The second step towards calculating the atom-solid dispersion coefficient C3 is to determine the response g(iω) of the solid. Using the Kramers-Kronig relation Eq. (3.29) one has
The material of interest in this work is amorphous silicon nitride (SiNx) because the transmission gratings used in the new atom diffraction experiments are etched out of an approximately 100 nm thick layer of SiNx. Amorphous silicon nitride is an important compound in the field of microelectronics where it is used as an anti-reflection coating in lithographic processes. By varying the manufacturing method and the parameters involved it is possible to make SiNx films of well-defined properties. In particular, the the optical band gap width is a function of the stoichiometric coefficientx, which is controlled by the relative amount of gases from which the films are deposited [86]. An important step in the manufacturing process is to check the band gap by measuring the frequency-dependent index of refraction n(ω) and the absorption coefficient κ(ω). These are connected with the complex dielectric function (ω) by the well-known formulae
0(ω) =n2(ω)−κ2(ω) (3.38) and
00(ω) = 2n(ω)κ(ω). (3.39)
The transmission gratings for the atom diffraction experiments to be discussed in this work have been made by Savas and co-workers [2] at the MIT using a process called low pressure chemical vapour deposition (LPCVD). The authors have pro-vided a measurement [85] of the optical datan(ω) andκ(ω) in the range between 1 eV and 6 eV for silicon nitride films from which the transmission gratings are made. A plot of the data is shown in Fig. 3.2. With Eq. (3.38) it is possible to
3.4. RESPONSE OF THE SOLID 37
Figure 3.2: Energy-dependent refractive index n(E) and absorption coefficient κ(E) for two silicon nitride samples as communicated by Savas [85]. The onset of κ(E) at about 2.2 eV marks the width of the optical band gap ∆ inside the amorphous material. The static value of n(E) as arising from electronic excitations in the solid can be extrapolated asn(0)≈2.1 . The plot of κ(E) is cut off on the right as on the original communication.
extract the imaginary part of the dielectric function00(ω) in the given frequency range. However, in order to determine the dispersion coefficientC3 via Eq. (3.28) one needs(iω) at imaginary frequencies and this requires by the Kramers-Kronig relation Eq. (3.30) the knowledge of 00(ω) over the entire frequency range from zero to infinity.
In the past, people proceeded by fitting 00(ω) in the limited range with an ansatz that included one or more electronic oscillators representing the solid and so were able to extrapolate 00(ω) to high frequencies. While more-oscillator solutions proved difficult to handle because of the large number of parameters, the one-oscillator approach, similar to that described above for atoms, lacks some accuracy. The crucial point is that a simple oscillator model does not account for the optical band gap ∆, i.e. a range of frequencies 0 < ω < ∆ where no electronic transitions are likely to occur within the solid. This leads to errors in the calculation of C3 Eq. (3.28) if the characteristic atomic transition frequency falls in the range of the optical band gap of the solid as is the case for metastable rare-gas atoms and silicon nitride (cf. Table 3.1).
In a recent publication Zollner and Apen of the Motorola company have stud-ied the optical properties of LPCVD-made amorphous silicon nitride layers in detail. They point out that the imaginary part 00(ω) of the dielectric function is well parameterized by the Tauc-Lorentz formula [97]
00(ω) = Θ(ω−ΩT) AΩΓ(ω−ΩT)2
[(ω2−Ω2)2 + Γ2ω2]ω, (3.40) where the optical band gap appears explicitly as ∆ ≡ ¯hΩT. Θ is the step
func-tion and A,Ω,Γ are the strength, frequency, and spectral width, respectively, of one characteristic electronic transition within the solid. The measurement of refractive indexn(ω) and the absorption coefficientκ(ω) communicated by Savas [85] bears a convincing resemblance to that published by Zollner and Apen for LPCVD anti-reflection coatings, while it differs from those for silicon nitride made by another technique in terms of the magnitude of n and the onset of κ which marks the width of the optical band gap. By comparison with other sources in the literature [87, 88, 89, 90, 91] it turns out that the latter two features can in fact be used to identify the manufacturing process by looking at the optical data in the given range. As a result, it is plausible that the measurement of Savas has been performed on a LPCVD silicon nitride sample similar to the anti-reflection coatings tested by Zollner and Apen. Savas asserts that the measured material is identical with that of the transmission gratings that have been used in the diffraction experiments to be discussed.
From the foregoing observations it is concluded that the best way to ex-trapolate 00(ω) for the transmission gratings is to use the Tauc-Lorentz formula Eq. (3.40). It is expected to be more correct for low frequencies in the range of the optical band gap, while in the limit of large frequencies it is comparable to usual extrapolation methods. On extracting00(ω) via Eq. (3.38) from Savas’ data and fitting to it the Tauc-Lorentz formula Eq. (3.40) one obtains good agreement as is illustrated in Fig 3.3.
By inserting the fitted00(ω) into Eq. (3.37) and subsequent numerical integra-tion the electronic response g(iω) of silicon nitride is obtained. Combining this result with the atomic polarizability as in Eq. (3.33), finally, the atom-SiNx dis-persion coefficientC3 is determined by numerical integration of Eq. (3.28). Table 3.2 shows the appropriate values for ground-state and metastable rare-gas atoms.
The deuterium molecule D2 that also appears in the table is hereby and in the following counted among the ground-state atoms because due to its small size it can be treated like an atom in the present diffraction experiments.
atom He Ne D2 Ar Kr Ne* He* Ar* Kr*
C3 0.136 0.274 0.412 0.936 1.346 3.624 3.841 5.146 5.551 Table 3.2: Dispersion interaction strength C3 in units of meV nm3 for ground-state and metastable rare-gas atoms with silicon nitride. The calculations are based on the non-retarded Lifshitz formula Eq. (3.28), into which the one-oscillator approximation of the atomic polarizability Eq. (3.33), and the response of the solid, as found with the help of the Tauc-Lorentz parameterization Eq. (3.40), have been inserted.