The role of Hurst exponent on cold field electron emission from conducting materials: from electric field distribution to Fowler-Nordheim plots

This work considers the effects of the Hurst exponent (H) on the local electric field distribution and the slope of the Fowler-Nordheim (FN) plot when considering the cold field electron emission properties of rough Large-Area Conducting Field Emitter Surfaces (LACFESs). A LACFES is represented by a self-affine Weierstrass-Mandelbrot function in a given spatial direction. For 0.1 ≤ H < 0.5, the local electric field distribution exhibits two clear exponential regimes. Moreover, a scaling between the macroscopic current density () and the characteristic kernel current density (), , with an H-dependent exponent , has been found. This feature, which is less pronounced (but not absent) in the range where more smooth surfaces have been found (), is a consequence of the dependency between the area efficiency of emission of a LACFES and the macroscopic electric field, which is often neglected in the interpretation of cold field electron emission experiments. Considering the recent developments in orthodox field emission theory, we show that the exponent must be considered when calculating the slope characterization parameter (SCP) and thus provides a relevant method of more precisely extracting the characteristic field enhancement factor from the slope of the FN plot.


Supplementary Information
Numerical method for electric field calculations Assuming that the region between the emitter surface and the anode is vacuum, this implies the numerical solution of Laplace's equation, in a discretized space, imposing suitable boundary conditions. In our case, they correspond to Dirichlet conditions both at the cathode (Φ = 0) and the anode (Φ = Φ A ), and periodic lateral conditions, so that for a domain formed by L x × L y × L z points (in the directions x, y, z, respectively), we have Φ i,j,k = Φ i+Lx,j,k = Φ i,j+Ly,k = Φ i+Lx,j+Ly,k .
Equation (1) is then solved iteratively using a second order finite difference scheme, in which the electric potential is given by That is, at each iteration the new potential Φ (m+1) i,j,k is just the average of its magnitude on the surrounding points at the previous iteration, (m). These values are then refined iteratively starting from the first step, Φ i,j,k = 0, until a satisfactory convergence criterion is met. In this work, the error is set to be ξ = 10 −6 . In Eq. (3), ω is the length of lattice parameter in the cubic lattice.
In this work, we refine the grid in such that ω = 1/50u, where u is the basic unit distance. However, no significatively deviations has been detected using ω = u. Once convergence has been achieved, the intensity of the electric field, at a any point (not necessarily on the grid), r P = (x P , y P , z P ), can be evaluated by linear interpolation from the values at the eight vertices of the circumscribing grid cube. If such a cube "starts" at the grid location (i, j, k), the corresponding cartesian components, F x,y,z (r), are given by where: and r(x P,αi , y P,αj , z P, FIG. S1 -Characteristic FEF, evaluated numerically by the solution of the Laplace equation, as a function of the anode electric potential, Φ A , considering a semi-sphere and a cone protuberances superimposed to a planar conducting surface. To test the accuracy of our method, we consider calculate the characteristic FEF, γ C , solving the electric field in three dimensions around a conducting sphere (and conical) structure superimposed in the center of a planar conducting substrate. The electric potential at the surface of the conducting sphere was fixed to Φ = 0, while at the anode a electric potential Φ A . The results for γ C of the two structures are presented in Fig. S1, for ∆x i , ∆x j L, as a function of the electric potential Φ A . It's clear the very good agreement between our numerical results and the analytical solution which predicts γ C = 3 for the semi-sphere on a plane [1]. In the case of conical structure, our results are in accordance with those presented on reference [2].

