Thin Solid Films xxx (2013) xxx–xxx

TSF-32999; No of Pages 6

Contents lists available at ScienceDirect

Thin Solid Films

j ourna l homepage: www.e lsev ie r .com/ locate / ts f

Broadening of dielectric response and sum rule conservation

Daniel Franta a,b,⁎, David Nečas a,b, Lenka Zajíčková a,b, Ivan Ohlídal a

a Department of Physical Electronics, Faculty of Science, Masaryk University, Kotlářká 2, 611 37 Brno, Czech Republicb CEITEC —Central European Institute of Technology, Masaryk University, Kamenice 5, 625 00 Brno, Czech Republic

⁎ Corresponding author.E-mail address: franta@physics.muni.cz (D. Franta).

0040-6090/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.tsf.2013.11.148

Please cite this article as: D. Franta, et al., Th

a b s t r a c t

a r t i c l e i n f o

Available online xxxx

Keywords:Dielectric responseDispersion modelBroadeningSum rule

Different types of broadening of the dielectric response are studiedwith respect to the preservation of the Thomas–Reiche–Kuhn sum rule. It is found that only the broadening of the dielectric function and transition strength functionconserve this sum rule, whereas the broadening of the transition probability function (joint density of states)increases or decreases the sum. The effect of different kinds of broadening is demonstrated for interband andintraband direct electronic transitions using simplified rectangular models. It is shown that the broadening of thedielectric function is more suitable for interband transitions while broadening of the transition strength functionis more suitable for intraband transitions.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

The dielectric response can be equivalently described by variousfunctions of photon energy E, such as the imaginary part of the dielectricfunction εi, the transition probability function J and the transitionstrength function F:

εi≈JE2

¼ FE; ð1Þ

where symbol≈ represents the dipole approximation [1]. The functionF is defined as a continuous condensed-matter equivalent of the oscilla-tor strength known for discrete transitions in atomic spectra [1]. It wasshown that this function satisfies the integral form of Thomas–Reiche–Kuhn sum rule [2,1]:Z ∞

0F Eð ÞdE ¼ N; ð2Þ

where the constantN is called the total transition strength. Note that J in(1) is proportional to the transition density J [1] or the joint density ofstates Jvc [3,4] if the transition probability is constant but it is not equiv-alent in general. In the frame of the dipole approximation the real part ofconductivity is related to the transition strength function according to

σ r≈ϵ0ℏ

F; ð3Þ

where ϵ0 is the vacuum permittivity and ℏ is the reduced Planckconstant. The functions F or J can be used to construct dispersionmodelsfulfilling the sum rule (2) [1,5,6]. In the case of crystalline materials,

ghts reserved.

in Solid Films (2013), http://d

direct interband transitions and two-phonon absorption are modeledusing a piecewise function exhibiting sharp features, i.e. Van Hovesingularities [1,7].

In reality, more complex phenomena, such as electron–phononinteraction and isotopic effects, cause broadening of the dielectricresponse that have to be taken into account in the dispersion models.The broadening is introduced phenomenologically as the convolutionwith a normalized broadening function. The broadening function usuallyhas the form of a Lorentzian or Gaussian [3,4,8–14] and its selectiondepends on the broadening phenomena [4]. The most applicable, fromour experiences, is the Gaussian broadening function. The Lorentzianbroadening or an approximation of Gaussian broadening is often chosento avoid computational difficulties.

The convolution is applied to different quantities, usually J or εi. Kimet al. showed that the broadening of εi is preferable because it does notlead to spurious static conductivity in the broadened dielectric response[4].

The broadening needs to be judged also from the sum rule preserva-tion point of view. This work discusses the broadening of εi and σr thatcorrespond to the Fermi golden rule and the Kubo formula, respectively.These two approaches represent the two starting points for the expres-sion of the dielectric response for quantum systems. It will be shownthat the broadening of εi is more suitable for interband transitionswhile the broadening of F or σr is better for intraband transitions.

