Fundamental Theory of Transmission Electronic Microscopy
by Yong Ding
The smallest distance between two points that we can resolve by our eyes is about 0.1-0.2 mm, depending on how good our eyes are. This distance is the resolution or resolving power of our eyes. The instrument that can show us pictures revealing detail finer than 0.1 mm could be described as a microscope.
The Rayleigh criterion defines the resolution of light microscope as:
where λ is the wavelength of the radiation, μ is the refractive index of the view medium and β is the semi-angle of collection of the magnifying lens. The variable of refractive index and semi-angle is small, thus the resolution of light microscope is mainly decided by the wavelength of the radiation source. Taking green light as an example, its 550nm wavelength gives 300nm resolution, which is not high enough to separate two nearby atoms in solid-state materials. The distance between two atoms in solid is around 0.2nm.
Based on wave-particle duality, we know that electron has some wave-like properties:
If an electron is accelerated by an electrostatic potential drop eU, the electron wavelength can be described as:
If we take the potential as 100keV, the wavelength is 0.0037nm. The resolution of electron microscope should be better than that of light microscope.
Because the average distance between two successive incident electrons is around 0.15mm (taking 100keV as the accelerate potential), which is far great than the TEM specimen thickness (~100-500 nm), we can consider the interaction between electrons and specimen as single electron scattering event. From the wave-particle duality point of view, the incident electron can be expressed as a plane wave . Resolving the Schrödinger equation, we can get the departure electron wave function as:
Taking Mutt approximation:
the amplitude of the scattering beam is the Fourier transform of the specimen's potential.
If we considering perfect crystal, its potential can be described as:
Sp(r) is the shape factor of crystal.
The Fourier transform of the potential is:
where V g is the structural factor and
The diffraction intensity can be calculated as:
There are two basic modes of TEM operation, namely the bright-field mode, where the (000) transmitted beam contributes to the image, and the dark-field imaging mode, in which the (000) beam is excluded. The size of the objective aperture in bright-field mode directly determines the information to be emphasized in the final image. When the size is chosen so as to exclude the diffracted beams, one has the configuration normally used for low-resolution defect studies, so-called diffraction contrast . In this case, a crystalline specimen is oriented to excite a particular diffracted beam, or a systematic row of reflections, and the image is sensitive to the differences in specimen thickness, distortion of crystal lattices due to defects, strain and bending.
Diffraction contrast is a dominant mechanism for imaging dislocations and defects in the specimen. However, the resolution of this imaging technique is limited to 1-3 nm. Diffraction contrast mainly reflects the long-range strain field in the specimen and it is unable, however, to provide high-resolution information about the atom distribution in the specimen.
The diffraction of electrons is purely a result of the wave property of particles. The wavelength ?of an electron is a typical quantity for characterizing an incident plane wave.
The calculation of electron wavelength has been performed in section 1, however, without consideration the perturbation of the crystal potential on the electron kinetic energy. If the electron is traveling in a crystal, which is characterized by an electrostatic potential field V(x,y,z), the equation should be modified as
Therefore, the structure perturbed electron wavelength can be obtained. The effective wave number is
where an approximation of V << U0 is made. We now consider a case in which the specimen covers only half space and leave the other half as vacuum. Thus the relative phase shift of the wave traveling in the crystal field relative to the wave traveling in the absence of a field for a specimen thickness d is
The projected potential of the crystal is
Therefore, the effect of the potential field is represented by multiplying the wave function by a phase grating function
This is known as the phase object approximation . (POA), in which the crystal acts as a phase grating filter. From this expression, it can be seen that the effect of the crystal potential is to modify the phase of the incident electron wave. The variation of the projected crystal potential results in the change of electron phase. The contrast produced by this mechanism is called phase contrast .