Looped carbon nanotube fibers as cathodes with giant field enhancement factors

Carbon nanotube (CNT) fibers are a class of one-dimensional macroscopic materials consisting of tightly packed and highly aligned CNTs. These CNT fibers are made by a wet spinning process, have diameters ranging from 10 to 100 μm, and can be spooled into long lengths. They have high electrical and thermal conductivities and have demonstrated excellent performance when used as wire conductors.^1,2 Additionally, they have demonstrated remarkable performance when used as field electron emission (FE) cathodes.^1,3–11

Field emitters consist of sharp structures that amplify the macroscopic field, F_M, producing a local field at the apex needed to eject electrons at reasonably low applied voltages. The fraction $γa≡Fa/FM$ is known as the apex field enhancement factor (aFEF), regarded as one important parameter to assess the quality of the emitter.^12–20 To achieve high aFEFs, an emitter must be lengthy and have a high curvature at its tip. However, these characteristics make the emitter mechanically fragile, and so a two-stage emitter is a good solution for long and sharp emitters.^21–33 The first stage, being thick and sturdy, self-sustains high above the cathode, but it is not sharp enough to generate FE. A sharp protruding second-stage structure generates the FE, which is also much shorter, not to compromise its firmness. We stress that, in a two-stage emitter, the validity of Schottky's conjecture (SC),³⁴ i.e., when the aFEF of a multistage structure would be the product of the aFEFs of each of the individual stages, is not guaranteed. Actually, in a two-stage emitter, SC is expected to function when a protrusion has characteristic dimensions much larger than the ones of the protrusion above it.^23,24,26,35 However, even when SC does not work, it is possible to achieve high aFEFs, as shown hereafter.

Recently, Zhang et al.⁴ took advantage of the high flexibility of CNT fibers to produce a cathode in a looped geometry. In this case, FE occurs from the apex of the looped structure rather than the rough tip of a cut CNT fiber.¹¹ Additionally, Zhang et al. showed that there is added thermal management benefit to a looped fiber geometry.⁴ When compared to a vertical emitter of the same height, the temperature at the apex of a looped CNT fiber emitter is approximately at half the temperature at the apex of the tip of a vertical CNT fiber when producing the same amount of emitted current.⁴ 

The goal of this paper is to provide an explanation for the giant aFEFs we recently extracted from the FE characteristics of FE cathodes made from looped CNT fibers. The fabrication technique as reported by Behabtu et al.¹ produces a CNT fiber that consists of fibrillar substructures. These thin fibrils are 10–100 nm in diameter and >50 μm in length and highly aligned and densely packed. The details of looped CNT fiber cathode fabrication and the FE measurement experimental details are given in the supplementary material. Our FE data pass the orthodoxy test designed by Forbes³⁶ to check the reliability of FE parameters inferred from Fowler–Nordheim (FN) plots.^37,38 The orthodoxy analysis leads to extremely high effective FEFs (⁠$104 ≲ γeff ≲ 105$⁠) (see the supplementary material). The Scanning Electron Microscopy (SEM) image of the apex of a looped CNT fiber taken after FE measurements is shown in Fig. 1. It was found that the looped CNT fibers with protruding fibrils were good field emitters.

It was conjectured that the protruding fibrils play an important role in the FE process from looped CNT fibers, with each fibril having its own FE characteristics, with adjacent fibrils potentially experiencing electrostatic depolarization effects (commonly called “shielding” or “screening”).^39–43 However, these effects are relatively less pronounced when one fibril is significantly taller than the others. Therefore, it is anticipated that the tallest fibril contributes to the majority of the emitted current. Furthermore, different fibrils are expected to contribute to FE at different applied voltages. To test the validity of this assertion, this Letter proposes a two-stage model of a looped CNT fiber aFEF to account for their impressive experimentally observed γ_eff. A scaling is found for γ_a in terms of the geometrical parameters of a featureless looped fiber. Then, we analyze the importance of the fibrils composing looped CNT fibers to reproduce their high aFEFs.

Hereafter, we first simulated a looped CNT fiber, i.e., without accounting for the presence of fibrils at its surface, as observed experimentally through SEM analysis of its surface after FE measurements (see Fig. 1). The looped CNT fiber was modeled as a parabolic arc of height h, base length b, and diameter $2rfiber$⁠, as shown in the inset of Fig. 2. The looped CNT fiber in Fig. 1 has a height of $∼4$ mm, a base length of $∼2$ mm, and a diameter of ∼90 μm. By using a finite element technique,²⁰ the electrostatic potential distribution was calculated numerically using COMSOL (v. 5.3) to solve the Laplace equation in a 3D domain, taking advantage of the fact that the electrostatic potential distribution is symmetric with respect to the plane that cuts the fiber in two equal halves. Therefore, it was not necessary to simulate the entire fiber. Only one quarter of the looped CNT fiber was simulated using so-called symmetry boundaries. These boundaries act as mirrors by imposing the local electrostatic field to be perpendicular to their normal direction (⁠$Fl.n=0$⁠) at the surface. The looped CNT fiber and the bottom plate on which it rests were assumed to be grounded (i.e., $Φ$ = 0 V). The top boundary, located at a distance d = H–h from the apex of the looped CNT fiber, where H is the distance between the anode and the emitter plate (see the inset of Fig. 2), was assumed at potential $ΦA$⁠. The looped fiber was placed in a Hall of mirrors that mimics the solution from an infinite square array of loops. For the looped CNT fiber to be considered isolated, it must be placed in a domain of size L sufficiently large. A minimum lateral length, L, must be used to ensure that the solution differs from the solution of an isolated emitter by an error, at most, ε. This value is given by $L=6h/ε1/3$⁠, as was found previously.²⁰ In this work, we present the results for ε = 10%.