2. Dielectric response

This sectionwill summarize the approaches to the construction of theelectronic part of the dielectric response. The incorporation of thenucleonic part of the dielectric response is described in [1]. Within thedipole approximation, the dielectric function can be obtained usingdifferent approaches leading to the same unbroadened εi. One possibilityis to start from the Fermi golden rule that expresses the transition

x.doi.org/10.1016/j.tsf.2013.11.148

2 D. Franta et al. / Thin Solid Films xxx (2013) xxx–xxx

probability for transitions from an initial ground state |i ⟩ to s finalstate | f⟩ of a closed system [15]:

εi ¼πϵ0V

Xf≠i

f

j ⟨ f jdxeji ⟩ j2 δ E f−Ei−E� �

−δ Ei−E f−E� �h i

; ð4Þ

whereV is the volume of the system, Ei and Ef are the energies of the ini-tial and final states, respectively. The symbol dxe denotes the total dipoleoperator of electrons along the external electric field and can beexpressed as follows [15]:

j ⟨ f jdxeji ⟩ j2 ¼ e2j ⟨ f jxeji ⟩ j2 ¼ eℏme

� �2j ⟨ f jpxeji ⟩ j2

E f−Ei� �2 ; ð5Þ

where me and e denote the electron mass and the elementary charge,respectively. Operators pxe and xe are the total momentum and positionoperators of electrons. Therefore, the dielectric function can bewritten as

εi ¼ehð Þ2

4πϵ0m2eV

Xf≠i

f

j ⟨ f jpxeji ⟩ j2

E f−Ei� �2 δ E f−Ei−E

� �−δ Ei−E f−E� �h i

: ð6Þ

The relation εi = J/E2 is obtained if the transition probability func-tion J is introduced as

J ¼ ehð Þ24πϵ0m

2eV

Xf≠i

f

⟨ f jpxeji ⟩ j2hδ E f−Ei−E� �

−δ Ei−E f−E� �i��� ð7Þ

and the properties of the δ-function

δ E f−Ei � E� �

Ei−E f

� �2 ¼δ E f−Ei � E� �

E2ð8Þ

are utilized.Another possible starting point is the Kubo formula derived for

an open non-interacting system [16,17]. It expresses the real part ofconductivity σr as

σ r ¼πℏV

Xf≠i

i; f

expΩ−EikBT

� �j ⟨ f j jxeji ⟩ j2E f−Ei

hδ E f−Ei−E� �

þ δ Ei−E f−E� �i

ð9Þ

using the electric current operator jxe. Itsmatrix element is related to themomentummatrix element as follows:

j ⟨ f j jxeji ⟩ j2 ¼ e2

m2ej ⟨ f jpxeji ⟩ j2: ð10Þ

The symbols kB and T denote the Boltzmann constant and tempera-ture, respectively, and Ω is the thermodynamic potential satisfying

Xi

expΩ−EikBT

� �¼ 1: ð11Þ

The summation over initial states can be avoided for low tempera-tures, when only the ground state is considered. Then, the expressionfor the transition strength function is obtained from Eqs. (3) and (9) [1]:

F ¼ ehð Þ24πϵ0m

2eV

Xf≠i

f

j ⟨ f jpxeji ⟩ j2E f−Ei

δ E f−Ei−E� �

þ δ Ei−E f−E� �h i

: ð12Þ

All three expressions (6), (7) and (12) are equivalent and related byEq. (1).

