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An exact solution of the phase retrieval problem is described as applied to in-situ X-ray reflectometry of a growing layered film. The following statement is proved: if the reflectivity R and the derivative dR/dt are known at the time t, then the real and the imaginary parts of the amplitude reflectivity r(t) are found uniquely at this point t.
©2008 Optical Society of America
1. Introduction
Nowadays, X-ray reflectometry is widely used to determine the depth distribution of the dielectric constant within a layered system [1, 2]. However, the solving of the inverse problem of X-ray reflectometry is often hampered by the ambiguity in the solution, essentially due to the absence of information about the phase of the amplitude reflectivity (see, e.g., Ref. [3] and references therein). Manifestly, the phase retrieval problem, i.e., the determination of the phase of the reflected wave from X-ray reflectivity measurements, cannot be solved uniquely in the general case. Moreover, in the phase retrieval method commonly used, which is based on the analysis of the reflectivity curve in the frame of the simplified Born or Distorted-Wave Born approximations, such an ambiguity on the evaluation of the phase persists [1]. Therefore, the possibility of inferring an exact solution of the problem, even in specific cases, is of special interest.
Majkrzak, et al. (see Ref. [4] and references therein) have proposed an exact theoretical approach to the phase retrieval problem in neutron and X-ray reflectometry known as the reference layer method or closely-related surround method. It is based on the following statement: for three samples composed of the same unknown layered structure and of different, but known, reference layers (substrates), the complex amplitude reflectivity of the unknown structure can be found exactly by measuring the reflectivity from all three samples. The approach, although correct from a mathematical point of view, has a number of practical disadvantages. (a) It is only valid for non-absorbing materials. (b) The optical properties of the different substrates or reference layers should be known a priori. (c) It is necessary to guarantee the absolute identity of the unknown structure within the 3 samples in spite of the presence of different substrates or reference layers.
In the present work, we describe a method of exact phase retrieval that is free from the shortcomings mentioned above. The method is applicable to a layered film for which the reflectivity has been measured in-situ during growth, so that, at a point t in time, both the reflectivity R and the derivative dR/dt are known. In this case both the real ℜ[r(t)] and imaginary ℑ[r(t)] parts of the amplitude reflectivity r can be found exactly.
2. Theory
Below we will consider a film growing by deposition of a material on the top surface of either a bulk substrate or of a more complex layered-structured substrate. The chemical composition of the incident flux of particles may vary with time, so that the dielectric permeability of the film may change with depth. Using the imbedding method [5], one can deduce the equation describing the variation of the amplitude reflectivity r with varying the film thickness h:
where k=ω/c is the wave number, θ the grazing angle of an incident beam, and ε(h) the dielectric permeability at the top of the film. (In the Appendix, Eq. (1) is derived using a technique somewhat different from that given in [5]).
Equation (1) implies that the dielectric permeability inside the film remains unchanged. However, there are cases where the deposition of material on top of a film affects the film structure underneath. Typical examples include the cases of implantation and diffusion of atoms [6]. In the temporal intervals where such processes take place, and only during these very short intervals, Eq. (1) is invalid.
The behavior of the intensity reflectivity R=|r|2 can also be established from Eq. (1):
where the asterisk denotes the complex conjugation. At first glance, Eq. (2) seems unusable, as it contains ε(h), the unknown solution to the inverse problem. Moreover, the derivative dR/dh is not a quantity measured directly in the experiment, because the film thickness is not necessarily a linear function of the time. However, the temporal derivative dR/dt=(dR/dh)·(dh/dt) is a well-defined function, which can be obtained directly from the experimental curve R(t). The deposition rate is equal to dh/dt=qµ/ρ, where q is the incident flux of particles (per unit area and time), µ is the mass of an incident particle and ρ the material density at the top of the film. In its turn, the dielectric permeability in the X-ray region is written as ε=1-Aρ/µ·(f1-if2), where f1-if2 is the complex atomic scattering factor determined by the chemical composition of the incident flux. The constant A is expressed in terms of the classical electron radius a0 and of the radiation wavelength λ:A=a0λ2/π. Finally, we find an expression for the temporal derivative of the reflectivity (Eq. (3)) that does not explicitly contain the films parameters:
When R and dR/dt are known, Eq. (3) is not anymore a differential equation of the reflectivity R, and becomes an algebraic equation establishing a linear relation between the real and imaginary parts of the amplitude reflectivity r, while the other parameters in Eq. (3) are known or experimentally measured. In general, the flux of particles q and the chemical composition (f 1 and f 2) of these particles vary with time.
