the Fermi level, the unnormalized DOS distribution of conductive band N j⁎(S) is expressed by a symmetrical function [10]: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 32Q j S−Eg j =2 Ehj =2−S for 2 Nj4 ðSÞ = π Ehj −Eg j > : 0 otherwise:
Eg j =2bSb−Ehj =2
ð5Þ
The parameters Qj are defined by following integrals ∞
∞
−∞
−∞
∫ Nj ðSÞdS = ∫ Nj4 ðSÞdS = Q j :
ð6Þ
The real part of the dielectric function is given by the Kramers–Kronig (KK) relation [12] as er ðEÞ = 1 +
2 ∞ Sei ðSÞ H dS; π 0 S2 −E2
ð7Þ
where unity is the contribution of vacuum. Symbol H denotes the Cauchy principal value of the integral. The optical constants, i.e.,
nðEÞ + ikðEÞ =
pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be ðEÞ = er ðEÞ + iei ðEÞ:
ð8Þ
Although the integrals in Eqs. (3) and (7) have to be calculated numerically today's computers can calculate the dielectric function within 1 ms when using appropriate numerical algorithms. 2.2. Transitions π → σ⁎ and σ → π⁎ The contributions of π → σ⁎ and σ → π⁎ transitions can be calculated using Eqs. (3)–(5) with j ≠ k. If the symmetry of the valence and conduction bands is taken into account it holds ei;πYσ 4 ðEÞ = ei;σ Yπ4 ðEÞ:
ð9Þ
The complete imaginary part of dielectric function is then expressed as ei ðEÞ = ei;πYπ4 ðEÞ + ei;σYσ 4 ðEÞ + 2ei;πYσ 4 ðEÞ:
ð10Þ
Since the probability of π → σ⁎ and σ → π⁎ transitions differs from π → π⁎ and σ → σ⁎, an additional parameter Cπσ expressing the ratio of these probabilities has to be introduced. The contribution εi,π → σ⁎ can be then written in the following form: ei;πYσ 4 ðEÞ =
Cπσ ∞ ∫ Nπ ðSÞNσ 4 ðS + EÞds: E2 −∞
ð11Þ
2.3. High energy transitions The transitions from σ band to the states above σ⁎ modify the dielectric function in the IR/VIS/UV range only through KK relations, i.e. by a slight change in the real part of this function. In order to keep the number of model parameters low it is necessary to find a parameterization that, in ideal case, does not introduce any new parameters. Therefore, the following model of high energy bands was chosen. The r-th band above σ⁎ has the same shape as σ⁎ but it is shifted to higher energies by ΔEr. The contributions of the transitions from σ to the r-th high energy band are calculated as σ → σ⁎ contributions but the Eg and Eh in Eqs. (3)–(5) are replaced with Emin and Emax expressed as follows Emin = Eg + ΔEr
and
Emax = Eh + ΔEr :
ð12Þ
The shift of the first band above σ⁎ was chosen as ΔE1 = Eh/2. Similarly, the shift of the r-th band of higher excited states was ΔEr = r
Eh : 2
ð13Þ
2.4. Asymmetry of bands
Fig. 1. Schematic diagram of electronic structure of the DLC. Shaded area depicts occupied electronic states. Symbol EF denotes the Fermi energy.
Since the dielectric function depends only on the convolution of density of valence and conduction states only one additional parameter is needed for each asymmetric pair of valence and conduction bands. It means that the asymmetric pair is described by four parameters. Three parameters, Egi, Ehj and Qj, have the same meaning as in the base PDOS model and the fourth, Yj, equals to the
D. Franta et al. / Diamond & Related Materials 18 (2009) 413–418
ratio of conduction and valence band widths. Therefore, the DOS of valence band is parameterized as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < 32Q j −S−Eg j =2 Evj =2 + S for 2 Nj ðSÞ = π Evj −Eg j : 0 otherwise:
−Evj =2bSb−Eg j =2;
ð14Þ
2Ehj + Yj −1 Eg j : 1 + Yj
ð15Þ
Similarly for conduction band: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < 32Q j S−Eg j =2 Ecj =2−S for 2 Nj4 ðSÞ = π Ecj −Egj : 0 otherwise:
Eg j =2bSbEcj =2
ð16Þ
where 2Yj Ehj + 1−Yj Eg j : Ecj = 1 + Yj
ð17Þ
2.5. Phonon absorption The phonon absorption is modeled by discrete energy spectrum. Hence, the contribution to the imaginary part of the dielectric function is written ei;p ðEÞ =
1 ∑ Ai Gi ðEÞ; E2 i
ð18Þ
where Ai is parameter proportional to the density of corresponding bonds and probability of photon–phonon interaction. The functions Gi represent the Gaussian absorption peaks. If the antisymmetry of εi is taken into account they have to be written in the following form: Gi ðEÞ = g ðBi ; E−Ei Þ−g ðBi ; E + Ei Þ;
ð19Þ
where Ei and Bi denote the peak energy and broadening factor, respectively. The function g is a normalized Gaussian function 1 E2 g ðB; EÞ = pffiffiffiffiffiffi exp − 2 : 2B 2π B
photometer in spectral range 1.1–1.3 eV (900–1100 nm) and by an FTIR Bruker Vertex 80v spectrophotometer in spectral range 0.045–0.93 eV (370–7500 cm− 1). The Bruker spectrophotometer was used with the parallel beam transmittance accessory for correct absolute transmittance measurement. 4. Data processing
where Evj is calculated as Evj =
415
ð20Þ
