Local structure analysis of disordered materials via contrast variation in scanning transmission electron microscopy Open Access

Diffractometry techniques, which utilize x rays, neutrons, and electrons as incident beams, are indispensable for the analysis of crystal structures in materials science. In this regard, the diffraction intensity, I(g), depends on the atomic scattering factor, f(g), of the incident x ray/electron (or scattering length of neutron) as well as a phase term, exp(2πigr), where g and r are the scattering and position vectors of each atomic arrangement, respectively.¹ Although crystallographic analysis primarily depends on the diffraction intensity, there are certain complications when amorphous or disordered materials are considered. Disordered materials of interest often consist of multiple elements (i.e., non-monoatomic), and there are insufficient values to solve the equation of the diffraction intensity, $Ig=∑mfm(g)exp(2πigrm)2$⁠; therefore, the elemental information of f_m(g) and the atomic arrangement, r_m, of the phase term cannot be fully refined. A proposed solution to overcome this difficulty is the contrast variation scheme in which the x-ray diffraction (XRD) and neutron diffraction (ND) intensities are simultaneously refined.^2,3 Based on the different atomic number dependencies of the scattering properties in XRD and ND, the r_m values can be estimated. Anomalous (resonance) x-ray scattering is another similar approach used for differentiating the x-ray scattering factor of a specific element of interest.⁴ 

Scanning transmission electron microscopy (STEM) is one of various electron microscopy techniques,⁵ and direct observation of atomic arrangements has been realized using an aberration corrector.⁶ The STEM instrument enables simultaneous capture of bright-field (BF), annular bright-field (ABF), and annular dark-field (ADF) images⁷ [see Fig. 1(c)]. It has been confirmed that the contrast of each STEM image has a different dependence on the atomic number, Z. For instance, ADF imaging is routinely employed because of its intuitive imaging features⁸ and its intensity being roughly proportional to the power of the atomic number,⁹ i.e., Z^1.5–2.0. Conversely, ABF imaging is used for visualizing light (low Z) elements, as a relatively moderate atomic-number-dependence has been reported^10,11(e.g., Z^1/3). Contrary to ADF and ABF, the contrast of BF images highly depends on the imaging conditions (e.g., defocus of the objective lens), while a Z^0.6–0.7 dependence has been reported for crystal-structure images¹² acquired under specific optimum conditions (i.e., an ultrathin specimen with the Scherzer defocus). In view of the varying Z-dependencies discussed earlier, by employing a contrast variation scheme between different STEM imaging techniques, advanced crystal structure analysis strategies can be established. The purpose of this study is to explore a novel method for the local crystallographic analysis of disordered materials via contrast variation in STEM imaging.

The amorphous structure model was generated through a melt-quenching procedure in a classical molecular dynamics (MD) simulation using the LAMMPS code¹³ (released on March 16, 2018). The simulation box was a cube with an edge length of 200 Å. The atom density in this system was 2.2 g/cm³, and 529 242 atoms (176 414 Si and 352 828 O) were contained within the NVT ensemble. A time step of 1 fs was used in the Verlet algorithm, and the system was described by pair potentials with short-range Born–Mayer repulsions and long-range Coulomb interactions as follows:

where r is the interatomic distance between atoms i and j; B_ij and ρ_ij, respectively, define the magnitude (10⁻¹⁶J) and softness (Å) of the Born–Mayer term; E_i is the effective charge on atom i (E_Si = 2.4, E_O = −1.2); e is the elementary charge; and ɛ₀ is the permittivity of vacuum. The coefficients used in the simulation were B_ij = 21.39 and ρ_ij = 0.174 for Si–O atomic pairs and B_ij = 0.6246 and ρ_ij = 0.362 for O–O atomic pairs. The interactions between Si atoms were ignored. The initial atom configuration was set to random, and then the system was equilibrated at 4000 K for 100 000 steps. Subsequently, it was cooled to 300 K for 5 000 000 steps and annealed at 300 K for 100 000 steps.

