Comment on 'Metallic nanowire–graphene hybrid nanostructures for highly flexible field emission devices'

A recent paper [1] described cold field electron emission (CFE) from arrays of gold posts grown on a flexible graphene-like substrate. This is exciting technological progress, but the paper has theoretical problems. It has been chosen for comment because it concerns metal emitters (for which CFE theory is well understood) and appears in a UK journal that publishes engineering-related material. Some issues raised below are also relevant to other types of large area field emitter (LAFE).

The discussion here uses quantities defined more carefully elsewhere [2]: a and b denote the Fowler–Nordheim (FN) constants [3], ϕ the emitter work function, F_C the characteristic local barrier field, F_M the macroscopic field, γ_C the characteristic macroscopic field enhancement factor (γ_C ≡ F_C/F_M), J_M the macroscopic (i.e., 'LAFE-average') emission current density, λ_M the macroscopic pre-exponential correction factor, and ${S}_{\mathrm{M}}^{\mathrm{expt}}$ the measured slope of an FN plot of J_M(F_M). The equivalences between the Arif et al [1] symbols and mine (theirs first) are: B ≡ b, Φ ≡ ϕ, E ≡ F_M,J ≡ J_M, $m\equiv {S}_{\mathrm{M}}^{\mathrm{expt}}$; their A sometimes means +a, sometimes −a; interpretation of their β (my β^expt) is for discussion.

Current densities, fields and related quantities are taken positive here, even though they are negative in conventional electrostatics; a is always positive. Approximate FN-constant values are [3] a ≈ 1.541 × 10⁻⁶ A eV  V ⁻², b ≈ 6.831 × 10⁹ eV ^−3/2 V  m⁻¹ (typographical errors occur in [1]).

• (1)  
  The first comment concerns figure 3 in [1]. The FN-plot slope ${S}_{\mathrm{M}}^{\mathrm{expt}}$ yields an elementary slope characterization parameter β^expt given by

  $$\begin{eqnarray} &&{\beta }^{\mathrm{expt}}=-b{\phi }^{3/2}/{S}_{\mathrm{ M}}^{\mathrm{expt}}. \end{eqnarray} \tag{ 1 }$$

  The plots in figure 3 of [1] have been re-measured, giving the slope values in table 1. Using the clean gold value ϕ = 5.1 eV, these yield the β^expt values shown. Values reported in [1] (below figure 3) are also shown. Discrepancies exist for all three metal-post diameters, for no obvious physical reason. Perhaps data error has occurred.
• (2)  
  The second comment concerns FN-plot analysis. It is important to know whether the measured current is controlled solely by CFE at the emitter/vacuum interface and is physically described by an FN-type equation with a constant field enhancement factor and with ϕ equal to the expected value. If these things are not true, then the interpretation of β^expt may be uncertain. It is often tacitly assumed that a straight experimental FN plot proves these things. In fact, this test is weak: near-linearity is necessary but not sufficient. Reference [2] developed a stronger test that checked quantitatively for inconsistency between the slope value ${S}_{\mathrm{M}}^{\mathrm{expt}}$ and the 'orthodox' assumption—used in deriving equation (5) below—that the current is controlled by tunnelling through a Schottky–Nordheim (SN) barrier [4]. This test [2] uses FN-plot data to find the scaled-barrier-field value f_Cw [2, 4] related to a macroscopic field F_Mw near the middle of the experimental range of 1/F_M values, and checks whether f_Cw is outside the range 0.22–0.32. When applied to figure 3 in [1], taking (1/F_Mw) as 0.48 μm V⁻¹, and using the re-measured slopes ${S}_{\mathrm{M}}^{\mathrm{expt}}$, the test generates the f_Cw values shown in table 1. These indicate that systematic inconsistency exists. An improved test, introduced here, extracts f_C-values in a different way and checks the whole range of experimental data. For an FN plot of J_M(F_M) for an SN barrier, the f_C-value relating to the macroscopic field F_M is given theoretically by

  $$\begin{eqnarray} &&{f}_{\mathrm{C}}=-s({f}_{\mathrm{t}}){\eta }^{\mathrm{SN}}{F}_{\mathrm{ M}}/{S}_{\mathrm{M}}^{\mathrm{expt}}, \end{eqnarray} \tag{ 2 }$$