3. Experimental The DLC film was prepared by plasma enhanced chemical vapor deposition (PECVD) on double polished silicon single crystal (c-Si) substrate in capacitively coupled glow discharge (13.56 MHz) in constant bias mode from methane. The dc self-bias voltage and flow rate of methane were kept at Ub = −367 V and QCH4 = 1.8 sccm, respectively. The pressure and the rf power corresponding to this flow rate and bias were p = 10.5 Pa and approximately P = 50 W, respectively. The deposition time was 25 min. Prior to the deposition the substrate was cleaned in hydrogen plasma for 5 min (QH2 = 1 sccm, P = 50 W, Ub = −410 V, p = 6–7 Pa). A detailed description and schematic drawing of the reactor is given by Zajíčková et al. [13]. The optical characterization of the DLC film was carried out using variable angle spectroscopic ellipsometry, UV/VIS/NIR spectrophotometry and Fourier transform infrared (FTIR) transmittance spectroscopy. Ellipsometric quantities were measured by an ellipsometer Jobin Yvon UVISEL at five angles of incidence from 55° to 75° within the spectral range 1.2–6.5 eV (190–1000 nm). Reflectances were measured using a spectrophotometer PerkinElmer Lambda 45 within the same spectral range at an angle of incidence of 6°. Transmittance at normal incidence was measured by the PerkinElmer spectro-
The optical measurements described above provided reflectances R, transmittance T and associated ellipsometric parameters Is, Ic,II and Ic,III. The associated ellipsometric parameters represent three coordinates on Poincaré sphere defined as Is = sin 2 Ψ sin Δ, Ic,II = sin 2 Ψ cos Δ and Ic,III = cos 2 Ψ if no depolarization effects is present, where Ψ is azimuth and Δ is phase change. The fitting procedure utilized leastsquares method, i.e., minimized squares of differences between measured optical quantities R, T, Is, Ic,II and Ic,III and their theoretical values calculated by 2 × 2 matrix formalism [14]. First, it was necessary to determine the optical constants of the c-Si substrate in the entire spectral range in order to separate the IR absorption in film and substrate. The model of c-Si included both interband electronic transitions and phonon absorption. However, the modeling of silicon substrate is out of the scope of this article. For correct interpretation of experimental data it was also necessary to include inhomogeneous transition layer in the structural model of DLC film. This transition layer represents the amorphized Si layer originated from ion bombardment during initial stage of the deposition [7,15]. The optical constants of the transition layer were modeled using the six parameter PDOS model [16,17] containing two absorption bands. 5. Results and discussion It was found that if the transitions π → σ⁎ and σ → π⁎ are added to the base PDOS model the parameter Cπσ converges to zero. It means this extended model degenerates to the base model. Therefore, the resulting fit parameters are not shown in this section. In Table 1, the material parameters Q π, Egπ, Ehπ, Q σ, Egσ, Ehσ and Yπ corresponding to the interband transitions calculated by all other PDOS models are summarized. Symbols dfE, dfR and dfT denote the mean film thicknesses in the ellipsometry, reflectometry and FTIR transmittance measurement spots, respectively. The values of remaining fit parameters, characterizing the amorphized transition layer and the DLC film in IR region, will not be discussed in this paper. Besides the fit parameters, there is also quantity χ in Table 1. It quantifies the differences between the experimental and theoretical values of the optical data with respect to their estimated errors. If χ = 1 the fit is optimum, greater values indicate worse agreement [18]. In Fig. 2 the spectral dependences of the associated ellipsometric parameters Is, Ic,II, Ic,III and reflectance R in the UV/VIS/NIR region are plotted. One can see a sound agreement between experimental and
Table 1 The parameters of the PDOS dispersion models corresponding to chosen DLC film High energy transitions
No
Yes
No
Yes
asymmetric bands
No
No
Yes
Yes
Q π [eV3/2] Egπ [eV] Ehπ [eV] Q σ [eV3/2] Egσ [eV] Ehσ [eV] Yπ dFE [nm] dfR[nm] dfT [nm] Χ α = Q π/Q σ
3.81 1.03 7.09 150 1.61 55.3 – 205.9 205.0 204.5 1.79 0.0254
3.73 1.03 7.04 133 1.61 51.1 – 205.8 204.9 204.4 1.81 0.0280
4.19 1.07 7.10 146 1.68 53.9 0.670 205.5 204.6 204.8 1.77 0.0287
4.12 1.07 7.05 130 1.68 49.7 0.665 205.4 204.5 204.7 1.78 0.0318
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D. Franta et al. / Diamond & Related Materials 18 (2009) 413–418