The STEM image simulation was performed using a multi-slice software (HREM Research, Inc., xHREM).¹⁴ The acceleration voltage was set at 300 kV (electron wavelength λ = 1.97 pm). The convergence semiangle, α, was 20 mrad, and the corresponding diffraction limit, 0.6λ/α, was 59 pm.¹² The full width at half maximum of the simulated incident probe was 50 pm. The slice width and scanning step of the multi-slice simulation were 0.2 nm and 25 pm, respectively. The STEM detection angle ranges of the BF, ABF, and ADF images were 0–1, 10–20, and 50–150 mrad, respectively. These observation settings are typical and practical for actual high-resolution STEM imaging with an aberration corrector. Further details of the structure modeling process and STEM image simulation are provided in the supplementary material.

Various diffractometry-based and imaging-based TEM techniques have been developed to analyze amorphous and disordered materials. One of modern diffractometry-based techniques is nanobeam (or Angstrom-beam) electron diffraction (NBED or ABED)¹⁵ in which a fine probe scans over a target specimen and spatially resolved diffraction patterns are acquired [Fig. 1(a)]. By observing the crystalline spots in the diffraction patterns, the crystalline clusters in the amorphous matrix can be successfully detected. However, the maximum achievable spatial resolution of this technique for observing small diffraction spots depends on the diameter of the incident probe, d_probe (∼0.6λ/α), which is limited by the small-convergence semiangle α.¹⁶ Accordingly, imaging-based TEM techniques have also been widely applied to observe disordered materials because TEM is capable of producing images with atomic-scale resolution.¹⁷ A prominent example is fluctuation electron microscopy (FEM)¹⁸ in which dark-field images are acquired by varying the scattering angle, θ_in [Fig. 1(b)]. Both short- and medium-range orders have been analyzed using FEM. As mentioned earlier, the diffraction intensities, I(g), in all these methods is governed by both the factors of atomic number and atomic arrangement; therefore, it is difficult to analyze element-dependent spatial distributions using conventional diffractometry-based and imaging-based TEM techniques.

Figures 1(c) and 1(d) schematically illustrate the method used in the present study. Atomic-resolution BF, ABF, and ADF STEM images are simultaneously acquired using a large convergence angle, and the Fourier transforms of each STEM image are converted to one-dimensional reciprocal profiles via rotational averaging [Fig. 1(d), left]. Although the Fourier transform of STEM images produces similar results with electron diffraction imaging, it better reflects the Z-dependence and contrast transfer properties of the imaging technique used. We also calculate the autocorrelations and rotational-averaged profiles of the STEM images [Fig. 1(d), right]. Autocorrelation is a mathematical operation used to deduce repeating patterns, and in STEM imaging, it reflects the real-space atomic arrangements. Because the present autocorrelation procedure is normalized by the amplitude of the original image, all autocorrelation profiles show +1 at the origin, even in the case of the ABF image, which consists of dark dots.

In this step, we demonstrate our proposed contrast-variation-based analysis using simulated STEM images of amorphous SiO₂. The most critical parameter in atomic-resolution STEM imaging is the defocus Δz of the objective lens. The defocus in the simulation was set to in-focus (i.e., Δz = 0) in which the incident probe is focused on the mid-plane of the specimen thickness. Under this condition, the ADF image demonstrates the maximum contrast, whereas the BF image shows the minimum contrast (Fig. S2 of the supplementary material). More importantly, these two features are, respectively, consistent with the theoretical incoherent and coherent imaging properties. Figures 2(a)–2(c) show the BF, ABF, and ADF images under in-focus condition and their corresponding Fourier transforms and autocorrelations.