Considerations regarding two-dimensional electric field variation across the LACFES
In this Supplementary Information section, we wish to demonstrate that the same relevant results obtained in the case of the one-dimensional electric field variation across the LACFES (using orthodox cold field electron emission theory) are also obtained for a genuine rough three-dimensional LACFES setup (where a two-dimensional electric field variation across the LACFES is considered). For this purpose, we use fractional Brownian motion (fBm) algorithms [3] to generate random self-affine objects with specific Hurst exponents H. FBm algorithms can be generalized to higher dimensions using a multidimensional process such that for three dimensions, an irregular LACFES of height h(x, y) results in interfaces that are statistically equivalent in all directions. We consider the self-affine sets that are constructed using the well-known midpoint displacement algorithm for H=0.1 [4].
In Fig. S2, we illustrate the rough LACFESs produced using the described fBm algorithm for a Hurst exponent of H=0.1 and with global roughnesses W ≈ 20 nm, 37.5 nm and 0.5µm. In Fig. S3 (top), several equipotential surfaces are shown for the case in which the electric potentials of the LACFES (with H = 0.1 and W ≈ 37.5 nm) and the far-away conducting anode are Φ S = 0 V and Φ A = 500 V, respectively, corresponding to Dirichlet conditions. We apply this approximation, following the same procedure used in the one-dimensional electric field variation across the LACFES, with the intent of rounding the sharpest projections. In this way, we replace the real fBm surface with an equipotential surface, Φ E = 1 V (very similar to the previous one), that represents an approximation of the field emitter surface (see Fig. S3 (bottom)). Again, following the same procedure using the WM function, this methodology ensures that on small scales (though larger than the atomic scale), the local roughness is negligible, whereas on large scales (though much less than the lateral size of the system, L), the local roughness scales with the same exponent H. In Fig. S4, we present the local FEF distributions, ρ(γ), for LACFESs shown in Fig. S2. Interestingly, exponential behaviors (two exponentially decaying regions as well as a characteristic FEF factor of the same order, for W ≈ 37.5 nm and W ≈ 0.5µm, as that observed in thin-film emitters with irregular surfaces -see Ref. [30] of the manuscript) are observed in all cases, the same behavior as that discussed in the case of WM surfaces, when the one-dimensional FIG. S4 -Local FEF distributions, ρ(γ), for LACFESs constructed using the fBm algorithm for H = 0.1 with global roughnesses of W ≈ 20 nm (red squares), W ≈ 37.5 nm (blue triangles) and W ≈ 0.5µm (green hexagons). Exponentially decaying behavior can be clearly observed in all cases. In particular, for W ≈ 37.5 nm and W ≈ 0.5µm, the same behavior previously observed, that of two exponentially decaying regimes, is apparent. The slope of the dashed (red) line is (−0.247 ± 0.008). The slopes of the solid (blue) lines are (−0.0324 ± 0.0001) (region (I)) and (−0.046 ± 0.001) (region (II)). The slopes of the dot-dashed (green) lines are (−0.00273 ± 0.00005) (region (I)) and (0.0090 ± 0.0001) (region (II)) (see Table  SI). The differences observed in the ρ(γ) distributions are essentially related to the difference in global roughness between the LACFESs.
local electric field variation across the emitter profile is considered. Thus, the one-dimensional case is an interesting tool that captures the main results experimentally obtained for rough LACFESs, as noted previously. 0.1 × log 10 (e) = (0.0090 ± 0.0001)] is found for interval (II). By contrast, for W ≈ 20 nm, only one exponentially decaying regime is observed, with δ I 0.1 × log 10 (e) = (0.247 ± 0.008). This finding indicates that the ρ(γ) distribution is also sensitive to changes in the global roughness of the LACFES. Differences between the theoretically obtained values of the characteristic FEF and those obtained experimentally can also be caused by differences in the global roughness. Table SI summarizes the values of the parameters extracted from the ρ(γ) distributions presented in Fig.  S4 and the γ C values for all values of W explored.
Finally, we show that the β W (note that the notation is now used with W for global roughness) correction, which includes the effect of the geometry of the real three-dimensional LACFES setup on the estimation of the characteristic FEF, must be considered to ensure more precise estimation of γ C (which was found to be γ C ≈ 97.24 [≈ 519.74] from the ρ(γ) distribution depicted in Fig. S4 for W ≈ 37.5 nm [≈ 0.5µm]). We restrict this discussion to H = 0.1 and for roughnesses W ≈ 37.5 nm and W ≈ 0.5µm, i.e., to the LACFESs represented in Figs. S2 (b) and (c), respectively. Fig. S5(a), shows the behavior of the corresponding J M -F M -type FN plots. The FN plots seems, again, to exhibit approximately linear behavior for the considered range of macroscopic electric field F M , although, in reality, the  Table SII). The work function of the LACFES is considered to be approximately constant and equal to φ  Table SII).
derivative is not constant, as observed in the corresponding inset. Fig. S5(b) shows the behavior of log 10 (J M ) as a function of log 10 (J kC ), considering a macroscopic electric field in the range 0.07V /nm ≤ F M ≤ 0.1V /nm and 12.5V /µm ≤ F M ≤ 20V /µm for W ≈ 37.5 nm and W ≈ 0.5µm, respectively. These results suggest, again, a scaling relation between J M and J kC with β W = (  Fig. S5(a), characteristic FEF γ T C , the βW values extracted from the results presented on Fig. S5(b) and γ β W ,σ C . γ T C and γ β W ,σ C were calculated using Eqs. estimation of the characteristic FEF), then the corrected slope characterization parameter is found to be γ β W ,σ C ≈ 93 [≈ 505.8], yielding an error of 4% [2.6%] with respect to the γ C value determined from the ρ(γ) distribution (see Table  SII for values of the slopes of the J M -F M -type FN plots shown in Fig. S5(a), the characteristic FEF values γ T C , the β W values extracted from the results presented in Fig. S5(b) and γ β W ,σ C ).
In conclusion, the scale that results in the exponent β W must be considered (including in the case of a twodimensional electric field variation across the LACFES and in addition to the generalized slope correction from the SN barrier) in the calculation of the slope characterization parameter that is used by experimentalists to extract a more precise characteristic FEF, such as that found using the one-dimensional WM function.