Physical Principles of Electron Microscopy: An Introduction to TEM ...

Appendix
MATHEMATICAL DERIVATIONS
A.1 The Schottky Effect
The reduction �� in the surface-barrier height with applied electric field
(Schottky effect) can be calculated by first considering the case where the
extraction field is zero. When an electron leaves the conducting cathode, it
generates (within the vacuum) its own electric field, which arrives
perpendicular to the surface of the conductor and induces an equal and
opposite charge +e that is distributed over the cathode surface (Fig. A-1a).
The field lines outside the cathode are the same as those generated by an
electrostatic dipole, obtained by replacing the cathode by a point charge +e
located at an equal distance z inside the surface; see Fig. A-1b. The electron
therefore experiences an attractive force given by Coulomb’s law:
F(z) = K (�e)(+e)/(2z) 2
(A.1)
in which K = 1/(4��0) = 9.0 � 10 9 N m 2 C -2 is the Coulomb constant. The
negative sign of F(z) indicates that the force acts in the –z direction, toward
the surface.
With voltage applied to an extraction electrode, an additional field –Ee is
created (in the –z-direction) that gives rise to a force Fe = –e(–Ee) = +eEe.
The positive sign of Fe denotes that this extraction force acts in the +z
direction. The total force is therefore:
F = F(z) + Fe = �(K/4) (e 2 /z 2 ) + e Ee
(A.2)