Please cite this article as: D. Franta, et al., Thin Solid Films (2013), http://d

3. Broadening

Broadening of the dielectric response is introduced phenomenolog-ically by replacing the δ-functions in the formulas (6), (7) and (12)above with a normalized broadening functions β, e.g. the Gaussian

δ xð Þ→β xð Þ ¼ 1ffiffiffiffiffiffi2π

pBexp − x2

2B2

!: ð13Þ

Here B N 0 denotes the broadening parameter. However, the previ-ously equivalent formulas become non-equivalent after this transfor-mation. Depending on how the inverse square of energy in theformulas is split between the powers of (Ef − Ei) and E, different typesof broadening are obtained:

eεi Eð Þ ¼ 1E2

Z ∞

−∞β E−tð Þ J tð Þdt J−broadening;

eεi Eð Þ ¼ 1E

Z ∞

−∞β E−tð Þ J tð Þ

tdt F−broadening;

eεi Eð Þ ¼Z ∞

−∞β E−tð Þ J tð Þ

E2dt ε−broadening;

ð14Þ

where eεi denotes broadened εi. The different broadening types can bewritten more succinctly if we denote convolution by * and understandthat the independent variable is energy E:

eεi ¼ 1E2

β � Jð Þ J−broadening;

eεi ¼ 1E

β � JE

� �¼ 1

Eβ � Fð Þ F−broadening;

eεi ¼ β � JE2

¼ β � εi ε−broadening:

ð15Þ

Note that the symmetries of J, F and ε imply thatβmust be an even func-tion so that the broadening preserves the symmetries.

4. Sum rule conservation

In order to study the conservation of the sum rule by different typesof applied broadening, it is advantageous to expand F in the followingform

F Eð Þ ¼ Em

2

Z ∞

−∞δ E−tð Þ þ −1ð Þmδ E þ tð Þ�

F tð Þt−mdt; ð16Þ

where m is an integer. This identity clearly holds because F is an evenfunction. The factor (−1)m in the integrand follows from the identity

Emδ E � tÞ ¼ ∓tð Þmδ E � tÞ:ð ð17Þ

When δ in Eq. (16) is replaced with β, different values of the powerm lead to different types of broadening introduced above. The broad-ened transition strength corresponding to the m-th power, denotedeFm, can therefore be written as

eFm Eð Þ ¼ Em

2

Z ∞

−∞β E−tð Þ þ −1ð Þmβ E þ tð Þ�

F tð Þt−mdt; ð18Þ

wherem = −1, 0, and 1 correspond to J-, F- and ε-broadening, respec-tively. The sum rule is conserved by the broadening if

12

Z ∞

−∞eFm Eð ÞdE ¼ 1

2

Z ∞

−∞F Eð ÞdE ¼ N: ð19Þ

x.doi.org/10.1016/j.tsf.2013.11.148

ratio t/B

GaussianLorentzian

Fig. 1. Dependence of factor I−1,β(t) on ratio t/B for J-broadening with the Gaussian andLorentzian broadening function β.

3D. Franta et al. / Thin Solid Films xxx (2013) xxx–xxx

Substituting Eq. (18) to the left hand side and exchanging the order ofintegration leads to

12

Z ∞

−∞eFm Eð ÞdE ¼ 1

2

Z ∞

−∞F tð ÞIm;β tð Þdt; ð20Þ

where

Im;β tð Þ ¼ 12tm

Z ∞

−∞Em β E−tð Þ þ −1ð Þmβ E þ tð Þ�

dE: ð21Þ

The sum rule is conserved for a specific broadening type if Im,β(t) ≡ 1.It is obvious that this holds for m = 0, i.e. F-broadening

I0;β tð Þ ¼ 12

Z ∞

−∞β E−tð Þ þ β E þ tð Þ½ �dE ¼ 1; ð22Þ

because the convolution with any normalized broadening functionpreserves the integral. However, it also holds for m = 1:

I1;β tð Þ ¼ 12t

Z ∞

−∞E β E−tð Þ−β E þ tð Þ½ �dE

¼ 12t

Z ∞

−∞E þ tð Þ− E−tð Þ½ �β Eð ÞdE

¼ 1:

ð23Þ

The shift of the integration variable requires β(E) vanishingfaster than 1/E for E → ∞ which is satisfied by all normalizablebroadening functions. Therefore, ε-broadening also conserves thesum rule.