The BF image [Fig. 2(a)] displays multiple random bright and dark dots, and its feature resembles the in-focus high-resolution TEM (HRTEM) images of amorphous materials. In terms of Helmholtz reciprocity, this result is consistent with those reported in previous studies on coherent imaging.¹⁹ Furthermore, the Fourier transform of the BF image is truncated at a frequency of ∼10 nm⁻¹ [inner broken line in Fig. 2(a)], which is explained by the 20 mrad STEM objective aperture corresponding to a spatial distance of 0.1 nm at an acceleration voltage of 300 kV. The outer weak intensity reaching up to 20 nm⁻¹ (dotted line) is the nonlinear imaging term, which denotes the interference intensity between the diffracted waves and is usually negligible in BF images of weak phase objects.²⁰ Considering the strong dependence of the BF image contrast on the defocus and thickness parameters,⁵ we focus on the ABF and ADF images in the following discussion.

The ABF and ADF images, respectively, demonstrate dark and bright spots, corresponding to the atomic sites [Figs. 2(b) and 2(c)]. The Fourier transforms of both the ABF and ADF images exhibit intensities reaching up to 20 nm⁻¹, confirming the incoherent imaging features of these two techniques. The three white arrows in the Fourier transforms indicate the specific spatial frequencies of three scattering vectors, Q₁ [first sharp diffraction peak (FSDP)],²¹ Q₂ [principal peak (PP)],²² and Q₃, which are generally identified as the diffraction peaks of non-monoatomic amorphous materials in XRD and ND.³ It suggests the validity of structure modeling and STEM image simulations. We tabulate the three peaks and major atomic distances (e.g., d_Si–Si) in Table I. The spatial frequency in Table I is given by the scattering vector, |Q| = 4π sin θ/λ = 2π/d [Å⁻¹], which is standardly employed in XRD and ND.²³ In-depth studies have been made for the structural origins of the three peaks [e.g., see the recent review (Ref. 24)], particularly for the FSDP (see Fig. S4). The peak Q₂ is considered to reflect orientational correlations of SiO₄ tetrahedra, and the peak Q₃ stems from pair correlations of neighboring atoms.²⁵ 

To clarify the spatial frequency distribution of each STEM image, we calculate their rotational-averaged Fourier-transform profiles (Fig. 3). This process is also followed for the simulated diffraction amplitudes (see Fig. S1) with the three peaks, Q₁, Q₂, and Q₃ [Fig. 3(a)]. A notable difference in terms of intensity is observed between the ABF and ADF Fourier-transform profiles, particularly around Q₂. Although the ADF profile [Fig. 3(c)] exhibits an intense peak only at Q₁, the ABF profile [Fig. 3(b)] demonstrates an additional intensity peak around Q₂. It should be noted that the same pattern has been reported for XRD and ND, where Q₂ is observed only in ND data.²⁴ In the case of both XRD and ED, the atomic scattering factor of silicon, f_Si, is substantially larger than that of oxygen, f_O; however, the neutron scattering length of silicon is smaller than that of oxygen. In the case of STEM imaging, the ADF signal, which is roughly proportional to Z², is dependent on the silicon distribution. Contrary to ADF imaging, the ABF signals of silicon and oxygen atoms do not differ significantly; therefore, the disparity in the Fourier-transform profiles of these elements is theoretically attributed to their difference in terms of Z-dependence.

STEM imaging visualizes the atomic arrangements directly in real-space, which is a considerable advantage over conventional diffractometry techniques, such as XRD, ND, and ED (which require a high-angle scattering intensity, e.g., |Q| > 20 Å⁻¹), for determining fine real-space information. It should be noted that a typical STEM scanning step of 25 pm corresponds to a high-angle diffractometry of |Q| = 25.1 Å⁻¹. As such, the real-space arrangements of adjacent atoms can be estimated directly from the STEM image using an autocorrelation function [right panels in Figs. 2(a)–2(c)]. Here, to discuss the arrangements of adjacent atoms, we calculate the rotational-averaged autocorrelations (Fig. 4).

