Fowler–Nordheim-type equations Introduction Fowler–Nordheim-type equations, in the
J–
F form, are (approximate) theoretical equations derived to describe the local current density
J emitted from the internal electron states in the conduction band of a bulk metal. The
emission current density (ECD)
J for some small uniform region of an emitting surface is usually expressed as a function of the local work-function
φ and the local barrier field
F that characterize the small region. For sharply curved surfaces,
J may also depend on the parameter(s) used to describe the surface curvature. Owing to the physical assumptions made in the original derivation, {{NumBlk|:| \int_{-\infty}^{\infty} \frac{ {\mathrm{e}}^x } { 1+ {\mathrm{e}}^{wx} } \mathrm{d}x = \int_{0}^{\infty} \frac{\mathrm{d}u}{1+u^w} = \frac{\pi} {w\sin(\pi/w)}. |}} This is valid for (i.e., ). Hence – for temperatures such that : {{NumBlk|:| \lambda_T = \frac{\pi k_{\mathrm{B}} T/d_{\mathrm{F}} }{ \sin(\pi k_{\mathrm{B}} T / d_{\mathrm{F}})} \approx 1 + \frac{1}{6} \left( {\frac{\pi k_{\mathrm{B}} T}{ d_{\mathrm{F}}}} \right) ^2, |}} where the expansion is valid only if (. An example value (for , , ) is . Normal thinking has been that, in the CFE regime,
λT is always small in comparison with other uncertainties, and that it is usually unnecessary to explicitly include it in formulae for the current density at room temperature. The emission regimes for metals are, in practice, defined, by the ranges of barrier field
F and temperature
T for which a given family of emission equations is mathematically adequate. When the barrier field
F is high enough for the CFE regime to be operating for metal emission at 0 K, then the condition provides a formal upper bound (in temperature) to the CFE emission regime. However, it has been argued that (due to approximations made elsewhere in the derivation) the condition is a better working limit: this corresponds to a
λT-value of around 1.09, and (for the example case) an upper temperature limit on the CFE regime of around 1770 K. This limit is a function of barrier field. By defining an overall supply correction factor
λZ equal to , and combining equations above, we reach the so-called physically complete Fowler–Nordheim-type equation: {{NumBlk|:| J \;= \lambda_Z a \phi^{-1} F^2 P_{\mathrm{F}} \exp[- \nu_{\mathrm{F}} b \phi^{3/2} / F ], |}} where {\nu}_{\mathrm{F}} [= {\nu}_{\mathrm{F}}(
φ,
F)] is the exponent correction factor for a barrier of unreduced height
φ. This is the most general equation of the Fowler–Nordheim type. Other equations in the family are obtained by substituting specific expressions for the three correction factors {\nu}_{\mathrm{F}},
PF and
λZ it contains. The so-called elementary Fowler–Nordheim-type equation, that appears in undergraduate textbook discussions of field emission, is obtained by putting , , {{nowrap|{\nu}_{\mathrm{F}} → 1}}; this does not yield good quantitative predictions because it makes the barrier stronger than it is in physical reality. The so-called standard Fowler–Nordheim-type equation, originally developed by Murphy and Good, Note that the variable
f (the scaled barrier field) is not the same as the variable
y (the Nordheim parameter) extensively used in past field emission literature, and that "
v(
f)" does NOT have the same mathematical meaning and values as the quantity "
v(
y)" that appears in field emission literature. In the context of the revised theory described here, formulae for
v(
y), and tables of values for
v(
y) should be disregarded, or treated as values of
v(
f1/2). If more exact values for
v(
f) are required, then that formulates the problem differently and then uses first
Kp and then
εn (or a related quantity) as the variables of integration: this is known as "integrating via the normal-energy distribution". This approach continues to be used by some authors. Although it has some advantages, particularly when discussing resonance phenomena, it requires integration of the Fermi–Dirac distribution function in the first stage of integration: for non-free-electron-like electronic band-structures this can lead to very complex and error-prone mathematics (as in the work of Stratton on
semiconductors). Further, integrating via the normal-energy distribution does not generate experimentally measured electron energy distributions. In general, the approach used here seems easier to understand, and leads to simpler mathematics. It is also closer in principle to the more sophisticated approaches used when dealing with real bulk crystalline solids, where the first step is either to integrate contributions to the ECD over
constant energy surfaces in a
wave-vector space (
k-space),); (3) no significant "patch fields" exist, and in some circumstances might be a significant function of temperature. Because
Ar has a mathematical definition, it does not necessarily correspond to the area from which emission is observed to occur from a single-point emitter in a
field electron (emission) microscope. With a large-area emitter, which contains many individual emission sites,
Ar will nearly always be very very much less than the "macroscopic" geometrical area (
AM) of the emitter as observed visually (see below). Incorporating these auxiliary equations into eq. (30a) yields {{NumBlk|:| i = \; A_{\mathrm{r}} a {\phi^{-1}} {\beta}^2 V^2 \exp[- v(f) \;b \phi^{3/2} / \beta V ], |}} This is the simplified standard Fowler–Nordheim-type equation, in
i–
V form. The corresponding "physically complete" equation is obtained by multiplying by ''λ'
Z'P''F.