  where η^SN = 9.836 238(eV /ϕ)^1/2. Equation (2) follows from equation (A8) in [2]. Here, s(f) is the slope correction function for the SN barrier [4] and f_t is the f-value at which the tangent to the theoretical FN plot is parallel to the line fitted to the experimental FN plot (see [5]). The exact value of s(f_t) is always unknown, but a good approximation (usually valid to within a few per cent or better) is 0.95. Hence, for orthodox emission as defined in [2], the f_C-value relating to F_M is

  $$\begin{eqnarray} &&{f}_{\mathrm{C}}^{\ast 0.9 5}=0.9 5{\eta }^{\mathrm{SN}}{F}_{\mathrm{ M}}/{S}_{\mathrm{M}}^{\mathrm{expt}}. \end{eqnarray} \tag{ 3 }$$

  The notation ${f}_{\mathrm{C}}^{\ast 0.9 5}$ shows that s has been taken as 0.95. Whereas my earlier approach [2] used the formula s(f) ≈ 1 − f/6 to estimate s(f_t) from (possibly inappropriate) experimental data, the approach here uses a theoretically predicted value for s(f_t). This revised approach will often be more accurate, particularly when the emission is not orthodox. In the Arif et al work [1], the experimental (1/F_M)-range is from 0.33 to 0.63 μm V⁻¹; the related F_M-range is approximately 1.6 to 3 V μm⁻¹. Re-measured values are used for ${S}_{\mathrm{M}}^{\mathrm{expt}}$. The related f_C-values are written as ${f}_{\text{low}}^{\ast 0.9 5}$ and ${f}_{\text{high}}^{\ast 0.9 5}$; table 1 shows derived values for the three post diameters. Since the test is for lack of orthodoxy, the issue is whether any f-value in the derived range is physically unreasonable by orthodox standards. Re-analysis [6] of experiments by Dyke and Trolan [7] found the safe upper limits for steady and pulsed emission as f = 0.34 and f = 0.60, respectively. For each post diameter in table 1, the whole extracted f-value range is above both these safe limits. Thus, all these results would be physically unreasonable if the emission were orthodox. This finding confirms that systematic inconsistency exists. In such circumstances the extracted f-values have no well-defined physical meaning, but merely show that a problem exists. The most obvious explanations of this inconsistency are: (a) undetected series-resistance or 'saturation' effects, or other measuring-circuit misbehaviour; (b) tunnelling not through an SN barrier; (c) some mutual-screening effect; (d) field dependence in the array geometry, or other neglected voltage dependence in the field-enhancement effects. As Arif et al [1] point out, saturation effects often lead to FN-plot curvature, but this was not observed. Further, metal emitters of the radii used in [1] should have tunnelling barriers that can be treated as SN barriers. The detected inconsistencies thus remain unexplained. Possibly, options (c) and (d) merit careful exploration. When inconsistency is found, it may be incorrect to interpret β^expt as the value of a field-enhancement factor. It is possible that β^expt is just an experimental slope characterization parameter, and that it is not known how to relate it to a physical field-enhancement factor. I concur with Arif et al that one should be cautious in interpreting β^expt, though for reasons more complex than theirs.
• (3)  
  The third comment relates to equation (1) in [1], which can be written

  $$\begin{eqnarray} &&{J}_{\mathrm{M}}=a{\phi }^{-1}{\gamma }_{\mathrm{ C}}^{2}{F}_{\mathrm{ M}}^{2}\exp [-b{\phi }^{3/2}/{\gamma }_{\mathrm{ C}}{F}_{\mathrm{M}}]. \end{eqnarray} \tag{ 4 }$$

  This equation is widely used in LAFE papers. In reality, the simplest FN-type equation suitable for describing CFE from a metal-post array is [2]

  $$\begin{eqnarray} &&{J}_{\mathrm{M}}={\lambda }_{\mathrm{M}}a{\phi }^{-1}{\gamma }_{\mathrm{ C}}^{2}{F}_{\mathrm{ M}}^{2}\exp [-{v}_{\mathrm{ F}}b{\phi }^{3/2}/{\gamma }_{\mathrm{ C}}{F}_{\mathrm{M}}]. \end{eqnarray} \tag{ 5 }$$

  Equation (5) has two correction factors: v_F is a particular value of the principal SN barrier function v [4], and λ_M is a macroscopic pre-exponential correction factor [2]. Although equation (4) works adequately when deriving equation (1), the omission of λ_M is a serious flaw, because λ_M is believed to usually lie in the approximate range 10⁻⁹–10⁻³ [2].