symmetry of integral (3) does not permit to assign widths to the valence and conduction bands specifically, only the degree of asymmetry can be determined). However, the dependence of the sum of squares on Yπ was also quite weak, therefore, this result cannot be considered completely reliable. It can be concluded that these two extension of PDOS model neither improved the fits nor brought much new information about the material structure. In Fig. 3 the spectral dependences of the transmittance T measured by FTIR and conventional spectrophotometer is plotted. It can be seen that the PDOS model, extended with the model of phonon absorption below 0.5 eV (4000 cm− 1), can be successfully applied to optical characterization in the entire spectral range from IR to UV. The simultaneous processing of optical data in UV/VIS/NIR and IR spectral ranges allowed to calculate directly the dielectric function of the film also in the IR region. It supersedes often used inaccurate procedure of film interference subtraction, so called background subtraction, that neglects the dispersion of film optical constants. For determining the film dielectric function it was necessary to measure the correct absolute transmittance using parallel beam accessory of FTIR spectrophotometer. The spectral dependences of the film optical constants in IR region calculated using the fitted parameters of the IR-extended PDOS model are shown in Fig. 3. Several absorption peaks, that originate from both the DLC film and amorphized Si transition layer, were identified as labeled in this figure. [3,19–22]. The corresponding optical constants in the entire spectral range, i.e. from IR to UV, are plotted in Fig. 4.
Fig. 2. Spectral dependences of the associated ellipsometric parameters Is, Ic,II, Ic,III and reflectance R for chosen DLC film: the lines and symbols denote theoretical and experimental data, respectively. The ellipsometric dependences correspond to the angle of incidence of 65°.
theoretical data that corresponds to χ close to unity. Based on the comparison of χ values, it can be concluded that the qualities of all the fits are practically identical. Even though the extension of the model by high energy transitions leads to slightly worse fits and the addition of band asymmetry has the opposite effect, the fits are equivalent with respect to both the calculated optical quantities and optical constants in the measured range. However, differing values of material parameters indicate that the optical constants obtained from the individual models differ outside the measured range. Consequently, the important material characteristic, the ratio of π-to-σ electrons defined as α = Qπ/Qσ differs approximately by 20% (see Table 1). If the asymmetry of the bands is introduced it turns out that the sum of squares does not depend practically on parameter Yσ around unity, therefore, it was not possible to determine whether or how it differs from unity. Hence, the value of Yσ was fixed in unity during the fitting. Concerning the π band asymmetry, the best fit corresponds to Yπ = 0.67, i.e., the ratio of band widths approximately 2:3 (the
Fig. 3. Spectral dependences of the transmittance T, refractive index n and extinction coefficient k for chosen DLC film.
D. Franta et al. / Diamond & Related Materials 18 (2009) 413–418
417
Fig. 6. Details of the optical constants n and k of the c-Si substrate in IR region. Fig. 4. Spectral dependences of the refractive index n and extinction coefficient k for chosen DLC film.
For reference, the optical constants of the substrate and transition layer are given in Figs. 5 and 6. It can be seen that the optical constants of transition layer correspond to amorphous silicon [17]. Its thickness was 18 nm. 6. Conclusion Several extensions of previously published PDOS model of DLC dielectric response were tested on a typical DLC film on Si. It was found that the model extended by π → σ⁎ and σ → π⁎ transitions
degenerates to the base PDOS model. The other two model extensions, i.e., the addition of high energy transitions and band asymmetry, lead to equivalent agreement between the experimental and theoretical data as well as equivalent optical constants, whereas the material parameters differ. Therefore, the suggested model extensions are not needed for determining optical constants in the spectral range from IR to 6.5 eV. Since the individual models give different material parameters they are determined with relatively high errors. Hence, it is necessary to measure at higher energies, ideally using synchrotron radiation. In this case here presented extended models will probably show their importance. Moreover, it was shown that the PDOS models can be successfully combined with models of phonon absorption in IR region. The application of the combined model in the wide spectral range allows correct calculation of absorption peaks because the interference and dispersion effects are taken into account. Acknowledgements This work was supported by Ministry of Education of the Czech Republic under contract MSM 0021622411, by Czech Science Foundation under contract 202/07/1669 and by Czech Academy of Sciences under contract KAN311610701. References
Fig. 5. Spectral dependences of the refractive index n and extinction coefficient k of the transition layer and c-Si substrate.
