Simulated SFG Spectra

(7.4)

while the non-resonant background is defined as

(7.5)

7.3.3 Simulated SFG Spectra

In order to elucidate the effect of the thickness dependence of the local field factors on the intensity of the SFG PPP signal, equation 7.2 was plotted as a function of ωIR. The SFG

spectra simulations follow the method developed by Tong et al.63 Simulated spectra were plotted using Mathematica 9.0 software (Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2012)). Complex local field factors were calculated based on previously developed theoretical modeling of interference effects in SFG spectra collected from a thin polymer film on a reflective substrate.63-65 For convenience, expressions for the local field factors and the

nonlinear susceptibilities of the methyl symmetric and asymmetric stretching vibrational modes ( and , respectively) can be found in chapter 1. The values of βaac, βccc, and βaca were

set as βaac:βccc:βaca = 4:1:4 where βaac ≈ βccc and βccc = 1.37,84 The initial orientation of methyl

groups at the surface and buried interface was 37° and 75°, respectively, based on the interface selective SFG results discussed above (The silicon substrate has a native SiO2 layer, therefore we

believe that the methyl orientations at the PMSQ/silicon and PMSQ/fused silica interfaces are similar). In addition, the molecular densities at the PMSQ/air and Si/PMSQ interfaces were assumed to be equal. SFG spectra were then simulated by replacing

in equation 7.2 with a single complex number | | . The modulus of the non-resonant susceptibility was initially

set as | | | | where | | ≈ 0.1. The modulus was then raised in small increments while varying the phase φ from 0-2π until the simulated spectra matched the observed SFG spectra. The modulus of the non-resonant susceptibility that provided the closest match to the experimental spectra was 0.15. In addition, | | was kept at a constant value at all simulated thicknesses due to the relatively constant modulus of the local field factors at the Si/PMSQ interface. The simulated peak width (Γ) was 8 cm-1 based on fitting of the experimental spectra.

Simulated SFG spectra for films of thickness 20, 70, 100, 130, and 150 nm are shown in Figure 7.7. The change of the asymmetric vibrational mode from a dip to a peak while the symmetric vibrational mode remained a dip was reproduced by changing φ. Furthermore, the best fit occurred when the methyl orientation at the buried interface was 70° and

, i.e. when the methyl groups at the PMSQ/air and Si/PMSQ interfaces had opposite absolute orientations.

Slight differences between the structure of the native oxide on a silicon wafer and the structure of fused silica may have influenced the optimized methyl orientation at the buried Si/SiO2 interface relative to the fused silica interface. Different silica structures may have

interacted differently with PMSQ which slightly altered the interfacial methyl orientation. In addition, the molecular structure of PMSQ at the interface between fused silica and a drop cast PMSQ film may be slightly different than at the interface between a silicon wafer with a native oxide layer and a thin PMSQ film. However, even considering these differences, the optimized orientation of methyl groups at the buried Si/SiO2/PMSQ interface was very similar to the

Figure 7.7. (a) Experimental and (b) simulated film thickness-dependent PPP SFG spectra collected from PMSQ films on silicon substrates.

The phase of is expected to be dependent on the film thickness due to the thickness dependence of the phase of the local field factors at the Si/PMSQ interface.63 Each non-resonant susceptibility component of ( , , , ) was modeled as a constant complex number.72,85 Therefore, the overall phase of will be determined by the linear combination of the four contributing effective non-resonant susceptibilities. Accurate determination of the non-resonant susceptibilities, however, would require the relationships between the four susceptibilities to be known.

In addition to showing a change of the asymmetric peak from a dip to a peak, the simulated spectra at different film thicknesses also show similar changes in peak positions as observed in the experimental spectra. For example, at 100 and 130 nm, the experimental peaks appear to occur at different peak centers even though both peaks centers were ~2975 cm-1. The apparent shift in peak center is due to the interference between the resonant and non-resonant susceptibility components which determine the observed peak shape. The effect of the non- resonant susceptibility on the observed peak shape is similar to the effect at metal interfaces which has been reported85,86. However, unlike gold substrates where dominates the non-resonant signal, the modulus and phase of all four non-resonant susceptibilities should contribute to the PPP non-resonant background from silicon substrates as discussed above.

The effect of on the experimental spectra can be inferred by simulating the spectra from the surface and buried interface separately. Specifically, the surface, buried interface, and SFG signal can be simulated as:

| | (7.6)

| | (7.8)

respectively.

Parameter Value Parameter Value Parameter Value

θVIS 60° n2,VIS 1.42 | | 0.15 θIR 51° n2,IR 1.39 nm2,3 1.42 λVIS 532 nm n2,SFG 1.42 nm1,2 1.20 λIR 3367 nm n3,VIS87 4.150 + 0.044i θI 37° λSFG 459 nm n3,IR87 3.433 θII initial 75° ωss 2920 cm-1 n3,SFG87 4.585 + 0.130i θII optimized 70° ωas 2975 cm-1 8 cm-1

Table 7.2. List of Parameters and Values for Local Field Factor Calculations and Simulated SFG Spectra.

Figure 7.8. Simulated SFG spectra of (a) 20 and (b) 100 nm thick PMSQ films and contributions to the simulated SFG spectra from the surface and buried interface signal.

As shown in Figure 7.8, the opposite orientations of the methyl group at the surface and buried interface contributed to the peak shape and intensity at each interface. In addition, the high tilt angle of the methyl group at the buried interface resulted in the buried interface contributing more to the asymmetric stretching peak than to the symmetric stretching peak. The larger contribution of the buried interface to the asymmetric stretching peak than to the symmetric stretching peak is consistent with the experimental SFG spectra where the asymmetric stretching peak shape was affected by the film thickness while the symmetric peak was relatively constant. For example, Figure 7.8 shows that in the simulated SFG spectra of the PMSQ/air

interface of a 20 nm thick film the asymmetric stretching peak intensity is larger than the intensity of the symmetric stretching peak. However, in the SFG signal, the intensity of the asymmetric stretching peak is decreased due to a destructive contribution from the Si/PMSQ interface. Different from the 20 nm thick film, simulated SFG spectra of a 100 nm thick film show that the asymmetric stretching peak in the SFG signal is enhanced relative to the intensity at the PMSQ/air interface due to a constructive contribution from the Si/PMSQ interface. Furthermore, the simulated SFG spectra could not be approximated as the sum of the surface and buried interface signal (i.e. | | | | ) due to cross terms in the squared expression of equation 7.8.71

More generally, resonant and non-resonant contributions to the overall SFG signal were deduced by adjusting the phase of in thickness dependent simulated SFG spectra until the simulated spectra matched the experimental spectra. Separation of the resonant and non-resonant contributions to the SFG signal enabled the interfacial resonant susceptibilities to be obtained. The deduced interfacial resonant susceptibilities, which are determined by interfacial molecular structures, can then be used to characterize the molecular structure of the surface and buried interface.