Quantitative analysis of lattice disorder and crystallite size in organic semiconductor thin films

The crystallite size and cumulative lattice disorder of three prototypical, high-performing organic semiconducting materials are investigated using a Fourier-transform peak shape analysis routine based on the method of Warren and Averbach (WA). A thorough incorporation of error propagation throughout the multistep analysis and a weighted fitting of Fourier-transformed data to the WA model allows for more accurate results than typically obtained and for determination of confidence bounds. We compare results obtained when assuming two types of column-length distributions, and discuss the benefits of each model in terms of simplicity and accuracy. For strongly disordered materials, the determination of a crystallite size is greatly hindered because disorder dominates the coherence length, not finite size. A simple analysis based on trends of peak widths and Lorentzian components of pseudo-Voigt line shapes as a function of diffraction order is also discussed as an approach to more easily and qualitatively assess the amount and type of disorder present in a sample. While applied directly to organic systems, this methodology is general for the accurate deconvolution of crystalline size and lattice disorder for any material investigated with diffraction techniques.

Figure 1

Different mechanisms responsible for diffraction peak broadening. A collection of crystallites of different shapes and sizes is sketched in (a), showing a slight variation of the interplanar distance $d_{\mathit{hkl}}$ from one crystallite to another (lattice-parameter fluctuation). The column length $M$ in a crystallite can depend on its shape as shown in (b). Noncumulative vs cumulative disorder is sketched for a small number of diffracting planes in (c). Noncumulative disorder (top) involves random fluctuations about the ideal lattice positions (gray); for cumulative disorder (bottom), statistical fluctuations about the lattice spacing occur, which accumulate to destroy the ideal lattice. The histograms shown are the lattice positions of 10 000 one-dimensional (1D) simulated lattices, showing the loss of predictive ability due to distortion propagation that is the signature of paracrystalline disorder. Sketches of disordered two-dimensional (2D) lattices are shown in the far right, with the ideal lattice indicated by dotted gray lines.

Figure 2

Data processing and weighted Warren-Averbach full-fit analysis of P(NDI2OD-T2) along the lamellar stacking, specular diffraction direction. (a) Corrected data with measurement error, on a semilogarithmic scale, with coarse fit functions for individual peaks (solid orange), background (dotted gray), and the combined coarse fit (dashed red); inset: chemical structure of P(NDI2ODT2). The isolated peaks for $m$ $=$ 1 to $m$ $=$ 5 (b) and their respective normalized Fourier transforms (c) with the associated propagated error. The peak contribution from the cosine (solid cyan) and sine (dashed green) terms are shown with the raw isolated peaks in (b). The large errors for higher diffraction order peaks [especially (400), (500)] are shown here to reflect the uncertainty associated with multiple processing steps with these poorly resolved peaks. The weighting used in FT fits is based on these error values, thus, the FT data points with excessive error (those with the least certainty) play little role in the final FT fit. (d) The normalized Fourier transforms with the weighted WA full fits assuming a column-length distribution following a delta function (dashed blue) and a gamma function (solid red); inset: log scale representation of FT fits. (e) Resynthesis of the peaks in reciprocal space using the fit results shown in (d) match well with the raw data.

Figure 3

WA full-fit analysis of TIPS-pentacene specular diffraction. (a) Corrected specular data; inset: TIPS-Pn chemical structure. (b) Fourier transforms of the isolated peaks, with error, and fits utilizing delta (dashed blue) and gamma (solid red) distribution functions (inset: from narrowest to broadest, the 001, 002, 003 peaks).

Figure 4

WA full-fit analysis of π-stacking grazing diffraction of PBTTT. (a) Corrected grazing incidence data with associated errors, as a reference, we include grazing incidence data in the orthogonal direction (dotted line) showing the peak positions of the chain backbone-related contributions; inset: PBTTT chemical structure. (b) FTs of isolated peaks, with error, and fits utilizing delta (dashed blue) and gamma (solid red) distribution functions (inset: from narrow to broad, the 010 and 020 peaks). Note that the fits to the FT data overlap regardless of the assumed size distribution.

Figure 5

Robustness of full-fit analysis approach on P(NDI2OD-T2) data set. The fit results for $M$, $g$, and $e_{\mathrm{rms}}$ are shown for fitting routines incorporating weighted fits and compared to those without weights. Fits utilizing the $\gamma$ size distribution are shown as solid lines and filled markers, while δ distributions are shown with dashed lines and open markers. $X$ represents a fit incorporating peaks of order 1 through order $X$. The symbols at $X$ $=$ 1($+$1) and 1($+$2) are fits for just the first peak ($m$ $=$ 1), with one or two of the lowest frequency orders from the second peak, which can give solutions nearing that of the full-fit five-peak analysis.

Figure 6

Size distributions determined from fit results of P(NDI2OD-T2) data set. The gamma distribution function (shaded red area) provides an average column length ($M_{\mathrm{\gamma }}$) and variance (±$w_{\mathrm{\gamma }}$) indicated by the arrows, while the delta function defines a column length of $M_{\mathrm{\delta }}$ (blue vertical line). The relevant Fourier transforms of the (100) peak are shown as an inset, assuming each of these distributions (solid red for gamma function distribution, dashed blue for delta function distribution).

Figure 7

Relation of pseudo-Voigt mixing parameter (η) to $g$ and $e_{\mathrm{rms}}$. (a) Depiction of pseudo-Voigt line profile in the full range of $\eta$'s. $\eta$ $=$ 0 represents a Gaussian line shape, and $\eta$ $=$ 1 represents a Lorentzian. (b) $\eta$ maps showing the mixing parameter associated with various $g$ and $e_{\mathrm{rms}}$ values for an average column length of 10 repeat units (left) and 100 repeat units (right). For each of the panels in (b), the top left corner tends to $\eta$ $=$ 1 and the bottom right corner tends to $\eta$ $=$ 0.

Figure 8

Peak parameters determined from isolated peaks of TIPS-Pn (blue circle), P(NDI2OD-T2) (green triangle), and PBTTT (red square). (a) Peak width, Δ${}_q$, as a function of peak order (the lines are guides to the eye). (b) Pseudo-Voigt mixing parameter, η, as a function of peak order. The predictions of η from maps such as those in Fig. 7 are shown as shaded areas.