[1] J. Robertson, Diamond Relat. Mater. 6 (1997) 212. [2] W. Kulisch, Deposition of Diamond-Like Superhard Materials, Springer-Verlag, Berlin, 1999. [3] J. Robertson, Mater. Sci. Eng., R 37 (2002) 129. [4] J. Tauc, in: F. Abelès (Ed.), Optical Properties of Solids, North–Holland, Amsterdam, 1972, p. 277. [5] S. Adachi, Optical Properties of Crystalline and Amorphous Semiconductors: Matrials and Fundamental Principles, Kluwer, Boston, 1999. [6] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, Springer, Berlin, 2001. [7] D. Franta, I. Ohlídal, V. Buršíková, L. Zajíčková, Diamond Relat. Mater. 12 (2003) 1532. [8] D. Franta, L. Zajíčková, V. Buršíková, I. Ohlídal, Acta Phys. Slovaca 53 (2003) 373. [9] D. Franta, D. Nečas, L. Zajíčková, Opt. Express 15 (2007) 16230. [10] D. Franta, V. Buršíková, D. Nečas, L. Zajíčková, Diamond Relat. Mater. 17 (2008) 705. [11] S.C. Ray, T.I.T. Okpalugo, P. Papakonstantinou, C.W. Bao, H.M. Tsai, J.W. Chiou, J.C. Jan, W.F. Pong, J.A. McLaughlin, W.J. Wang, Thin Solid Films 482 (2005) 242. [12] R.de L. Kronig, J. Opt. Soc. Am. & Rev. Sci. Instrum. 12 (1926) 547. [13] L. Zajíčková, V. Buršíková, D. Franta, Czechoslov. J. Phys. 49 (1999) 1213.
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[14] I. Ohlídal, D. Franta, in: E. Wolf (Ed.), Progress in Optics, vol. 41, Elsevier, Amsterdam, 2000, p. 181. [15] D. Franta, I. Ohlídal, V. Buršíková, L. Zajíčková, Thin Solid Films 455–456 (2004) 393. [16] D. Franta, I. Ohlídal, M. Frumar, J. Jedelský, Appl. Surf. Sci. 212–213 (2003) 116. [17] D. Franta, I. Ohlídal, P. Klapetek, P. Roca i Cabarrocas, Thin Solid Films 455–456 (2004) 399.
[18] D. Franta, L. Zajíčková, I. Ohlídal, J. Janča, K. Veltruská, Diamond Relat. Mater. 11 (2002) 105. [19] B. Dischler, A. Bubenzer, P. Koidl, Solid State Commun. 48 (1983) 105. [20] Y. Hirose, Y. Terasawa, Jpn. J. Appl. Phys. 25 (1986) L519. [21] S.F. Durrant, R.P. Mota, W.A.B. deMoraes, Vacuum 47 (1996) 187. [22] K. Aumaille, C. Vallée, A. Granier, A. Goullet, F. Gaboriau, G. Turban, Thin Solid Films 359 (2000) 188.
