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

The Transmission Electron Microscope 63
only about 1800 K to provide adequate electron emission. The tip protrudes
about 0.3 mm outside the hole in the Wehnelt, so an accelerating field exists
at its surface, created by an extractor electrode biased positive with respect to
the tip. Because the tip is very sharp, electrons are emitted from a very small
area, resulting in a relatively high current density (Je � 10 7 A/m 2 ) at the
surface. Because the ZrO is easily poisoned by ambient gases, the Schottky
source requires a vacuum substantially better than that of a LaB6 source.
Field emission
If the electrostatic field at a tip of a cathode is increased sufficiently, the
width (horizontal in Fig. 3-4) of the potential barrier becomes small enough
to allow electrons to escape through the surface potential barrier by
quantum-mechanical tunneling, a process known as field emission. We can
estimate the required electric field as follows. The probability of electron
tunneling becomes high when the barrier width w is comparable to de
Broglie wavelength � of the electron. This wavelength is related to the
electron momentum p by p = h/� where h = 6.63 � 10 -34 Js is the Planck
constant. Because the barrier width is smallest for electrons at the top of the
conduction band (see Fig. 3-4), they are the ones most likely to escape.
These electrons (at the Fermi level of the cathode material) have a speed v of
the order 10 6 m/s and a wavelength � = h/p = h/mv � 0.5 � 10 -9 m.
E F
�
vacuum
level
cathode
metal
conduction
band
E
x
Schottky
emission
vacuum
field
emission
e- Figure 3-4. Electron-energy diagram of a cathode, with both moderate ( � 10 8 V/m) and high
(� 10 9 V/m) electric fields applied to its surface; the corresponding Schottky and field
emission of electrons are shown by dashed lines. The upward vertical axis represents the
potential energy E of an electron (in eV) relative to the vacuum level, therefore the downward
direction represents electrostatic potential (in V).
w
��
e -