Modified equations for large-area emitters The equations in the preceding section apply to all field emitters operating in the CFE regime. However, further developments are useful for large-area emitters that contain many individual emission sites. For such emitters, the notional emission area will nearly always be very very much less than the apparent "macroscopic" geometrical area (
AM) of the physical emitter as observed visually. A dimensionless parameter
αr,
the area efficiency of emission, can be defined by {{NumBlk|:| A_{\mathrm{r}} = \; \alpha_{\mathrm{r}} A_{\mathrm{M}}. |}} Also, a "macroscopic" (or "mean") emission current density
JM (averaged over the geometrical area
AM of the emitter) can be defined, and related to the reference current density
Jr used above, by {{NumBlk|:| J_{\mathrm{M}} = \; i/A_{\mathrm{M}} = \alpha_{\mathrm{r}} (i /A_{\mathrm{r}}) = \alpha_{\mathrm{r}} J_{\mathrm{r}}. |}} This leads to the following "large-area versions" of the simplified standard Fowler–Nordheim-type equation: {{NumBlk|:| J_{\mathrm{M}} = \alpha_{\mathrm{r}} a {\phi^{-1}} F^2 \exp[- v(f) \;b \phi^{3/2} / F ], |}} {{NumBlk|:| i = \; \alpha_{\mathrm{r}} A_{\mathrm{M}} a {\phi^{-1}} {\beta}^2 V^2 \exp[- v(f) \;b \phi^{3/2} / \beta V ], |}} Both these equations contain the area efficiency of emission
αr. For any given emitter this parameter has a value that is usually not well known. In general,
αr varies greatly as between different emitter materials, and as between different specimens of the same material prepared and processed in different ways. Values in the range 10−10 to 10−6 appear to be likely, and values outside this range may be possible. The presence of
αr in eq. (36) accounts for the difference between the macroscopic current densities often cited in the literature (typically 10 A/m2 for many forms of large-area emitter other than
Spindt arrays Equation (40) implies that versions of Fowler–Nordheim-type equations can be written where either
F or
βV is everywhere replaced by \gamma F_{\mathrm{M}}. This is often done in technological applications where the primary interest is in the field enhancing properties of the local emitter nanostructure. However, in some past work, failure to make a clear distinction between barrier field
F and macroscopic field
FM has caused confusion or error. More generally, the aims in technological development of large-area field emitters are to enhance the uniformity of emission by increasing the value of the area efficiency of emission
αr, and to reduce the "onset" voltage at which significant emission occurs, by increasing the value of
β. Eq. (41) shows that this can be done in two ways: either by trying to develop "high-
γ" nanostructures, or by changing the overall geometry of the system so that
βM is increased. Various trade-offs and constraints exist. In practice, although the definition of macroscopic field used above is the commonest one, other (differently defined) types of macroscopic field and field enhancement factor are used in the literature, particularly in connection with the use of probes to investigate the
i–
V characteristics of individual emitters. In technological contexts field-emission data are often plotted using (a particular definition of)
FM or 1/
FM as the
x-coordinate. However, for scientific analysis it usually better not to pre-manipulate the experimental data, but to plot the raw measured
i–
V data directly. Values of technological parameters such as (the various forms of)
γ can then be obtained from the fitted parameters of the
i–
V data plot (see below), using the relevant definitions.
