Thin-film limit formalism applied to surface defect absorption

The thin-film limit is derived by a nonconventional approach and equations for transmittance, reflectance and absorptance are presented in highly versatile and accurate form. In the thin-film limit the optical properties do not depend on the absorption coefficient, thickness and refractive index individually, but only on their product. We show that this formalism is applicable to the problem of ultrathin defective layer e.g. on a top of a layer of amorphous silicon. We develop a new method of direct evaluation of the surface defective layer and the bulk defects. Applying this method to amorphous silicon on glass, we show that the surface defective layer differs from bulk amorphous silicon in terms of light soaking. ©2014 Optical Society of America OCIS codes: (310.6860) Thin films, optical properties; (300.1030) Absorption. References and links 1. R. 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Deangelis, “Surface states and in-depth inhomogeneity in a-Si:H thin films: Effects on the shape of the PDS sub-gap spectra,” J. Non-Cryst. Solids 114(2), 750–752 (1989). #221938 $15.00 USD Received 25 Sep 2014; revised 14 Nov 2014; accepted 17 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031466 | OPTICS EXPRESS 31466 17. F. Becker, R. Carius, J.-T. Zettler, J. Klomfass, C. Walker, and H. Wagner, “Photothermal deflection spectroscopy on amorphous semiconductor heterojunctions and determination of the interface defect densities,” Mater. Sci. Forum 173–174, 177–182 (1995). 18. J. Holovský, M. Schmid, M. Stuckelberger, M. Despeisse, C. Ballif, A. Poruba, and M. Vaněček, “Time evolution of surface defect states in hydrogenated amorphous silicon studied by photothermal and photocurrent spectroscopy and optical simulation,” J. Non-Cryst. Solids 358(17), 2035–2038 (2012). 19. J. 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Introduction
The so-called thin-film limit (TFL) or thin-film approximation is consistent with the concept of effective thickness that does not distinguish between thickness and absorption coefficient [1].The thickness d and dielectric function ε of an atomic monolayer lose their usual physical meaning and are rather defined as tensors, related to each other as ( )  , where N and ρ  is the density of dipoles and the vector of polarizability [2].Similarly, Drude theory of inhomogenous ultrathin films predicts optical properties depending only on integral values of dielectric function over the film thickness [3].Importantly, if the layer is parameterized by its absorption coefficient α, thickness d and refractive index n, the measurable optical properties A, R and T do not -in the FTL -depend on the parameters α, d or n individually, but only on their product αdn.Neither do they depend directly on the wavelength.The derivation of the TFL is usually based on a linear approximation of the Fresnel equations in the limit of thickness going to zero [1,2,[4][5][6][7].These equations have appeared recently in a simple form for transmittance of freestanding graphene [6,8], but their general derivation also for reflectance is lacking in literature [7,9].Here we show a new, simple and instructive derivation of these equations in an accurate and useful form that will be used to a new method of surface defect absorption, e.g. in hydrogenated amorphous silicon (a-Si:H).
Samples of a-Si:H are usually deposited as thin layers.Low absorptance measurements such as photothermal deflection spectroscopy (PDS) [10,11], constant photocurrent method (CPM) or Fourier-transform photocurrent spectroscopy [12,13] (FTPS) are used to evaluate defect absorption.Defect absorption may be elevated at the surfaces [11,14] enhancing interference pattern of absorptance (and hindering its smoothening by normalization by transmittance), depending on the side of illumination [15][16][17].The evaluation of surface defect is complex and may be done either by varying sample thickness [11] or by comparison of absorptance measurements from layer and substrate side and complex simulations as done in our previous work [18].However, under conditions of the TFL the defective layer can be parameterized only by only one "effective product" comprising of the product of its (virtual) thickness, refractive index and absorption coefficient.This significantly reduces the number of unknowns and the equations under TFL are also much simpler.Hence, the surface and bulk defects can be calculated directly without fitting.