To understand the influence of h/b and $rfiber/h$ on the aFEF of a single looped CNT fiber, simulations were performed for the following set of geometrical parameters: 10 μm $≤rfiber≤$ 100 μm and 0.4 $≤h/b≤$ 2, with h = 4 mm, the latter being similar to those reported experimentally in Ref. 4 and in Fig. 1. We start our analysis with the anode located far away from the looped CNT fiber apex (i.e., a typical FE setup where $H≫h$⁠), a situation simulated using the Neumman boundary condition.²⁰ In this case, Fig. 2 shows that γ_a decreases as $rfiber/h$ increases. In addition, γ_a increases with h/b, for a fixed r_fiber, as expected. Interestingly, our numerical results are consistent with the “floating cylinder” model (FCM),¹⁶ which can be interpreted as a lower estimate of the aFEFs of the looped CNT fiber model (i.e., in the limit $h/b→0$⁠). For $rfiber/h≪1$⁠, the aFEF of the FCM is¹⁶ 

Our numerical results of the aFEF for the FCM model, shown in Fig. 2, are in clear agreement with the analytical ones, computed from Eq. (1). Discrepancies between the two approaches are expected to be larger in the limit of higher r_fiber-values. Indeed, we found discrepancies ranging from 1% to 5% for r_fiber varying from 10 to 100 μm, reflecting the accuracy of our results. The excellent accuracy of our numerical results for other models, like the “ellipsoid on a plane” and “hemisphere on a post” (HCP) ones,¹⁵ was shown in Ref. 20. The latter models give an upper estimate of the aFEF of the looped CNT fiber model (i.e., for the limit $b→0$⁠, see Fig. 2).

We can use the “floating sphere at emitter plane potential” (FS) model¹⁵ as a higher bound than the HCP model (⁠$b→0$⁠). The advantage of using the FS model is that it has analytical solution for the aFEF. The aFEF for the FS model is $γaFS≈3.5+h/rfiber$⁠.¹⁵ Combining the limits defined by $γaFCM$ and $γaFS$⁠, the aFEF of a looped CNT fiber can be approximated as

where f is given by

and θ and α are fitting parameters.

As shown in Fig. 3(a), Eqs. (2) and (3) give an excellent fit to the data shown in Fig. 2 for all h/b-values. Figures 3(b)–3(d) show that this fitting holds for different values of d, with some small discrepancies for d = 1 mm. Figure 4 shows that the fitting parameters θ and α are basically constant for $d/h≥1$ for the range of h/b values investigated here. A least mean squares fit of the θ and α dependence on h/b for $d/h≥1$ leads to the following scaling behavior: $θ=2.348[(h/b)−0.353]0.458$ (with error not larger than 15%) and $α=0.621[(h/b)−0.379]0.086$ (with error not larger than 6%). The increase in θ and α observed for $d/h<$ 1 reflects that, in this range, the aFEF should be dependent on d/h, which is consistent with experimental results reported for the effective gap field enhancement factor when CNTs were grown on metal tips.²¹ 

The above results for the aFEF of a straight looped CNT fiber do not produce values of γ_eff in agreement with experimental results since, for a typical range of looped CNT fiber parameters, we found 10$≲ γa ≲$ 100. Actually, as will be shown next, the large aFEF of looped CNT fibers is due to FE from the CNT fibrils that pull up from the fiber surface to align with the applied field during FE, as shown in the SEM image in Fig. 1.

Next, we examine the role of the fibrils standing on the surface of a looped CNT fiber. We show how the aFEF due to a combined two-stage FE model of fibrils protruding from the top of the straight looped CNT fiber helps generate the high local FEFs needed to explain the observed FE data. To investigate the importance of the fibrils, we simulated the two-stage emitter model schematically illustrated in Fig. 5, which consists of an additional quarter of an ellipsoid on a cylindrical post emitter (ECP) placed at the apex of the main fiber. Note, however, that the SEM image in Fig. 1 shows a much more intricate structure than our simplified model. The fibrils (and also the primary fiber) present kinks, gnarls, and thickness nonuniformities; they are off-center, and there are many of them. Nevertheless, we do not expect the simulation predictions to vary radically, for as long as the simulator faithfully represents the vicinity of the looped CNT fiber tip. The distance from any imperfections near the emitting tip smoothens out the potential distribution so that the imperfections should have little influence over the emission. The presence of other fibrils will cause a screening effect that diminishes the aFEF of the dominant emitter. Still, the variations are small, typically 5% to 10%, for spacings that are approximately equal to the height of the emitters.⁴⁴ In contrast, small irregularities on the tip sensitively alter the aFEF.