Contents lists available at ScienceDirect
Diamond & Related Materials j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / d i a m o n d
Limitations and possible improvements of DLC dielectric response model based on parameterization of density of states Daniel Franta ⁎, David Nečas, Lenka Zajíčková, Vilma Buršíková Department of Physical Electronics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
a r t i c l e
i n f o
a b s t r a c t
Available online 5 November 2008 PACS: 71.23.Cq 71.55.Jv 78.20.Ci 78.66.Jg Keywords: Diamond-like carbon Amorphous hydrogenated carbon Optical properties Band structure
Several extensions of previously published PDOS model of DLC dielectric response are tested on a typical DLC film on Si. The extended models are applied to the processing of optical measurements, namely UV/VIS/NIR ellipsometry and spectrophotometry as well as FTIR transmittance spectroscopy. The tested extensions of PDOS approach, that originally took into account only π → π⁎ and σ → σ⁎ interband transitions, are (i) the inclusion of π → σ⁎ and σ → π⁎ transitions, (ii) the addition of transitions from σ to bands above σ⁎, (iii) the removal of the assumption that the bands are symmetrical with respect to the Fermi level. However, it is found that these extensions are not needed for the processing of optical data in the measured range, although they could become important in the VUV and XUV regions. It is shown that the PDOS models can be successfully combined with models of phonon absorption in IR region. The application of the combined model in the wide spectral range allows correct calculation of absorption peaks because the interference and dispersion effects are taken into account. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Diamond-like carbon (DLC) is composed of carbon in two different bonding configurations, sp3 and sp2, and various amount of hydrogen [1–3]. The sp3 bonded carbon has four σ electrons, whereas sp2 bonded carbon has three σ and one π electrons. The hydrogen forms bonds only with σ electrons. Therefore, π and σ bands are distinguished in the electronic structure of DLC films. Dielectric response of DLC material above 1 eV, i.e., in UV, visible and near-IR (UV/VIS/NIR) spectral region is caused by excitations of electrons from occupied to empty states, namely from valence to conduction states. The complete electronic structure of DLC is schematically plotted in Fig. 1. The contribution of the j → k⁎ interband transitions to the imaginary part of the dielectric function εi is given by the following convolution [4–6]: ei;jYk4 ðEÞ =
eh mE
2 jp
jYk4
j2
4πe0 B0
∞
× ∫ N j ðSÞN k4 ðS + EÞdS; −∞
ð1Þ
where E, e, h, m, ε0, B0 and |pj → k⁎|2 are photon energy, electron charge, Planck's constant, electron mass, permittivity of vacuum, volume of certain part of Brillouin zone of corresponding crystalline material and squared momentum-matrix element, respectively. The
⁎ Corresponding author. Tel.: +420 54949 3836; fax: +420 541211214. E-mail address: [email protected] (D. Franta). 0925-9635/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2008.10.037
functions N j(S) and N k (S + E) represent energy distributions of the density of states (DOS) in j occupied and k⁎ empty bands, respectively, where S is the energy of the electronic states with respect to the Fermi level. Not all the transitions have a significant influence on the dielectric function in the UV/VIS/NIR region due to very different transition probabilities and the factor 1/E2 in Eq. (1). In the previously published model of DLC [7–10] based on parameterization of density of states (PDOS) the σ → π⁎ and π → σ⁎ interband transitions were disregarded because of their low probability due to the high localization of π electrons [3]. This model, here referred as the base PDOS model, used also some additional simplifications as listed below. The transitions from the core level and to the excited states above σ⁎ were omitted because their energies lie in VUV and XUV spectral range, far from the measured spectral range. The transitions from the valence bands to higher excited states can be expected for energies above 25 eV while the transitions from the core level to the conduction bands start at 285 eV [11]. Therefore, only π → π⁎ and σ → σ⁎ transitions were considered in the base PDOS model. In addition, the symmetry of valence and conduction bands with respect to the Fermi level was assumed, i.e., the π⁎ and σ⁎ conduction bands are mirror images of the valence π and σ bands, respectively. This paper will investigate the influence of above mentioned simplifications on the interpretation of optical measurements. Furthermore, the PDOS model will be extended with Gaussian absorption peaks describing the phonon absorption in order to interpret transmittance data in IR region below 0.5 eV (4000 cm− 1).
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D. Franta et al. / Diamond & Related Materials 18 (2009) 413–418
refractive index n and extinction coefficient k, are calculated from the complex dielectric function as
2. Model of DLC dielectric function 2.1. Base PDOS model The imaginary part of DLC dielectric function is composed of only two terms in case of the base model [7–10]: ei ðEÞ = ei;πYπ4 ðEÞ + ei;σYσ 4 ðEÞ;
