Rapid thickness reading of CH₃NH₃PbI₃ nanowire thin films from color maps

3.1 Calculation of color components

MAPbI₃ nanowires can have different morphologies and dimensions according to the parameters involved in the synthesis process (solvent concentration, temperature, sliding speed, etc.). The slip-coated nanowires can have rectangular or hollow cross section and thickness ranging between few to hundreds of nanometers, as seen in the three-dimensional AFM reconstruction images shown in Fig. 1b. Using a silicon substrate coated with a dielectric layer (e.g., SiO₂, TiO₂) the difference in height between the nanowires corresponds to a different color contrast in the optical images taken under white light illumination. Slip-coated perovskite nanowires exhibit a variety of colors according to their thickness and the dielectric layer, as shown in Fig. 1a, providing thus an ideal testbed for our optical method to measure nanowire thicknesses.

The different colors shown in Fig. 1a and their dependence upon the material thickness can be understood and reproduced by standard thin film optics theory. To this end, we follow Ref. 22 which describes a procedure to calculate the dependence of the red, green, blue (RGB) parameters on the thicknesses and optical properties of multilayer systems. We express the reflectance of the multilayer system MAPbI₃/dielectric/Si as follows 14, 16, 23:

(1)

where λ is the wavelength of the incident light, r₁, r₂, and r₃ are the amplitudes of the light reflected at air/MAPbI₃, MAPbI₃/dielectric, and dielectric/silicon interfaces, respectively. In Eq. 1, δ₁ = (2πd₁/λ)(n₁ − ik₁)cosθ₁ and δ₂ = (2πd₂/λ)(n₂ − ik₂)cosθ₂ are the phase changes across MAPbI₃ and the dielectric, where d_j, n_j + ik_j, and θ_j are the layer thickness, the complex index of refraction, and the propagation angle for MAPbI₃ (j = 1) and dielectric (j = 2), respectively (see scheme of Fig. 1c). The effect of the numerical aperture (NA) of the microscopic objective can be taken into account by performing an arithmetic average of the incident angle from θ = 0 to θ = sin⁻¹(NA). From the angle-averaged reflectance, we calculate numerically the tri-stimulus components as follows:

(2)

where P(λ) is the power spectrum of the light source and x(λ), y(λ), and z(λ) are the CIE color-matching functions. Finally, we convert Eq. 2 to the RGB color scheme using (R, G, B)^T = (X, T, Z)^T M⁻¹, where the transformation matrix M depends on the white reference of the light source and other instrumental parameters 22. Using the same method, and redefining the phase changes and the reflected amplitudes, we calculate also the RGB parameters, denoted R₀, G₀, and B₀, for dielectric/Si multilayer systems without MAPbI₃. Further details are given in the Supporting Information (online at: www.pss-a.com).

In Fig. 2, we show the calculated perceived colors of MAPbI₃/dielectric/Si as a function of the MAPbI₃ and of the dielectric layer thicknesses, respectively, d₁ and d₂, under an incident halogen light. In the calculations, we varied the thickness of the dielectric layer from d₂ = 100 nm to d₂ = 500 nm for the case of SiO₂, Fig. 2a, and from d₂ = 50 nm to d₂ = 300 nm for TiO₂, Fig. 2b. For all cases we have used the value NA = 0.8 for the numerical aperture of the microscope. From Fig. 2a we see that, for a given SiO₂ thickness, the calculated color of the multilayer changes distinctly with the thickness d₁ of MAPbI₃, at least for d₁ smaller than about 80–100 nm. Thicker MAPbI₃ films tend to have more saturated colors, with progressively less pronounced color variations as d₁ increases. A similar trend is observed for the MAPbI₃/TiO₂/Si multilayers shown in Fig. 2b. However, in this case color saturation sets in for somewhat smaller values of d₁ (about 50–70 nm), which is explained by the larger index of refraction n₂ of TiO₂ compared to that of SiO₂ (see Supporting Information, Fig. S2b). The difference between n₂ of TiO₂ and that of SiO₂ explains also that colors tend to saturate at thinner layers of TiO₂ compared to SiO₂.

From the calculated colors for d₂ = 280 nm of Fig. 2a, we see that the MAPbI₃ wires having blue color in Fig. 1a are expected to have thicknesses in the range between ≈10 and ≈40 nm, while the orange portions of the wires should indicate thicknesses larger than about 50 nm. These estimates are in fair qualitative agreement with the thicknesses values measured by AFM (Fig. 1b), suggesting that the calculated color charts of Fig. 2 may be exploited to characterize the size of MAPbI₃ wires.