To account for the uncertainties in the morphology of the fibril cap, we can adjust the sharpness of the ECP by varying the eccentricity of the ellipsoid. For the case where the ECP has a radius of $r2=50$ nm, an aspect ratio of $h2/r2=1500$⁠, r_fiber = 45 μm, h = 4 mm, b = 2 mm, H = 5 mm, and $h3=6r2$ (see Fig. 5 for the parameters used), we found a two-stage aFEF $γ2=56 121$⁠, similar to that extracted from experimental current–voltage characteristics in the form of the ordinary FN-like plots (i.e., $ln (ie/(FM)2)$ × $1/FM$⁠), where i_e is the emitted current.^4,11 These dimensions are compatible with SEM images of the emitter, showing the validity of the two-stage model to reproduce the experimental FE data. The r₂ and h₃ values were adjusted to achieve an aFEF of the two-stage emitter structure $γ2≈60 000$⁠. We tested the effect of an off-center emitter, as depicted in Fig. 5, to show that the fibril does not need to be on the axis of symmetry of the looped CNT fiber to generate large aFEFs (see the supplementary material). However, we found that an ellipsoidal cap with an adjustable aspect ratio at the fibril tip was necessary. We could obtain values of γ₂ observed experimentally when modeling the fibril tip as a hemispherical cap. In this case, Fig. 5 shows that γ₂ grows too slow as a function of the aspect ratio $h2/r2$⁠. We do not expect the fibril tips to have a regular shape. Hence, the aspect ratio of the ellipsoidal cap gives us a convenient degree of freedom to account for a jagged fibril end.

In the supplementary material, we show that theoretically simulated FN plots of the FE characteristics of different looped CNT fiber + fibril combinations have nearly identical slopes when varying the location of the fibril near the tip of the fiber. The FE current from an actual fiber with the many fibrils close to its tip, as shown in Fig. 1, could, therefore, be accounted for by adding the emission current contributions from all fibrils. We also assess the limit of validity of the SC for calculating the aFEF of a two-stage structure when varying the dimensions of the fibril. As numerical proof that SC is not valid when we obtained an aFEF of ∼60 000 for an ECP model at the top of the looped CNT fiber, we show the ratio between γ₂ of a two-stage structure (looped fiber + ECP) and the product of the aFEFs of the ECP and the corresponding looped fiber (both calculated independently), as a function of the ratio height of the ECP (fibril) h₂ over the fiber radius. Our results (see the supplementary material), presented for the data shown in Fig. 5 (considering ECP fibrils for d = 1 mm), show that, for $h2/rfiber≪1$⁠, SC is approximately valid, as expected. However, $γ2∼8400$ for $h2/rfiber∼0.1$ and $γ2∼1000$ for $h2/rfiber∼0.01$ reflect the strong depolarization effects (screening) of the CNT looped fiber on the fibrils, when SC works. Interestingly, when $h2∼rfiber$⁠, SC clearly does not work (as expected), but we achieve $γ2 ∼60 000$⁠. Therefore, our results suggest that the high aFEFs experimentally achieved may not be a consequence of the validity of the SC. In summary, FE characteristics of CNT-based looped fibers indicate that these cathodes have very high effective FEFs.^4,11 Based on SEM images and FE measurements from looped CNT fibers, we proposed a two-stage model for FE from these cathodes consisting of a looped fiber with CNT fibrils near its apex. Our simulations were used to analyze FE characteristics of several looped CNT fibers and the role of each stage in producing FE characteristics in agreement with experimental results. We have found a dependence of the apex field enhancement factor γ_a on the geometrical parameters for an isolated conductive looped CNT fiber. The aFEF was found to scale as $(γa−2)rfiber/(hf)≈[ln (2h/rfiber)]−1$⁠, with $f≡f(h,b,rfiber,θ,α)$ given by Eq. (3). Then, the use of a combined two-stage structure (looped CNT fiber + fibrils at its apex) allowed us to achieve a high aFEF in excellent agreement with the values extracted from an orthodoxy theory analysis of the experimental FE characteristics. Our simulations establish theoretical support for the use of looped CNT fibers as cathodes with impressive field enhancement factors.

See the supplementary material for a study of the orthodoxy test of the FE data in the Fowler–Nordheim regime of a single looped fiber to its morphology next to its tip. It also discusses the limit of validity of the Schottky conjecture for calculating the apex field enhancement factor of a two-stage structure when varying the dimensions of the fibril.

T. A. de Assis is grateful for the financial support of the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under Grant No. 308343/2017-4. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001. This work was also supported by the Air Force Office of Scientific Research under Award No. FA9550-17RXCOR428. J. Ludwick was supported through the Southwestern Ohio Council for Higher Education (SOCHE) under Contract No. FA8650-14-2-5800.