After solving Eq.(3) together with the obvious relation
we determine both ℜ(r(t)) and ℑ(r(t)) directly from the experimental data, without using any model for describing the reflection from the media. As the last equation has a quadratic form, there are two possible solutions of the phase retrieval problem, the correct one being chosen on the basis of additional physical considerations.
Let us formulate the conditions of validity of Eq. (3):
- The equation is valid for a growing film assuming that both the reflectivity R(t) and the derivative dR/dt are known at a certain time t. The knowledge of these two quantities is sufficient to find the complex amplitude reflectivity r(t) at the time t.
- The composition inside the film should not change during the temporal interval necessary to measure the derivative dR/dt. Therefore, Eq. (3) may be valid at certain stages of the film growth and invalid during others, in particular, when implantation or diffusion of atoms takes place.
- Equation (3) was deduced under the important assumption of the material polarizability (ε-1) to be proportional to the density. Hence, the equation can be applied to X-ray or neutron reflectometry but is invalid, e.g., in the case of visible light.
- The equation is valid for s-polarized radiation. However, the effect of the polarization can be neglected because the difference between s-and p-polarized X-ray or neutron reflectance is small under grazing incidence.
If these conditions are fulfilled, then the phase retrieval problem is solved exactly, in a very simple manner and in an explicit analytical form. Since Eq. (3) is a linear algebraic equation for the unknown functions ℜ(r(t)) and ℑ(r(t)) the phase of the reflected wave at a point t in time can be found without any knowledge of the pre-history of the structure growth. Hence, there is no necessity to monitor the reflectance during the whole deposition process for finding the phase of the amplitude reflectance of, e.g., a multilayer structure. It is sufficient to measure the reflectance during a short temporal interval at the last stage of growth, the goal being only to find the derivative dR/dt in the end of the deposition. Then the phase of the amplitude reflectance can be found via Eq. (3) independently of the chemical composition inside the structure, the presence of interlayers, the material of the substrate, etc. The only parameter that needs to be known is the value and the chemical composition of the flux of particles during the last stage of the growth.
Similarly, let us consider the case of a substrate composed of an unknown complex layered structure. If a thin film is deposited on top of this structure and the reflectivity measured in-situ in the last stage of the deposition, then the phase of the amplitude reflectance of the total system can be determined.
As the effect of absorption is contained in Eq. (3), the method is applicable to any wavelength of the probe beam from hard X-rays to EUV radiation, independently of the value of absorption and as long as the material polarizability remains proportional to its density.
3. Experimental results and discussion
Let us consider the result of the simplest experiment on the measurements of the reflectivity as a function of time during the growth of a tungsten film on a super-polished silicon substrate (Fig. 1, black dots).
Fig. 1. Reflectivity versus deposition time from a growing tungsten film measured at an X-ray energy of 17.5 keV and with a grazing angle θ=0.5° (dots). The solid curve was calculated assuming a constant density of the film.
The measurements were performed at the beamline BM5 of the ESRF. A special vacuum chamber intended for sputter deposition was installed at the beamline to allow measuring in-situ both the reflectance and the scattering from films grown from various material targets. A detailed description of the deposition system is given in Ref. [7]. The X-ray energy was set to 17.5 keV and the grazing angle of the incident beam set equal to 0.5°, i.e., to nearly twice the critical angle of total external reflection for bulk tungsten. The incident flux was estimated to q=7.26·1013 atom/cm2/s resulting in a typical growth rate for tungsten of about 12 pm/s. The reflectivity measurements were performed every 7 s, which corresponds to an increase of the film thickness of about 80 pm, equivalent to one third of a monolayer of tungsten.