ð2Þ
where εi,π → π⁎ and εi,σ → σ⁎ denote the contributions of π → π⁎ and σ → σ⁎ transitions, respectively. The individual contributions are calculated using following convolution ei;jYk4 ðEÞ =
1 ∞ ∫ Nj ðSÞNk4 ðS + EÞdS; E2 −∞
ð3Þ
where j = k = π,σ. Here, the functions N represent the unnormalized density of states [10] that differ from the density of states N because they include the constant factors in front of the integral in Eq. (1). From similarity with DOS distribution in crystals in vicinity of energy extrema of the bands the unnormalized density of valence states N j (S) can be modeled as a parabolic function [10]: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 32Q j −S−Eg j =2 Ehj =2 + S for 2 Nj ðSÞ = π Ehj −Egj > : 0 otherwise:
−Ehj =2bS b−Eg j =2
ð4Þ
The parameters E gj, E hj and Q j are the minimum energy limit of transitions (band gap), maximum energy limit of transitions and parameters proportional to the valence electron densities of the material, respectively. Assuming the symmetry with respect to the Fermi level, the unnormalized DOS distribution of conductive band N j⁎(S) is expressed by a symmetrical function [10]: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 32Q j S−Eg j =2 Ehj =2−S for 2 Nj4 ðSÞ = π Ehj −Eg j > : 0 otherwise:
Eg j =2bSb−Ehj =2
ð5Þ
The parameters Qj are defined by following integrals ∞
∞
−∞
−∞
∫ Nj ðSÞdS = ∫ Nj4 ðSÞdS = Q j :
ð6Þ
The real part of the dielectric function is given by the Kramers–Kronig (KK) relation [12] as er ðEÞ = 1 +
2 ∞ Sei ðSÞ H dS; π 0 S2 −E2
ð7Þ
where unity is the contribution of vacuum. Symbol H denotes the Cauchy principal value of the integral. The optical constants, i.e.,
nðEÞ + ikðEÞ =
pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be ðEÞ = er ðEÞ + iei ðEÞ:
ð8Þ
Although the integrals in Eqs. (3) and (7) have to be calculated numerically today's computers can calculate the dielectric function within 1 ms when using appropriate numerical algorithms. 2.2. Transitions π → σ⁎ and σ → π⁎ The contributions of π → σ⁎ and σ → π⁎ transitions can be calculated using Eqs. (3)–(5) with j ≠ k. If the symmetry of the valence and conduction bands is taken into account it holds ei;πYσ 4 ðEÞ = ei;σ Yπ4 ðEÞ:
ð9Þ
The complete imaginary part of dielectric function is then expressed as ei ðEÞ = ei;πYπ4 ðEÞ + ei;σYσ 4 ðEÞ + 2ei;πYσ 4 ðEÞ:
ð10Þ
Since the probability of π → σ⁎ and σ → π⁎ transitions differs from π → π⁎ and σ → σ⁎, an additional parameter Cπσ expressing the ratio of these probabilities has to be introduced. The contribution εi,π → σ⁎ can be then written in the following form: ei;πYσ 4 ðEÞ =
Cπσ ∞ ∫ Nπ ðSÞNσ 4 ðS + EÞds: E2 −∞
ð11Þ
2.3. High energy transitions The transitions from σ band to the states above σ⁎ modify the dielectric function in the IR/VIS/UV range only through KK relations, i.e. by a slight change in the real part of this function. In order to keep the number of model parameters low it is necessary to find a parameterization that, in ideal case, does not introduce any new parameters. Therefore, the following model of high energy bands was chosen. The r-th band above σ⁎ has the same shape as σ⁎ but it is shifted to higher energies by ΔEr. The contributions of the transitions from σ to the r-th high energy band are calculated as σ → σ⁎ contributions but the Eg and Eh in Eqs. (3)–(5) are replaced with Emin and Emax expressed as follows Emin = Eg + ΔEr
and
Emax = Eh + ΔEr :
ð12Þ
The shift of the first band above σ⁎ was chosen as ΔE1 = Eh/2. Similarly, the shift of the r-th band of higher excited states was ΔEr = r
Eh : 2
ð13Þ
2.4. Asymmetry of bands
Fig. 1. Schematic diagram of electronic structure of the DLC. Shaded area depicts occupied electronic states. Symbol EF denotes the Fermi energy.
Since the dielectric function depends only on the convolution of density of valence and conduction states only one additional parameter is needed for each asymmetric pair of valence and conduction bands. It means that the asymmetric pair is described by four parameters. Three parameters, Egi, Ehj and Qj, have the same meaning as in the base PDOS model and the fourth, Yj, equals to the
D. Franta et al. / Diamond & Related Materials 18 (2009) 413–418
ratio of conduction and valence band widths. Therefore, the DOS of valence band is parameterized as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < 32Q j −S−Eg j =2 Evj =2 + S for 2 Nj ðSÞ = π Evj −Eg j : 0 otherwise:
−Evj =2bSb−Eg j =2;
ð14Þ
2Ehj + Yj −1 Eg j : 1 + Yj
ð15Þ
Similarly for conduction band: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < 32Q j S−Eg j =2 Ecj =2−S for 2 Nj4 ðSÞ = π Ecj −Egj : 0 otherwise:
Eg j =2bSbEcj =2
ð16Þ
where 2Yj Ehj + 1−Yj Eg j : Ecj = 1 + Yj
ð17Þ
2.5. Phonon absorption The phonon absorption is modeled by discrete energy spectrum. Hence, the contribution to the imaginary part of the dielectric function is written ei;p ðEÞ =
1 ∑ Ai Gi ðEÞ; E2 i
ð18Þ
where Ai is parameter proportional to the density of corresponding bonds and probability of photon–phonon interaction. The functions Gi represent the Gaussian absorption peaks. If the antisymmetry of εi is taken into account they have to be written in the following form: Gi ðEÞ = g ðBi ; E−Ei Þ−g ðBi ; E + Ei Þ;