In case of J-broadening, on the other hand, the factor

I−1;β tð Þ ¼ t2

Z ∞

−∞

1Eβ E−tð Þ−β E þ tð Þ½ �dE; ð24Þ

generally differs from unity. The Gaussian β defined by Eq. (13)results in

I−1;Gauss tð Þ ¼ffiffiffi2

pt

BD

tffiffiffi2

pB

� �; ð25Þ

where D denotes the Dawson integral [18]. Since I−1,Gauss(t) can takevalues both greater and smaller than unity (I−1,Gauss(t) = 1 fort ≈ 1.307B), Gaussian J-broadening can both decrease and increasethe sum. For Lorentzian broadening, i.e. a broadening function ofthe form

β xð Þ ¼ Bπ

1

x2þB

2 ; ð26Þ

the value is

I−1;Lorentz tð Þ ¼ 11þ B=tð Þ2 b1: ð27Þ

Therefore, the sum is always decreased by Lorentzian J-broadening.The dependence of the factor I−1,β on t/B is illustrated for both thebroadening functions in Fig. 1. In summary, J-broadening generallydoes not conserve the sum rule. It depends on the specific form ofthe broadening and the broadened function, whether the sum increasesor decreases.

It is also possible to consider other powers of m that correspondto broadening of unusual quantities. For instance m = 2 would

Please cite this article as: D. Franta, et al., Thin Solid Films (2013), http://d

correspond to the broadening of EJ = E²F = E³εi. It is evident thatthe sum rule would not be conserved:

I2;β tð Þ ¼ 12t2

Z ∞

−∞E2 β E−tð Þ þ β E þ tð Þ½ �dE ¼ 1þ μ2;β

t2N1; ð28Þ

where μ2,β denotes the second central moment of β, which is pos-itive unless β is the δ-function. Clearly, the sum would be increasedby such type of broadening. Similar results can be obtained forother values of m.

Therefore, it can be concluded that only F- and ε-broadening conservethe sum rule.

5. Application of broadening

5.1. Direct interband transitions

The behavior of different broadening typeswill be demonstrated for asimple rectangular form of the unbroadened transition strength function

F Eð Þ ¼ const: : Eminb jEjbEmax0 : otherwise

�ð29Þ

that can represent direct interband transitions with a minimum tran-sition energy Emin N 0 and maximum transition energy Emax. The corre-sponding broadened dielectric response functions, J, F and εi, for all thethree types of Gaussian broadening are shown in Fig. 2. It can be seenthat J-broadening and F-broadening lead to a spurious Drude-like singu-larity in εi. Kim et al. discussed this effect for J-broadening and arguedthat ε-broadening is correct because it does not lead to this singularity[4]. Broadening by a Lorentzian gives similar results but leads to amuch more pronounced singularity because a Lorentzian approacheszero more slowly than a Gaussian. Moreover, F-broadening leads toa much less pronounced singularity than J-broadening. Therefore,the Gaussian F-broadening can be used in practice for broadening ofinterband transitions for typical values of bandgap and broadeningparameter B, even though ε-broadening is more correct.

Another effect that can be observed in Fig. 2 is the increase inthe transition strength (area below F) for J-broadening as opposedto F-broadening and ε-broadening which preserve the sum rule. Todemonstrate this effect more clearly, the relative cumulative transitionstrength nrel(E), defined as follows

nrel Eð Þ ¼ 1N

Z E

0F Xð ÞdX; ð30Þ

is plotted in Fig. 3. It should be noted that although the sum wasincreased by J-broadening for this specific set of parameters it can alsobe decreased as discussed in Section 4.

x.doi.org/10.1016/j.tsf.2013.11.148

-broadeningFT-broadening

F

0photon energy E

-broadening

Fig. 2. Illustration of the effect of J-, F- and ε-broadening using a simple rectangular modelof interband transitions. The units of the axes are arbitrary, Emax = 2.5Emin andB = 0.4Emin. The broadening width B is represented by a bar.