The ADF autocorrelation profile [Fig. 4(b)] displays one peak at 0.42 nm, which corresponds to the major peak Q₁ in diffractometry. This suggests that the ADF image reflects the amorphous SiO₂ network structure in accordance with ED and XRD, while being consistent with the similarities between the electron diffraction pattern and its Fourier transform (Fig. 3). On the contrary, the ABF autocorrelation profile [Fig. 4(a)] shows additional peaks at the short distances of 0.16 and 0.27 nm, which, respectively, corresponds to the d_Si–O (0.162 nm) and d_O–O (0.265 nm). The dependence of the ABF image contrast on both Si and O atoms enables us to analyze the short-range ordering of even light elements, a task that is otherwise challenging with XRD and ED. Furthermore, while performing similar simulations of the amorphous structure without oxygen atoms, the two peaks (0.16 and 0.27 nm) of the ABF autocorrelation profile disappear (see Fig. S3c), suggesting that they both originate from the oxygen signals.

Although we highlighted several notable advantages, there are certain theoretical limitations hindering the application potential of this method, such as (a) spatial resolution, (b) applicable specimen thickness, and (c) imaging conditions dependence.

• The spatial resolution of our method is limited by the incident probe size. Recently developed aberration-corrected microscopes allow for resolutions in the scale of several tens of picometers (e.g., 40.5 pm at 300 kV²⁶). However, when assuming a 50 pm incident probe, the corresponding Nyquist frequency (10 nm⁻¹) is smaller than that of the typical XRD range. Nevertheless, it should be noted that the major diffraction peaks (Q₁, Q₂, and Q₃) of disordered materials, whose Nyquist frequency is less than 10 nm⁻¹, can be fully covered by this method (Fig. 3).

• The applicable specimen thickness depends on the depth of focus (λ/α²) and chromatic defocus spread, which are roughly estimated to be a few tens of nanometers. In the case of thick specimens, the out-of-focus background in the STEM images does not pose a limitation, as it is summed at the center of the Fourier transforms.

• It has been well established that the imaging conditions in HRTEM and STEM BF experiments require careful control. Fortunately, the contrast in ABF and ADF imaging techniques is not highly dependent on the defocus parameter, as opposed to those in HRTEM or BF imaging. Additionally, rotational averaging can reduce the effect of residual aberrations on both the Fourier transforms and autocorrelations.

Since the atomic arrangement information from the Fourier transforms and autocorrelations are limited, the combination of this method with MD modeling or other XRD and ND results of known bulk materials is effective.

The contrast variation-based imaging technique presented herein is practical because it enables the simultaneous capturing of BF, ABF, and ADF images using the standard experimental settings of STEM instruments. Compared with conventional electron microscopy, where the dynamical diffraction effect hinders the intuitive interpretation of the specimens, this problem is not encountered when studying amorphous materials. Furthermore, the conjoined use of Fourier transforms and autocorrelation analyses significantly helps to accurately reflect the local atomic structures of the tested materials.

We proposed a new method to analyze the local atomic structures of non-monoatomic materials using a contrast variation scheme in conjunction with ADF and ABF STEM imaging. Through this strategy, the element-sensitive local atomic arrangement of SiO₂ samples was analyzed. This method can be applied to various materials, such as amorphous thin layers for semiconductor devices (e.g., dielectric layers for metal–oxide–semiconductors and transparent electrodes for display devices), which cannot be analyzed using conventional diffractometry techniques because of the high spatial resolution required. Our proposed method can also be utilized for investigating the crystal structures of disordered materials, e.g., inhomogeneous structural materials, high-entropy alloys, and metallic glass. Thus, we believe that our study will provide new insights into the local structural analysis of non-monoatomic disordered materials.

See the supplementary material for an amorphous SiO₂ structure model, STEM simulation details, defocus definition in STEM imaging, other STEM simulation results, and additional information about SiO₂ local structure.

This work was partly supported by the MEXT Elements Science and Technology Project and Elements Strategy Initiative to Form Core Research Center, JSPS KAKENHI Grant No. 17H01318. This work was also supported by KAKENHI Grant Nos. 20H02624 (to K.K., J.K., and O.C.), 20H05878 (to M.S. and S.K.), 20H05881 (to S.K. and Y.O.), 20H05884 (to M.S.), and 19K05648 (to Y.O.) of MEXT for materials and supply costs.