High-resolution transmission electron microscopy

High-resolution transmission electron microscopy is an imaging mode of specialized transmission electron microscopes that allows for direct imaging of the atomic structure of samples. It is a powerful tool to study properties of materials on the atomic scale, such as semiconductors, metals, nanoparticles and sp2-bonded carbon (e.g., graphene, nanotubes). While this term is often also used to refer to high resolution scanning transmission electron microscopy, mostly in high angle annular dark field mode, this article describes mainly the imaging of an object by recording the two-dimensional spatial wave amplitude distribution in the image plane, similar to a "classic" light microscope. For disambiguation, the technique is also sometimes referred to as phase contrast transmission electron microscopy. At present, the highest point resolution realised in high resolution transmission electron microscopy is around 0.5 ångströms (0.050 nm). At these small scales, individual atoms of a crystal and defects can be resolved. For 3-dimensional crystals, it is necessary to combine several views, taken from different angles, into a 3D map. This technique is called electron tomography. One of the difficulties with high resolution transmission electron microscopy is that image formation relies on phase contrast. In phase-contrast imaging, contrast is not intuitively interpretable, as the image is influenced by aberrations of the imaging lenses in the microscope. The largest contributions for uncorrected instruments typically come from defocus, spherical aberration and astigmatism of the objective lens. These values can be estimated from the so-called Thon ring pattern (named for Friedrich Thon) appearing in the Fourier transform modulus of an image of a thin amorphous film.

Image contrast and interpretation

The contrast of a high resolution transmission electron microscopy image arises from the interference in the image plane of the electron wave with itself. Due to our inability to record the phase of an electron wave, only the amplitude in the image plane is recorded. However, a large part of the structure information of the sample is contained in the phase of the electron wave. In order to detect it, the aberrations of the microscope (like defocus) have to be tuned in a way that converts the phase of the wave at the specimen exit plane into amplitudes in the image plane. The interaction of the electron wave with the crystallographic structure of the sample is complex, but a qualitative idea of the interaction can readily be obtained. Each imaging electron interacts independently with the sample. Above the sample, the wave of an electron can be approximated as a plane wave incident on the sample surface. As it penetrates the sample, it is attracted by the positive atomic potentials of the atom cores, and channels along the atom columns of the crystallographic lattice (s-state model). At the same time, the interaction between the electron wave in different atom columns leads to Bragg diffraction. The exact description of dynamical scattering of electrons in a sample not satisfying the weak phase object approximation, which is almost all real samples, still remains the holy grail of electron microscopy. However, the physics of electron scattering and electron microscope image formation are sufficiently well known to allow accurate simulation of electron microscope images. As a result of the interaction with a crystalline sample, the electron exit wave right below the sample φe(x,u) as a function of the spatial coordinate x is a superposition of a plane wave and a multitude of diffracted beams with different in plane spatial frequencies u (spatial frequencies correspond to scattering angles, or distances of rays from the optical axis in a diffraction plane). The phase change φe(x,u) relative to the incident wave peaks at the location of the atom columns. The exit wave now passes through the imaging system of the microscope where it undergoes further phase change and interferes as the image wave in the imaging plane (mostly a digital pixel detector like a CCD camera). The recorded image is not a direct representation of the samples crystallographic structure. For instance, high intensity might or might not indicate the presence of an atom column in that precise location (see simulation). The relationship between the exit wave and the image wave is a highly nonlinear one and is a function of the aberrations of the microscope. It is described by the contrast transfer function.

The phase contrast transfer function The phase contrast transfer function is a function of limiting apertures and aberrations in the imaging lenses of a microscope. It describes their effect on the phase of the exit wave φe(x,u) and propagates it to the image wave. Following Williams and Carter, assume the weak phase object approximation (thin sample), then the contrast transfer function becomes

C T F ( u ) = A ( u ) E ( u ) 2 sin ⁡ ( χ ( u ) ) {\displaystyle CTF(u)=A(u)E(u)2\sin(\chi (u))}

where A(u) is the aperture function, E(u) describes the attenuation of the wave for higher spatial frequency u, also called envelope function. χ(u) is a function of the aberrations of the electron optical system. The last, sinusoidal term of the contrast transfer function will determine the sign with which components of frequency u will enter contrast in the final image. If one takes into account only spherical aberration to third order and defocus, χ is rotationally symmetric about the optical axis of the microscope and thus only depends on the modulus u = |u|, given by

χ ( u ) = π 2 C s λ 3 u 4 − π Δ f λ u 2 {\displaystyle \chi (u)={\frac {\pi }{2}}C_{s}\lambda ^{3}u^{4}-\pi \Delta f\lambda u^{2}}

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