4 D. Franta et al. / Thin Solid Films xxx (2013) xxx–xxx

5.2. Direct intraband transitions

Concerning the direct intraband electronic transitions, the case ofvery low temperature T → 0 will be considered first. In this case, thedirect intraband transitions occur inmaterials with zero direct bandgap,such as graphene [19]. The transition strength function (conductivity)can be then assumed constant in the vicinity of zero energy. Therefore,the following simple unbroadened transition strength function can be

-broadeningJ-broadeningunbroadened

Fig. 3. Illustration of the effect of J-,F- and ε-broadening on the sum rule, i.e. relative cumu-lative transition strength nrel(E) defined in Eq. (30). The samemodel and broadening as inFig. 2 were used.

Please cite this article as: D. Franta, et al., Thin Solid Films (2013), http://d

used for the demonstration purposes showing the behavior of differentbroadening types:

F Eð Þ ¼ const: : jEj b Emax0 : otherwise;

�ð31Þ

where Emax is the maximum energy of transitions. The results ofGaussian J-, F- and ε-broadening are shown in Fig. 4. It can be seenthat J-broadening and F-broadening lead to similar broadened func-tions. Unlike in the preceding paragraph, J-broadening decreases thetransition strength which illustrates the observation in Section 4,that J-broadening can both decrease and increase the transitionstrength. See also Fig. 5 in which the relative cumulative transitionstrength nrel(E) is plotted for intraband transitions.

The effect of ε-broadening is considerably different from J- andF-broadening. The transition strength is redistributed to higherenergies, creating a hole in the transition strength function aroundE = 0. This artifact makes ε-broadening unsuitable for directintraband transitions despite of the sum rule preservation. Hence,F-broadening is more appropriate in this case.

One can argue that at a finite temperature, direct intraband transi-tions with the energy of E ≲ kBT do not contribute to the dielectricresponse because the occupation numbers of electron states aroundthe Fermi energy are approximately 0.5 and therefore the absorptionand emission probabilities are approximately equal. This effect alsocauses the vanishing of the transition strength of direct transitionsaround the zero energy and it could seem that the hole created byε-broadening is of the same nature. However, these two effects aredifferent. If the occupation numbers of electrons in the initial and

F

0photon energy E

Fig. 4. Illustration of the effect of J-, F-, ε and FT-broadening using a simple rectangularmodel of intraband transitions. The units of the axes are arbitrary, B = 0.2Emax, andkBT = 0.02Emax. The broadening width B and value of kBT are shown by bars.

x.doi.org/10.1016/j.tsf.2013.11.148

-broadeningJ-broadening

FT-broadeningunbroadened

Fig. 5. Illustration of the effect of J-,F-, ε- and FT-broadening on the sum rule, i.e. relativecumulative transition strengthnrel(E) defined in Eq. (30). The samemodel and broadeningas in Fig. 4 were used.

5D. Franta et al. / Thin Solid Films xxx (2013) xxx–xxx

final state are taken into account for a finite temperature, the transi-tion strength function has to be multiplied by the following temper-ature factor:

f T ¼ f FDe Eið Þ− f FDe ðE f Þ; ð32Þ

where fFD is the Fermi–Dirac statistics. In the simple case of symmet-rical bands, such as in graphene, the resulting dielectric function canbe written as [19]:

eεi Eð Þ ¼ f FDe −E=2ð Þ− f FDe E=2ð Þh i1

Eβ � Fð Þ: ð33Þ

The combination of F-broadening with the multiplication by thethermal factor described by Eq. (33) will be called FT-broadening. Thebroadened dielectric response functions corresponding FT-broadeningare also displayed in Fig. 4. It is clear from both figure and formula (33)that an FT-broadened transition strength function behaves as |E| aroundzero energy which corresponds to finite values of εi for E → 0. This isin contrast to ε-broadening which results in a quadratic dependenceof F and linear dependence εi in the vicinity of E = 0. Since fT ≤ 1 theFT-broadening does not conserve the transition strength as can alsobe seen in Fig. 5. This is expected because the transition strength isredistributed to indirect intraband transitions at finite temperatures.