Table 1.  Data associated with the FN plots shown in figure 3 of [1]. The neper (Np) is used as the unit of natural logarithmic difference.

Post diameter (nm)	Slope ${S}_{\mathrm{M}}^{\mathrm{expt}}$ (Np V  μm−1) (re-measured)	Extracted value of βexpt		Derived fC -values
				As given in [1]	Re-analysed	fCw	${f}_{\text{low}}^{\ast 0.9 5}$	${f}_{\text{high}}^{\mathrm{\ast 0.95 }}$
73	−2.4	3533	33 000	2.3	2.8	5.3
100	−8.1	2674	9 700	0.94	0.82	1.5
200	−5.4	1956	14 000	1.3	1.2	2.3

Extraction of λ_M-values from 'orthodox' experimental data was recently discussed [2]. A simple method involves finding the f_C-value relating to a particular experimental F_M-value (or vice-versa), and finding the experimental J_M-value relating to this F_M-value. The f_C-value is then used in a scaled FN-type equation [2] relating to the SN barrier: for the linked set of values {f_C,F_M,J_M} this equation yields a theoretical value for (J_M/λ_M). Hence, λ_M can be obtained.

LAFE papers often do not show the units in which ${J}_{\mathrm{M}}/{F}_{\mathrm{M}}^{2}$ is measured before taking the logarithm for an FN plot. Sometimes one can infer what measured J_M value corresponds to a given F_M; unfortunately, this cannot be done for figure 3 in  [1].

An alternative approach uses the J_M-values given on page 4 of [1] and shown in table 2 here. Corresponding f_C-values are not known, but one can calculate values of (J_M/λ_M) at the limits of the Dyke–Trolan steady operating range (0.20 ≤ f_C ≤ 0.34) [7] and obtain corresponding λ_M estimates. If the emission were orthodox, it could then be assumed that the actual value of ${\lambda }_{\mathrm{M}}^{\mathrm{expt}}$ would lie in the calculated range. All values of ${\lambda }_{\mathrm{M}}^{\mathrm{expt}}$ in table 2 are very much less than unity. These results imply that, if FN-type equations were to apply physically to the measured current–voltage data (and hence to the reported J_M − F_M data), then λ_M must be included in these equations.

Table 2.  Estimates of the range within which λ_M lies, made using the current-density (J_M) values on p 4 of [1], and equations given in [2], with ϕ = 5.1 eV. Note that all extracted values of ${\lambda }_{M}^{\mathrm{expt}}$ are very much less than unity.

Post diameter (nm)	JM (A m−2)	${\lambda }_{M}^{\mathrm{expt }}$for assumed f-values
				f = 0.20	f = 0.34
73	3	8.7 × 10−6	5.6 × 10−10
100	0.54	1.6 × 10−6	1.0 × 10−10
200	0.12	3.5 × 10−7	2.3 × 10−11

The factor λ_M should always be included when an FN-type equation is used to analyse LAFE data. However, when (as in the present case) a test suggests that the emission is not orthodox, then equation (5) may in practice be serving as a complicated fitting equation for a nearly linear semi-logarithmic plot, and may or may not describe a tunnelling process. A derived ${\lambda }_{\mathrm{M}}^{\mathrm{expt}}$ value must then be treated as an obscure empirical fitting parameter, and its physical interpretation as needing further investigation.

Using equation (4) does not usually impede LAFE development. However, using an equation that over-predicts current density by a very large factor is undesirable, because it can give a misleading impression of potential LAFE performance to non-experts (including funding providers), and might in special circumstances cause engineering regulatory difficulties.

Better re-analysis could have been made here if [1] had included information about the units of ${J}_{\mathrm{M}}/{F}_{\mathrm{M}}^{2}$. Incomplete reporting of experimental data occurs in many LAFE papers. Reference [2] makes suggestions for improved practice.

In the author's view, many published LAFE results may have theoretical problems broadly similar to those discussed in comments (2) and (3) above, particularly when the emitter is not a metal.

Acknowledgment