Modified equations for nanometrically sharp emitters Most of the theoretical derivations in the field emission theory are done under the assumption that the barrier takes the Schottky–Nordheim form eq. (3). However, this barrier form is not valid for emitters with radii of curvature
R comparable to the length of the tunnelling barrier. The latter depends on the work function and the field, but in cases of practical interest, the SN barrier approximation can be considered valid for emitters with radii , as explained in the next paragraph. The main assumption of the SN barrier approximation is that the electrostatic potential term takes the linear form \Phi = Fx in the tunnelling region. The latter has been proved to hold only if x \ll R. Therefore, if the tunnelling region has a length L, x for all x that determines the tunnelling process; thus if L \ll R eq. (1) holds and the SN barrier approximation is valid. If the tunnelling probability is high enough to produce measurable field emission, L does not exceed 1–2 nm. Hence, the SN barrier is valid for emitters with radii of the order of some tens of nm. However, modern emitters are much sharper than this, with radii that of the order of a few nm. Therefore, the standard FN equation, or any version of it that assumes the SN barrier, leads to significant errors for such sharp emitters. This has been both shown theoretically and confirmed experimentally. The above problem was tackled by Kyritsakis and Xanthakis, {{NumBlk|:| M^{KX} (x) = h - eFx \left[ 1-\frac{x}{R} + O \left(\frac{x}{R} \right)^2 \right] - \frac{e^2}{16\pi \epsilon_0 x} \left[ 1-\frac{x}{2R} + O \left(\frac{x}{R} \right)^2 \right] .|}} After neglecting all O(x/R)^2 terms, and employing the
JWKB approximation (4) for this barrier, the Gamow exponent takes a form that generalizes eq. (5) {{NumBlk|:| G(h,F,R) = \frac{b h^{3/2}}{F} \left(v(f) + \omega(f)\frac{h}{eFR} \right) |}} where f is defined by (30d), v(f) is given by (30c) and \omega(f) is a new function that can be approximated in a similar manner as (30c) (there are typographical mistakes in ref.,).
Empirical CFE i–V equation At the present stage of CFE theory development, it is important to make a distinction between theoretical CFE equations and an empirical CFE equation. The former are derived from condensed matter physics (albeit in contexts where their detailed development is difficult). An empirical CFE equation, on the other hand, simply attempts to represent the actual experimental form of the dependence of current
i on voltage
V. In the 1920s, empirical equations were used to find the power of
V that appeared in the exponent of a semi-logarithmic equation assumed to describe experimental CFE results. In 1928, theory and experiment were brought together to show that (except, possibly, for very sharp emitters) this power is
V−1. It has recently been suggested that CFE experiments should now be carried out to try to find the power (
κ) of
V in the pre-exponential of the following empirical CFE equation: {{NumBlk|:| i = \; C V^{\kappa} \exp[-B/V], |}} where
B,
C and
κ are treated as constants. From eq. (42) it is readily shown that {{NumBlk|:| - \mathrm{d}\ln i / \mathrm{d} (1/V) = \; \kappa V + B, |}} In the 1920s, experimental techniques could not distinguish between the results (assumed by Millikan and Laurtisen) A first experimental test of this proposal has been carried out by Kirk, who used a slightly more complex form of data analysis to find a value 1.36 for his parameter
κ. His parameter
κ is very similar to, but not quite the same as, the parameter
κ used here, but nevertheless his results do appear to confirm the potential usefulness of this form of analysis. Use of the empirical CFE equation (42), and the measurement of
κ, may be of particular use for non-metals. Strictly, Fowler–Nordheim-type equations apply only to emission from the conduction band of bulk
crystalline solids. However, empirical equations of form (42) should apply to all materials (though, conceivably, modification might be needed for very sharp emitters). It seems very likely that one way in which CFE equations for newer materials may differ from Fowler–Nordheim-type equations is that these CFE equations may have a different power of
F (or
V) in their pre-exponentials. Measurements of
κ might provide some experimental indication of this. == Fowler–Nordheim plots and Millikan–Lauritsen plots ==