Note that the same thermal factor should also be applied to directinterband transition described in Section 5.1, especially for tempera-tures T ≳ Eg/kB, where Eg is the bandgap energy. In both cases, thetransition strength redistributed to indirect intraband transitions(free carrier contribution) can be calculated as follows:

Nfc ¼Z ∞

01− f FDe −E=2ð Þ þ f FDe E=2ð Þh ieF Eð ÞdE; ð34Þ

where eF Eð Þ is obtained using F-broadening in the case of intraband tran-sitions and ε-broadening in the case of intraband transitions.

5.3. Multi-phonon absorption

Multi-phonon absorption does not occur at a discrete energy spec-trum as one-phonon absorption does. The absorption structures formbands with Van Hove singularities similar to absorption bands formedby direct electronic transitions. Since the dielectric response of phononscannot contain a Drude-like singularity corresponding to the responseof free carriers, the ε-broadening is appropriate.

Please cite this article as: D. Franta, et al., Thin Solid Films (2013), http://d

6. Combination with Kramers–Kronig transform

So far, the broadening has been discussed for the functions related tothe imaginary part of the dielectric function. Although εi itself containscomplete information about the dielectric response, the constructionof dispersion models applicable in optical characterization requires tocomplement it with the real part of the dielectric function εr using theKramers–Kronig relation [15]:

εr Eð Þ ¼ 1π

Z ∞

−∞

εi xð Þx−E

dx ≡H εi½ �; ð35Þ

where crossed integral denotes the Cauchy principal value and H[f]denotes the Hilbert transform [20] of function f. Combining it with thebroadening thus seems to require double integration which may presenta difficulty, especially if the integrations must be carried out numerically.

In the case of ε-broadening, the double integration can be avoidedtrivially. The substitution of the expression for ε-broadened εi fromEq. (15) leads to

eεr ¼ H eεi� ¼ H β � εi� ¼ H β½ � � εi;½� ð36Þ

i.e. the real part is obtained by broadening with the Hilbert transform ofβ. If β is Gaussian (13) its Hilbert transform is expressed using theDawson integral

H β½ � xð Þ ¼ −ffiffiffi2

p

πBD

xffiffiffi2

pB

� �: ð37Þ

For a polynomial εi the broadened real part eεr can be expressed in aclosed form using special functions in a relatively simple waywhich hasalso been utilized for construction of dispersion models [11].

It is also possible to avoid the double integration for F-broadeningalbeit using a slightly less straightforward procedure. The real part ofthe dielectric function corresponding to an F-broadened dielectricresponse can be expanded as follows

eεr¼H eFE

" #¼ 1

π

Z ∞

−∞

1x−E

eF xð Þx

dx ¼ 1π

Z ∞

−∞

1E

1x−E

−1x

� eF xð Þdx: ð38Þ

Since eF xð Þ=x is even, its integral vanishes in the sense of Cauchyprincipal value and therefore

eεr ¼ 1πE

Z ∞

−∞

eF xð Þx−E

dx ¼ 1EH β � F½ � ¼ 1

EH β½ � � Fð Þ: ð39Þ

The complexity of F-broadening is, therefore, the same as for ε-broadening.Obviously, both the expressions (36) and (39) for the real part can

be combined with the corresponding expression for the imaginarypart in Eq. (15) to a single formula, expressing the complex dielectricfunction using a convolution with the complex function

β ¼ H β½ � þ iβ: ð40Þ

Nevertheless, for numerical calculation, it is usually more efficient toperform the broadening of the real and imaginary parts separatelybecause of different asymptotic behaviors of Gaussian and Dawsonfunctions for large arguments and different precision requirements onthe real and imaginary parts.