Thin-film limit
We base our derivation on the conservation of energy, the continuity of the parallel components of an electric field across the layer and the assumptions of a low-absorbing medium ( n k  ) and a small thickness ( 1 d α  , dn λ  ).These approximations imply a linear dependence of the absorbed energy I A in a layer of an absorbing medium of thickness d, where I eff is the "effective" energy flux.Note that the flux I eff is treated as a constant because Eq. ( 1) neglects its attenuation.The energy flux is related to its respective electric field through the time-averaged Poynting vector S, defined by Eq. (2).
From this, it follows that also the effective electric field E eff is constant inside the layer.We define the measurable optical absorptance A as A = I A /I 0 by normalization to the energy flux of the incident wave I 0 propagating in the overlayer (refractive index n 0 ): To calculate the absorptance A, the value of E eff has to be known.In the same manner reflectance and transmittance are defined as R = I R /I 0 and T = I T /I 0 .Employing the law of energy conservation 1 = A + R + T for the whole system, we can then write: Again, based on our assumptions we neglect the evolution of the electric field throughout the ultrathin layer and assume the continuity of parallel components of electric fields: This derivation does not rely on the electric field attenuation between two distinct borders of the thin film, but assumes only the presence of an "absorbing interface" where the value of the effective field E eff has to fulfill the conditions of Eqs. ( 4) and ( 5).Assuming n k  , it follows that the Fresnel coefficients E R /E 0 and E T /E 0 are real and the absolute-value brackets in Eq. ( 4) can be omitted.From Eqs. ( 4) and ( 5), we obtain a quadratic equation for E T /E 0 featuring only one non-zero root, from which we obtain the transmittance T TFL : ( ) Once E T is known, combining Eqs. ( 5) and (3), one obtains the absorptance A TFL : ( ) Reflectance then follows from energy conservation: To test of the TFL validity, especially in the case of graphene, is interesting as it points to the difficulty to directly measure its optical parameters.More detailed discussion as well as an experimental validation of the new TFL on graphene is published elsewhere [19].

Surface defect correction method
The surface defect correction method is based on the same set of approximations as the thinfilm limit.The situation is sketched in Fig. 1.A layer with optical parameters, indexed by α 1 , d 1 , n 1 , is deposited on glass with refractive index n 2 .The ultrathin defective surface layer, labeled '01', is described only by value of the effective product (αdn) 01 .
The absorptance in the defective layer can be calculated by Eq. ( 3) where we have to insert field E eff calculated by (5).We calculate E eff from reflected electric field for the top illumination and we calculate E eff from transmitted electric filed for the bottom illumination.To distinguish between E eff for surface and interface -will be discussed later -we use labeling E 01 and E 12 respectively.When the layer is illuminated from top we use labeling "+", conversely we use "-" for illumination from the substrate side.Assuming that the effect of the defective surface absorption has magnitude below 1% (usually it is much less) we can as well neglect the effect of the defective layer on the transmittance t 210 and reflectance t 210 of the whole stack.Symbols r 012 and t 210 indicate the amplitude (Fresnel) coefficients.The ascending order of the indices indicate the "+" direction of illumination and vice versa.
For the electric field at the interface E 01 we get ( ) where E 0+ and E 2-are electric fields outside the stack, to which everything is normalized.By application of Eq. ( 3) we obtain A 01+ and A 01-, describing the absorptance of the interface layer for light incident from top and bottom respectively:  (13) The total measured absorptances A tot+ and A tot-include both the absorptances of the bulk layer and the surface defective layer.The back reflectance R 02 = (n 0 -n 2 ) 2 /(n 0 + n 2 ) 2 of the back side of the substrate is also taken into account: ( ) ( ) In the low and medium absorption region, we can, assuming Using Eqs. ( 12) and ( 13) we can access the effective product (αdn) 01 as follows: N n α λ π = + and t ij , r ij are intensity Fresnel coefficients for perpendicular incidence on i/j interface.Once knowing (αdn) 01 , we can get to A 01-from Eq. ( 13) and to A 1-from Eq. ( 15): The absorptance in bulk A 1-divided by transmittance (an interference-free quantity), can be used to calculate the absorption coefficient α 1 by Eq. ( 6) in [20].The evaluation is two-step: Standard evaluation [20,21] gives α 1 (n 1 is simulated by Cauchy formula) in high absorption region, neglecting surface defects; then α 1 n 1 are inserted into c and the right side of Eq. ( 19).