ð19Þ
where Ei and Bi denote the peak energy and broadening factor, respectively. The function g is a normalized Gaussian function 1 E2 g ðB; EÞ = pffiffiffiffiffiffi exp − 2 : 2B 2π B
photometer in spectral range 1.1–1.3 eV (900–1100 nm) and by an FTIR Bruker Vertex 80v spectrophotometer in spectral range 0.045–0.93 eV (370–7500 cm− 1). The Bruker spectrophotometer was used with the parallel beam transmittance accessory for correct absolute transmittance measurement. 4. Data processing
where Evj is calculated as Evj =
415
ð20Þ
3. Experimental The DLC film was prepared by plasma enhanced chemical vapor deposition (PECVD) on double polished silicon single crystal (c-Si) substrate in capacitively coupled glow discharge (13.56 MHz) in constant bias mode from methane. The dc self-bias voltage and flow rate of methane were kept at Ub = −367 V and QCH4 = 1.8 sccm, respectively. The pressure and the rf power corresponding to this flow rate and bias were p = 10.5 Pa and approximately P = 50 W, respectively. The deposition time was 25 min. Prior to the deposition the substrate was cleaned in hydrogen plasma for 5 min (QH2 = 1 sccm, P = 50 W, Ub = −410 V, p = 6–7 Pa). A detailed description and schematic drawing of the reactor is given by Zajíčková et al. [13]. The optical characterization of the DLC film was carried out using variable angle spectroscopic ellipsometry, UV/VIS/NIR spectrophotometry and Fourier transform infrared (FTIR) transmittance spectroscopy. Ellipsometric quantities were measured by an ellipsometer Jobin Yvon UVISEL at five angles of incidence from 55° to 75° within the spectral range 1.2–6.5 eV (190–1000 nm). Reflectances were measured using a spectrophotometer PerkinElmer Lambda 45 within the same spectral range at an angle of incidence of 6°. Transmittance at normal incidence was measured by the PerkinElmer spectro-
The optical measurements described above provided reflectances R, transmittance T and associated ellipsometric parameters Is, Ic,II and Ic,III. The associated ellipsometric parameters represent three coordinates on Poincaré sphere defined as Is = sin 2 Ψ sin Δ, Ic,II = sin 2 Ψ cos Δ and Ic,III = cos 2 Ψ if no depolarization effects is present, where Ψ is azimuth and Δ is phase change. The fitting procedure utilized leastsquares method, i.e., minimized squares of differences between measured optical quantities R, T, Is, Ic,II and Ic,III and their theoretical values calculated by 2 × 2 matrix formalism [14]. First, it was necessary to determine the optical constants of the c-Si substrate in the entire spectral range in order to separate the IR absorption in film and substrate. The model of c-Si included both interband electronic transitions and phonon absorption. However, the modeling of silicon substrate is out of the scope of this article. For correct interpretation of experimental data it was also necessary to include inhomogeneous transition layer in the structural model of DLC film. This transition layer represents the amorphized Si layer originated from ion bombardment during initial stage of the deposition [7,15]. The optical constants of the transition layer were modeled using the six parameter PDOS model [16,17] containing two absorption bands. 5. Results and discussion It was found that if the transitions π → σ⁎ and σ → π⁎ are added to the base PDOS model the parameter Cπσ converges to zero. It means this extended model degenerates to the base model. Therefore, the resulting fit parameters are not shown in this section. In Table 1, the material parameters Q π, Egπ, Ehπ, Q σ, Egσ, Ehσ and Yπ corresponding to the interband transitions calculated by all other PDOS models are summarized. Symbols dfE, dfR and dfT denote the mean film thicknesses in the ellipsometry, reflectometry and FTIR transmittance measurement spots, respectively. The values of remaining fit parameters, characterizing the amorphized transition layer and the DLC film in IR region, will not be discussed in this paper. Besides the fit parameters, there is also quantity χ in Table 1. It quantifies the differences between the experimental and theoretical values of the optical data with respect to their estimated errors. If χ = 1 the fit is optimum, greater values indicate worse agreement [18]. In Fig. 2 the spectral dependences of the associated ellipsometric parameters Is, Ic,II, Ic,III and reflectance R in the UV/VIS/NIR region are plotted. One can see a sound agreement between experimental and
Table 1 The parameters of the PDOS dispersion models corresponding to chosen DLC film High energy transitions
No
Yes
No
Yes
asymmetric bands
No
No
Yes
Yes
Q π [eV3/2] Egπ [eV] Ehπ [eV] Q σ [eV3/2] Egσ [eV] Ehσ [eV] Yπ dFE [nm] dfR[nm] dfT [nm] Χ α = Q π/Q σ
3.81 1.03 7.09 150 1.61 55.3 – 205.9 205.0 204.5 1.79 0.0254
3.73 1.03 7.04 133 1.61 51.1 – 205.8 204.9 204.4 1.81 0.0280
4.19 1.07 7.10 146 1.68 53.9 0.670 205.5 204.6 204.8 1.77 0.0287
4.12 1.07 7.05 130 1.68 49.7 0.665 205.4 204.5 204.7 1.78 0.0318
416
D. Franta et al. / Diamond & Related Materials 18 (2009) 413–418
