Analysis of a capped carbon nanotube by linear-scaling density-functional theory.

The apex region of a capped (5,5) carbon nanotube (CNT) has been modelled with the DFT package ONETEP, using boundary conditions provided by a classical calculation with a conducting surface in place of the CNT. Results from the DFT solution include the Fermi level and the physical distribution and energies of individual orbitals for the CNT tip. Application of an external electric field changes the orbital number of the highest occupied molecular orbital (HOMO) and consequently changes its distribution on the CNT.

Imaging with straight-edge phase plates in the TEM.

The image of a simple phase object produced by a round lens with a Foucault or Hilbert phase plate can be determined with Abbe imaging theory and a 2D transform expressed in cylindrical coordinates. The contributions to the image amplitude from a uniform disc object and an azimuthally varying plate can then be distinguished and their phases relative to the incident wave can be compared. It appears that the usual choice of added phase for a Hilbert plate causes the image of a weak disc object to vanish as the plate edge approaches the axis, but a different choice of plate thickness can enable a weak phase object to provide a linear contribution to image intensity.

Imaging by Zernike phase plates in the TEM.

The images produced from simple phase objects, lenses and Zernike phase plates when all have rotational symmetry can be calculated by 1D Fourier-Bessel transforms. For a simple disc object producing a uniform phase shift over its diameter, the resulting image can be defined for any size of object phase change. The monotonic range of intensity variation with object phase is found to depend strongly on the phase change introduced by the phase plate; this property of the system is not well predicted by the weak phase approximation. The effect of spreading the phase transition at the plate over a range of radius is beneficial if the plate phase change is sufficiently small. Weak-phase calculations for a phase distribution more typical of a spherical object are also shown.

Imaging of weak phase objects by a Zernike phase plate.

Analysis of the imaging of some simple distributions of object phase by a phase plate of Zernike type shows that sharp transitions in the object phase are well transmitted. The low-frequency components of the complete object function are attenuated by the plate. The behaviour can be characterised by a cut-on parameter defined as the product of the cut-on frequency of the plate and a characteristic dimension of the object. When this parameter exceeds a value of the order of unity, a sharp boundary in the object is imaged by a Zernike plate as a dark lining inside the boundary with a white outline or halo outside the boundary, in agreement with reported observations. The maximum diameter of objects that can be imaged accurately is inversely proportional to the diameter of the hole for beam transmission in the phase plate.

Characterisation of ferromagnetic rings for Zernike phase plates using the Aharonov-Bohm effect.

Holographic measurements on magnetised thin-film cobalt rings have demonstrated both onion and vortex states of magnetisation. For a ring in the vortex state, the difference between phases of electron paths that pass through the ring and those that travel outside it was found to agree very well with Aharonov-Bohm theory within measurement error. Thus the magnetic flux in thin-film rings of ferromagnetic material can provide the phase shift required for phase plates in transmission electron microscopy. When a ring of this type is used as a phase plate, scattered electrons will be intercepted over a radial range similar to the ring width. A cobalt ring of thickness 20 nm can produce a phase difference of π/2 from a width of just under 30 nm, suggesting that the range of radial interception for this type of phase plate can be correspondingly small.

The transverse structure of cold field electron emission.

Solutions of Schrödinger's equation have been obtained for electron emission from a small area in a system of planar electrodes. In a uniform field, wave components with non-zero transverse phase constant see a greater height and width of the potential barrier than the normal component does, the extra energy being that corresponding to the transverse momentum. Also the total intensity shows a Gaussian decrease as beam radius increases, even for a point source. The characteristic Gaussian radius increases as the square root of the distance from the starting plane. In contrast, flow from a point source into a region of uniform potential does not show these effects, but spreads much more broadly.

Some comments on models for field enhancement.

To estimate the apex field-enhancement factor gamma(a)associated with a pointed protrusion on a flat planar surface, the simple physical models of a 'floating sphere at emitter-plane potential' and a 'hemisphere on a post' are often discussed. The corresponding mathematical expressions have the form: gamma(a)=m+h/rho, where rho is the sphere or hemisphere radius, h is its 'height above the emitter plane', and m is a constant variously taken as 0, 2 or 3. Recent numerical simulations for the 'hemisphere on a post' model, reported elsewhere by two of us (CJE and GV) and by Kokkaris, Modinos and Xanthakis, have shown that all of these simple formulae significantly overpredict gamma(a) if h/rho is large. This article first reexamines the basis of these simple formulae and confirms that they are less secure than is sometimes thought. The formulae reported elsewhere as fits to the numerical results are then quoted and compared with the simple formulae, and with the known exact analytical result for the 'hemi-ellipsoid on a plane' model. Discrepancies can be rationalised. Some general conclusions are drawn.

Fowler-Nordheim theory for a spherical emitting surface.

In tests on a field emitter whose dimensions and work function were known, Fowler-Nordheim (F-N) theory as usually stated for a planar emitter was found to give poor agreement with observations. The effect of curvature of the emitting surface has been modelled by including (a) non-linear variation of potential with distance from the surface; (b) the consequent changes in the exponent and pre-exponential terms in the F-N expression for current; and (c) the variation of current density over the surface, modelled by an effective solid angle. Application of the resulting expression to the measured data gives estimates for apex radius which agree much more closely with the measured value than the value from planar theory does.