Interface defects correction method
In [18] we have shown that, if the defect density is both at top surface and at the interface with substrate, the surface correction is practically impossible.However, when the defective layer is only at the interface (labeled "12"), represented by effective product (αdn) 12 , the Eqs.( 21), (22) analogical to (19), (20) Note that the Eqs.( 16), ( 17), ( 19)-( 22) simplify when the substrate back surface can be neglected (R 02 = 0).This is the case of PDS where refractive index of ambient is close to 1.5.

Results and discussion
We simulated the complete situation by the transfer-matrix method [22].We first defined the structure as in the Fig. 1 with d 1 = 360nm, n 1 = n a-Si:H , the thickness of the defective layer was 3nm and its refractive index was the same as the layer.We calculated A tot+ , A tot -and T by transfer matrix method.Then we extracted back the absorption of bulk A 1-and A 1+ and surface effective products (αdn) 01 and (αdn) 12 by Eqs. ( 19)-( 22), see Fig. 2.
The accuracy and robustness of the calculation depends on how far from zero are the values on left and right side of the Eq. ( 19) and ( 21).This depends on the refractive index n 2 : When we are in the region of low absorptance and if n 2 = n 1 then, every time the T is in maximum, right sides of ( 19) and ( 21) go to zero, which is a singularity in the calculation.On the other hand, when n 2 >n 1 no singularity occurs in the right side of (19) whereas the right side of (21) has even more singularities because it crosses zero many times.That is why the correction performs better for defective surface than defective interface, as we see in Fig. 2.
We applied the correction method to the experiment described in [18], where we had identified defective layer on the top surface.A 360 nm thick hydrogenated amorphous silicon was deposited on glass by plasma-enhanced chemical vapor deposition.The spectra of A tot+ , A tot-and T were measured by FTPS and PDS.The measurements were repeated in time and as the last step, the sample was light soaked.A significant evolution was observed in the curves around energy 1.2 eV where absorptance corresponds to defect density [12,21,23], see Fig. 3.The A tot-curves were multiplied by c and all curves were put into absolute scale to fit to PDS results at region around 1.7eV (FTPS is not an absolute method).This gave the left side of the Eq. ( 19) and (αdn) 01 was calculated.From Eq. (20) A 1-was obtained and absorption coefficient α 1 was calculated by [20] and bulk defect states assessed by [23] assuming density of atoms in bulk ~4×10 22 cm −2 .Surface defects were calculated by dividing (αdn) 01 by refractive index of bulk (n 1 ~3.5) and assuming density of surface atoms ~10 15 cm −2 .We can observe similar trend of decrease of bulk and surface states in time.After the light soaking step bulk defect density increases significantly whereas the surface defects keep decreasing.

Conclusion
Together with a new way of thin-film limit derivation we developed a simple and direct method of evaluation of defective layer at surface of thin layer or at interface of the layer with substrate.This method compares absorption measurement from layer side and glass side and works well if only one (either at the surface or at the interface with glass) defective layer thinner than 3nm is present.Separate evaluation of surface and bulk defect states is crucial.
Here it helped to reveal different behavior of bulk and surface during light soaking.

Fig. 1 .
Fig. 1.Sample of layer with surface defective layer at the top surface or at the interface with substrate.

Fig. 3 .
photon energy (eV) Then, if we calculate the difference A tot+ and cA tot-, where c fulfills equation Fig.2.Absorption in bulk material -represented by A 1 /T ratio, and in 3nm thick defective layer -represented by (αdn) ij , either or on surface or on glass-layer interface.Lines are directly simulated, symbols are extracted by the correction method presented here from rigorously simulated data of A tot+ /T, A tot-/T (thin black lines).