Speaker
Description
An electron within a piece of condensed matter is held into the material by a surface barrier. In
simple basic models, the force preventing electron escape is attributed to an image attraction between
the electron and the material surface. This gives rise to an energy barrier of zero-field height H that
prevents classical escape. The application of a classical electrostatic field of appropriate polarity and
magnitude lowers this barrier, and can reduce it to zero if the field magnitude is sufficiently large.
Effects of this general kind were discussed by Kelvin and by Maxwell, in the context of
conducting spheres. J. J. Thomson, when discussing (in his 1903 book [1]) electrical discharge effects
between closely spaced planar surfaces, suggested that a barrier lowering effect of this kind might
enable electron emission to cause the observed phenomena. However, relevant planar-geometry
equations were first clearly formulated by Schottky in 1913 [2], and hence the effect is known as the
(classical) Schottky effect.
Schottky's original treatment was in terms of "electric potentials", and used the Gaussian equation
system. Modern treatments involve the component of electron total energy in the direction (z) normal
to the material surface, and use the modern international equation system that has the vacuum electric
permittivity ε0 in Coulomb's Law [3]. For a good conductor, such as a metal, the energy barrier is
described by an energy-like quantity M(z) given (for a planar surface) by
M(z) = H – eFz – e2/16πε0z , (1)
where e is the elementary positive charge and F is the magnitude of the linear field outside the planar
conductor surface. (My term for M(z) is the motive energy.) This barrier is often now called the
Schottky-Nordheim barrier. It can be shown that the maximum height of the barrier is lowered by an
energy ΔS called the Schottky reduction (or Schottky lowering) and given by
ΔS = c F1/2 , (2)
where c is a universal constant given by:
c ≡ (e3/4πε0)1/2 ≅ 1.199985 eV V–1/2 nm1/2 = 3.794686×10–5 eV V–1/2 m1/2 . (3)
Clearly, the so-called zero-barrier field F0(H) needed to reduce a barrier of zero-field height H to
zero, via the Schottky effect, is
F0(H) = c–2H2 . (4)
where c–2 ≅ 0.6944615 eV–2 V nm–1 = 6.944615×108 eV–2 V m–1 . (5)
This universal constant c (alternatively denoted by cS) has been called the Schottky constant. The
numerical value of c (when fields are measured in V/cm) was first given by Schottky in 1914 (see
eq. (6) in [2]), with the numerical value of c–2 given in his 1923 paper [4] (see Table 1, on p. 83).
Although the Schottky constant plays a central role in the theories of field electron emission, thermal
electron emission and ionic field evaporation, particularly since the 1970s reforms in the international
system of measurement, it is not widely recognised as a useful universal constant.
This Poster provides a brief "tutorial" introduction to the Schottky constant, primarily for those
not familiar with it. It will include: a proof of equations (2) and (3); a demonstration of how the
equations need to be modified when the escaping entity is an ion of charge ne (where the chargenumber
n is a small integer); statements identifying the scientific contexts and equations in which the
Schottky constant is most commonly used; and a demonstration that the Schottky constant is a
"property of the world" that is represented by technically different physical quantities in different
equation systems (strictly, what is discussed above is the "ISQ Schottky constant" [3]).
[1] J.J. Thomson, Conduction of Electricity through Gases (1st ed., Cambridge Univ. Press, 1903), see p. 386.
[2] W. Schottky, Physik. Zeitschr. 15, 872–878 (1914).
[3] Since 2009, the modern equation system that uses ε0 has been called the "International System of
Quantities" (ISQ).
[4] W. Schottky, Z. Phys. 14, 63–106 (1923).
