Hierarchical Field Emission Devices - Hierarchical Nanostructures for Energy Devices - page 232

bombardment causing erosion and blunting of the emitting tips. Despite
the great progress, obtaining sharp, robust and chemically stable cathodic
materials and architectures that will enable the establishment of a FED
industry and boost FEDs to be the dominant display technology is an
ongoing effort.
The basic formulation of quantum mechanical tunneling of conduction
electrons from a flat metallic surface into a vacuum was introduced by
Fowler and Nordheim in 1928 3a resulting in the Fowler-Nordheim (FN)
equation. This equation provides the emission current density, J, corres-
ponding to electron tunneling from an exact triangular barrier without any
correction factors. Later, Gomer 3b derived the image potential corrected FN
equation which can be written as

d n 3 r 4 n g | 0

!
; J ¼ Aj 1 E 2 exp Bj 2

E

(10 : 1)

where j is the work function of the emitter and A, B are constants with
A ¼ 1.54 10 6 AeVV 2 and B ¼ 6.83 V 3/2 Vnm 1 . E ¼ V/d is the electric field
at the emitting surface, where V is the applied voltage bias between the
emitting cathode and a collecting anode and d is the spacing between them.
J is given by the measured emission current in Ambers times the emitting
area, S (I ¼ S J). In the case of semiconductors, the field electron emission
theory is complex. Various parameters including the contribution of the
effective electron mass, field penetration, surface states, doping and band
structure have to be taken into account. A detailed account of the theory field
emission from semiconductors is available. 4
To account for the FE behavior of non-planar structures with certain
geometries, one should consider that the local field at the emitting point,
E loc (Figure 10.1b), is different from the average field E of a flat emitter. In a
first approximation the relation between the two fields is:
E loc ¼ bE ¼ b V
d

.

(10 : 2)

where b is the geometric field enhancement factor and b/d is the local field
conversion factor. Accordingly, in terms of the macroscopic field E, the FN
eqn (10.1) becomes

!

J ¼ Aj 1 ð bE Þ 2 exp Bj 2
bE

(10 : 3)

or

¼ ln Ab 2 = j Bj 2 = bE

ln J = E 2

(10 : 4)

The factor b depends on the emitter geometry. In principle, the calculation
of b is complex and several models have been proposed for its estimation.
Such models are not universally applicable and may vary for different types