7. Conclusion

The phenomenological broadening of the dielectric response wasstudied on the basis of broadening of different dielectric responsefunctions, i.e. the transition probability function J, the transitionstrength function F and the dielectric function ε. It was found that

x.doi.org/10.1016/j.tsf.2013.11.148

6 D. Franta et al. / Thin Solid Films xxx (2013) xxx–xxx

only F-broadening and ε-broadening conserve the sum rule, whereasJ-broadening generally increases or decreases the sum. The applicabilityof F- and ε-broadening was demonstrated for interband and intrabanddirect electronic transitions using simplified rectangular models. It wasshown that ε-broadening is more suitable for interband transitionswhile F-broadening ismore suitable for intraband transitions. The appli-cation of F-broadening to direct interband transitions causes a spuriousDrude-like singularity in the dielectric response. If ε-broadening is ap-plied to direct intraband transitions, it causes vanishing of the transitionstrength around zero energy which is, however, of a different naturethan similar effect caused by the broadening of Fermi–Dirac statistics.Combining the broadening and the Kramer–Kronig transform, which isused to obtain the real part of dielectric function, was discussed and itwas shown that both ε- and F-broadening of the real part can beexpressed using a convolution with the Hilbert transform of the broad-ening function.

Acknowledgments

This work was supported by the Czech Ministry of Education, Youthand Sports of the Czech Republic; projects “R&D center for low-costplasma and nanotechnology surface modification” (CZ.1.05/2.1.00/03.0086) and “CEITEC—Central European Institute of Technology”(CZ.1.05/1.1.00/02.0068) from European Regional Development Fund.

Please cite this article as: D. Franta, et al., Thin Solid Films (2013), http://d

References

[1] D. Franta, D. Nečas, L. Zajíčková, Thin Solid Films 534 (2013) 432.[2] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,

Springer-Verlag, Berlin, 1957.[3] S. Adachi, Phys. Rev. B 35 (1987) 7454.[4] C.C. Kim, J.W. Garland, H. Abad, P.M. Raccah, Phys. Rev. B 45 (1992) 11749.[5] D. Franta, D. Nečas, L. Zajíčková, I. Ohlídal, J. Stuchlík, D. Chvostová, Thin Solid Films

539 (2013) 233.[6] D. Franta, D. Nečas, L. Zajíčková, I. Ohlídal, J. Stuchlík, Thin Solid Films 541 (2013)

12–16.[7] L. Van Hove, Phys. Rev. 89 (1953) 1189.[8] S. Adachi, Phys. Rev. B 41 (1990) 1003.[9] C.C. Kim, J.W. Garland, P.M. Raccah, Phys. Rev. B 47 (1993) 1876.

[10] C. Tanguy, Solid State Commun. 98 (1996) 65.[11] B.D. Johs, C.M. Herzinger, J.H. Dinan, A. Cornfeld, J.D. Benson, Thin Solid Films

313–314 (1998) 137.[12] A.B. Djurišić, E.H. Li, Phys. Status Solidi B 216 (1999) 199.[13] A.B. Djurišić, Y. Chan, E.H. Li, Semicond. Sci. Technol. 16 (2001) 902.[14] Y.S. Ihn, T.J. Kim, T.H. Ghong, Y. Kim, D.E. Aspnes, J. Kossut, Thin Solid Films 455–456

(2004) 222.[15] F. Wooten, Optical Properties of Solids, Academic Press, New York, 1972.[16] R. Kubo, J. Phys, Soc. Jpn. 12 (1957) 570.[17] P.B. Allen, in: M.L. Cohen, S.G. Louie (Eds.), Conceptual Foundations of Material

Properties: A Standard Model for Calculation of Ground- and Excited-State Proper-ties, Elsevier, Amsterdam, 2006, p. 165.

[18] In: M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables, National Bureau of Standards, 1964.

[19] A.B. Kuzmenko, E.B. Kuzmenko, E. van Heumen, F. Carbone, D. van der Marel, Phys.Rev. Lett. 100 (2008) 117401.

[20] R.N. Bracewell, The Fourier Transform& Its Applications,McGraw-Hill, Singapore, 1999.

x.doi.org/10.1016/j.tsf.2013.11.148