The phase φ(t) of the amplitude reflectivity r=|r|exp(iφ) extracted from the experimental curve R(t)=|r(t)|2 with the use of Eqs. (3)–(4) is presented in Fig. 2, circles.
As mentioned before, there are two possible solutions of the phase retrieval problem, as illustrated in curves 1 and 2. Curve 3 in Fig. 2 and the solid curve in Fig. 1 were calculated in the frame of the simplest model possible, i.e., assuming a constant film density over depth (18 g/cm3, i.e., a dielectric permeability ε=1-1.94·10-5+i·1.79·10-6). Since curve 1 and 3 (model) in Fig. 2 are close to each other, we retain curve 1 as the solution to the phase retrieval problem that corresponds to reality. Nonetheless, the phase inferred from the experimental data as well as the measured reflectivity shown in Fig. 1 differ noticeably from the model calculations, suggesting that the simplest model of a film with constant density is not fully correct. The discrepancy between the measured and the calculated curves may be caused by (a) the variation of the tungsten density with depth, (b) the presence of an interlayer at the substrate – film interface due to implantation and diffusion of tungsten atoms at the initial stage of a film growth, and, in general, (c) the presence of roughness, although estimations showed this effect to be negligible for the very smooth sample studied. The rms roughness in the [4·10-4 to 5·10-2 nm-1] range of spatial frequency was equal to 0.13 nm for the Si substrate and 0.20 nm for the external film surface [8].
Fig. 2. Two solutions of the phase retrieval problem (curves 1 and 2) found directly from the experimental curve shown in Fig. 1. Curve 3 shows the phase evolution calculated assuming a uniform tungsten film with constant density. For illustrative purpose, we assumed the phase to vary within the [-π/2, 3π/2] interval.
4. Conclusion
This work described and presented a very simple and exact approach to the phase retrieval problem in X-ray reflectometry. The approach requires in-situ reflectivity measurements during a layered film growth, so that both the reflectivity R(t) and the temporal derivative dR/dt are known. These two values are sufficient to find the phase of the amplitude reflectivity. The phase retrieval at a point t in time does not require any knowledge of the pre-history of the structure growth, including the chemical composition inside the structure and the substrate, and the presence of interlayers.
If the reflectivity curve R(θ,t)versus grazing angle could be measured at any point in time, we would be able to find the phase φ(θ,t) versus angle. On this basis, one could solve the inverse problem of X-ray reflectometry more correctly and analyze the variations of the dielectric constant profile with time. Such an angular-dispersive and time-resolved setup was built earlier for use with a laboratory X-ray source [6]. An equivalent setup combined with a powerful synchrotron source, would improve the temporal sampling rate.
Appendix
Let us direct the Z-axis towards the depth of the sample and consider a plane wave coming from the vacuum onto the growing film under a grazing angle θ. Then, the wave field, considered as a function of two variables z and h, complies with the following asymptotic conditions:
where r(h) and t(h) are the amplitude reflectance and transmittance, and ε s is the dielectric permeability of the substrate (at z→+∞).
In accordance with Eq. (A1),
On the other hand we can write that
From Eq. (A1) we obtain immediately that
To find the derivative ∂E/∂h we use the integral form of the wave equation:
where the wave field E(z,h1) corresponds to the reflection from a film of thickness h 1<h. Equation (A5) assumes that the dielectric permeability ε(z) remains unchanged at z>-h 1 during film growth from thickness h 1 to h.
The Green function in Eq. (A5)is written as where the field E̅(z,h), corresponding to the wave coming onto the film surface from the interior of the sample, behaves as and the Wronskian W(h)=E′E̅-EE̅′=2iκ 0 t̅(h)exp[i(κs-κ 0)h].
After differentiating Eq. (A5), we find the general equation for ∂E/∂h. Then, directing h 1→h and z→-h, we obtain
where ε(-h) is the dielectric permeability at the top of the film. Combining Eqs. (A2)–(A4) and (A6) we find the final Eq. (1) describing the variation of the amplitude reflectivity with the film thickness.
Acknowledgment
One of the authors (I.K.) acknowledges the support of the ISTC (project #3124).
References and links
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