# Spatial resolution

## Geometric effects

When working with a transmission electron microscope (TEM) in scanning (STEM) or focused probe mode, the spatial resolution depends of several effects. For probes greater than ~2 nm and thicker samples (greater than ~ 75 nm), you can approximate the resolution with simple geometric arguments that relate to the beam broadening in the specimen due to both elastic and inelastic scattering. As shown schematically in the figure below, the Auger electron signal is generated from a narrow region at the entrance and exit surfaces of the sample. The energy dispersive x-ray spectroscopy (EDS) signal is generated from the total interaction volume of the electron beam. This interaction volume is significantly broadened by electron scattering in the sample. On the other hand, the electron energy loss spectroscopy (EELS) signal, detects only the energy change in the primary electron beam that is predominately forward scattered. The broadening due to elastic scattering can affect the EELS signal, but you can limit this effect with an angle limiting aperture to reduce high angle scattering from entering the spectrometer.

## Secondary excitations

In addition to the direct excitation process, it is also probable to observe secondary, non-local excitations. In many cases, this results from the scattering of high energy electrons and x-rays in the system. The detection of trace amounts of a material in one region is highly suspect if that substance resides as a major component elsewhere in the system.

## Below 1 nm

The above kinematic arguments apply for high energy electrons down to about ~2 nm. To form a probe smaller than 1 nm, the probe convergence angle must increase. If the probe angle is too large, the effects of spherical aberration will create broad tails to the probe. The image resolution will come predominately from the sharp center of the probe, but if a significant fraction of the beam current is in the probe tails it will affect the microanalysis resolution. Even in the case of an optimized probe angle,

$$\alpha _{0} = (4\lambda /C_{s})^{1/4}$$

the effect of the convergence angle as the probe passes through the sample will increase the interaction volume in proportion to the specimen thickness and will be approximately ( $$\alpha _{0}t$$ ) for an amorphous sample. For crystalline material, the electron channeling will tend to concentrate the beam to a tighter diameter than predicted from purely geometric considerations.

With aberration correction, STEM probes at atomic dimensions can form. In this case, you need to consider the wave nature of electrons and the long range of coulomb interactions. It is often necessary to use simulations to understand the spatial extent of the interaction. A general trend is that low energy loss events tend to have lower spatial resolution. This can be understood either by impact parameter arguments (excite low energy losses from longer distances) or wave-optical considerations (low energy losses have a narrow angular range and therefore cannot be localized spatially according to the uncertainty principle). In either case, if the collection angle is too small, the signal will be delocalized by the diffraction limit of the collection process. These considerations led Egerton to propose a spatial resolution that contains 50% of the signal ($$d _{50}$$) given by:

$$(d_{50})^{2} = (0.5\lambda /\theta _{E}^{3/4})^{2} + (0.6\lambda /\beta )^{2}$$

Where $$\beta$$ is the EELS collection angle and $$\theta _{E}$$ is the characteristic scattering angle for the measured energy loss event ($$\theta _{E}\approx E/2E_{0}$$) with $$E_{0}$$ being the primary beam energy.

## Effect of noise

The spatial resolution you obtain in a measurement is typically limited not by the underlying scattering physics, but by the measurement noise and dose (per unit area) you can deposit on the sample. As a general consideration, you need a signal-to-noise ratio (SNR) of about 3 to see a feature. If you cannot achieve the required SNR with the allowed dose, you can increase the signal at the expense of spatial resolution by increasing the analyzed area either by enlarging the probe diameter or summing adjacent pixel.

The dose limit is a function of the induced physical damage to the specimen or stability of the sample and analysis system.

EELS EDS XPS Auger CL
Probe → Signal e- → e- e-$$\gamma$$ $$\gamma$$ → e- e- → e- e-$$hv$$
Spatial resolution Å – µm nm – mm µm – mm µm – mm nm – mm

• In thin specimens: limited by scattering geometry down to ~0.5 nm
• Below 0.5 nm, coherent scattering, Coulomb interaction and channeling must be considered
• In thin specimens: limited by SNR due to weak scattering
• Also limited by scattering geometry down to ~0.5 nm
• Below 0.5 nm, channeling must be considered
• Limited by electron collection optics in traditional systems
• Synchrotron STXM based systems limited to ~20 nm by x-ray optics
• Limited by interaction volume and low SNR
• Limited by carrier diffusion and low SNR

• In thick specimens, limited by signal-to-background ratio
• In thick specimens, limited by interaction volume and secondary fluorescence