3.2 RGB distance

Calculated color maps as those shown in Fig. 2 are useful in establishing a qualitative or even semi-quantitative correspondence between film thicknesses and observed colors of multilayer systems. This correspondence can be set in more quantitative terms by calibrating the calculated colors with known thicknesses of the deposited material measured for example by AFM 18, 19. Here we introduce a one-step, calibration-free method that enables to reconstruct the film thickness directly from optical images. To this end, we measure the deviation from the experimental and calculated RGB parameter as follows. We denote , , and the RGB components extracted at a given position of an optical image of MAPbI₃/dielectric/Si, and , , and the RGB components of the bilayer system dielectric/Si. Using the calculated and experimental RGB components we construct the vectors

(3)

from which we define the RGB distance as follows:

(4)

In the ideal situation in which calculated RGB parameters reproduce exactly the measured ones, the thickness d₁ for a given would be given by the solution of . There are, however, always some discrepancies between the calculated and measured RGB parameters, due to, for example, the nonperfect flat shapes of the nanowires, slight inhomogeneities of the dielectric thickness, and the approximated spectrum of the light source. We thus estimate the thickness of MAPbI₃ at a given position of an optical image by finding the value of d₁ for which has an absolute minimum. Since the components of and are measured with respect to the corresponding RGB values of the substrate, absolute minima of are expected to give a fair estimate of the actual wire thickness. From the color chart of Fig. 2, we see that in principle this method is able to estimate MAPbI₃ thicknesses smaller than about 80–100 nm for SiO₂/Si substrates and 50–70 nm for TiO₂/Si substrates, which are the regions in Fig. 2 where the calculated colors change more visibly with d₁.

3.3 Reconstruction of MAPbI₃ thicknesses

To reconstruct the wire thicknesses from optical images of MAPbI₃/dielectric/Si multilayers, we first calculate the RGB components (as described above) by increasing d₁ from 0 to 100 nm in steps of 1 nm, with dielectric thickness fixed at the measured value. Subsequently, for each pixel of the optical image we look for the value of d₁ such that is smallest. This process generally takes only a few seconds on a laptop computer even for images of about 1000 × 1000 pixels. We test this procedure on selected regions of the optical image of MAPbI₃/SiO₂/Si, shown in Fig. 1a, by comparing the values of d₁ obtained from minimization of with the heights extracted from AFM measurements, as shown in Fig. 3. From the d₁ maps shown in Fig. 3c, g, and m, and from the corresponding profiles reported in Fig. 3d, h, and n, we see that our method enables us to identify with overall good accuracy wire thicknesses comprised between 20 and 50 nm. Remarkably, even if the AFM profiles reveal that the upper surface of the wires is generally not perfectly flat, the reconstructed thicknesses are in good accord. The lower spatial resolution of the optical images compared to the AFM micrographs and the finite numerical aperture of the microscope contribute to round-off the fine details of the reconstructed thickness profiles. In this respect, the orientation of the nanowire surfaces does not appear to influence the overall performance of the method. Furthermore, wires thickness smaller than about 15 nm are generally underestimated by the optical reconstruction, as seen by comparing the AFM profile of the first (last) peak of Fig. 3d (3h) with the corresponding profile of the reconstructed thickness. Note also that MAPbI₃ thicknesses greater than about 70 nm are not recognized, as evidence for example in Fig. 3n, in which the peak of height ≈100 nm in the AFM profile is completely missing in the d₁ profile. By looking at optical image of Fig. 3i, we see that the missing peak comes from a wire that has colors not much different from that of the SiO₂/Si substrate. We expect thus, that the use of different thicknesses of the SiO₂ layer would possibly increase the visibility of wires also thicker that 70 nm.

To assess the effect of the dielectric thickness and of the deposition process, we have reconstructed the wire thicknesses from optical images of MAPbI₃ slip-coated on SiO₂/Si substrates with d₂ = 100 nm and d₂ = 500 nm, as shown in Fig. 4. From Fig. 4c, we see that the distribution of the reconstructed d₁ values for d₂ = 100 nm has a broad peak centered at d₁ ≈ 55 nm with a tail extending above 80 nm. Note also that a portion of wires results to have thicknesses of about 100 nm or larger, as indicated by the peak at d₁ ≈ 100 nm. A similar analysis of MAPbI₃/SiO₂/Si with d₂ = 500 nm, shown in Fig. 4d–f, suggests that the wires in this case have thicknesses broadly distributed up to about d₁ ≈ 60 nm. The bottom panels of Fig. 4 show our analysis for a dense film of MAPbI₃ nanowires slip-coated on 500 nm thick SiO₂ substrate: the reconstructed nanowire film thickness is narrowly distributed around approximately 15 nm, implying thus a fairly homogeneous coverage of the substrate. The results of Fig. 4 show also that our method can automatically reconstruct thicknesses profiles from large-scale optical images, giving thus the possibility of performing statistical analyses as those presented in Fig. 4c, f, and i.

In Fig. 5, we show the results of applying the minimization of to optical images of MAPbI₃ wires slip-coated on TiO₂/Si substrates with d₂ = 100 nm. Compared to the case with SiO₂/Si substrates, the AFM images (Fig. 5b and f) reveal a substantial amount of MAPbI₃ nanoparticles deposited on the substrate. These nanoparticles are much less visible in Fig. 5a and e due to the lower resolution of the optical imaging, and are thus largely not recognized by our method. Nevertheless, the reconstructed wire thicknesses shown in the maps and profiles of Fig. 5c and g and Fig. 5d and h are in overall good accord with the heights measured by AFM. We note that the AFM profiles of MAPbI₃/TiO₂/Si display wire cross sections that are much more corrugated than those shown in Fig. 3, with an average wire thickness which is, however, well captured by the reconstructed profiles shown in Fig. 5d and h.