symmetry of integral (3) does not permit to assign widths to the valence and conduction bands specifically, only the degree of asymmetry can be determined). However, the dependence of the sum of squares on Yπ was also quite weak, therefore, this result cannot be considered completely reliable. It can be concluded that these two extension of PDOS model neither improved the fits nor brought much new information about the material structure. In Fig. 3 the spectral dependences of the transmittance T measured by FTIR and conventional spectrophotometer is plotted. It can be seen that the PDOS model, extended with the model of phonon absorption below 0.5 eV (4000 cm− 1), can be successfully applied to optical characterization in the entire spectral range from IR to UV. The simultaneous processing of optical data in UV/VIS/NIR and IR spectral ranges allowed to calculate directly the dielectric function of the film also in the IR region. It supersedes often used inaccurate procedure of film interference subtraction, so called background subtraction, that neglects the dispersion of film optical constants. For determining the film dielectric function it was necessary to measure the correct absolute transmittance using parallel beam accessory of FTIR spectrophotometer. The spectral dependences of the film optical constants in IR region calculated using the fitted parameters of the IR-extended PDOS model are shown in Fig. 3. Several absorption peaks, that originate from both the DLC film and amorphized Si transition layer, were identified as labeled in this figure. [3,19–22]. The corresponding optical constants in the entire spectral range, i.e. from IR to UV, are plotted in Fig. 4.
Fig. 2. Spectral dependences of the associated ellipsometric parameters Is, Ic,II, Ic,III and reflectance R for chosen DLC film: the lines and symbols denote theoretical and experimental data, respectively. The ellipsometric dependences correspond to the angle of incidence of 65°.
theoretical data that corresponds to χ close to unity. Based on the comparison of χ values, it can be concluded that the qualities of all the fits are practically identical. Even though the extension of the model by high energy transitions leads to slightly worse fits and the addition of band asymmetry has the opposite effect, the fits are equivalent with respect to both the calculated optical quantities and optical constants in the measured range. However, differing values of material parameters indicate that the optical constants obtained from the individual models differ outside the measured range. Consequently, the important material characteristic, the ratio of π-to-σ electrons defined as α = Qπ/Qσ differs approximately by 20% (see Table 1). If the asymmetry of the bands is introduced it turns out that the sum of squares does not depend practically on parameter Yσ around unity, therefore, it was not possible to determine whether or how it differs from unity. Hence, the value of Yσ was fixed in unity during the fitting. Concerning the π band asymmetry, the best fit corresponds to Yπ = 0.67, i.e., the ratio of band widths approximately 2:3 (the
Fig. 3. Spectral dependences of the transmittance T, refractive index n and extinction coefficient k for chosen DLC film.
D. Franta et al. / Diamond & Related Materials 18 (2009) 413–418
417
Fig. 6. Details of the optical constants n and k of the c-Si substrate in IR region. Fig. 4. Spectral dependences of the refractive index n and extinction coefficient k for chosen DLC film.
For reference, the optical constants of the substrate and transition layer are given in Figs. 5 and 6. It can be seen that the optical constants of transition layer correspond to amorphous silicon [17]. Its thickness was 18 nm. 6. Conclusion Several extensions of previously published PDOS model of DLC dielectric response were tested on a typical DLC film on Si. It was found that the model extended by π → σ⁎ and σ → π⁎ transitions
degenerates to the base PDOS model. The other two model extensions, i.e., the addition of high energy transitions and band asymmetry, lead to equivalent agreement between the experimental and theoretical data as well as equivalent optical constants, whereas the material parameters differ. Therefore, the suggested model extensions are not needed for determining optical constants in the spectral range from IR to 6.5 eV. Since the individual models give different material parameters they are determined with relatively high errors. Hence, it is necessary to measure at higher energies, ideally using synchrotron radiation. In this case here presented extended models will probably show their importance. Moreover, it was shown that the PDOS models can be successfully combined with models of phonon absorption in IR region. The application of the combined model in the wide spectral range allows correct calculation of absorption peaks because the interference and dispersion effects are taken into account. Acknowledgements This work was supported by Ministry of Education of the Czech Republic under contract MSM 0021622411, by Czech Science Foundation under contract 202/07/1669 and by Czech Academy of Sciences under contract KAN311610701. References
Fig. 5. Spectral dependences of the refractive index n and extinction coefficient k of the transition layer and c-Si substrate.
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