Field emission from non-uniform carbon nanotube arrays
© Dall'Agnol and den Engelsen; licensee Springer. 2013
Received: 4 April 2013
Accepted: 29 June 2013
Published: 10 July 2013
Regular arrays of carbon nanotubes (CNTs) are frequently used in studies on field emission. However, non-uniformities are always present like dispersions in height, radius, and position. In this report, we describe the effect of these non-uniformities in the overall emission current by simulation. We show that non-uniform arrays can be modeled as a perfect array multiplied by a factor that is a function of the CNTs spacing.
Carbon nanotube (CNT) arrays for field emission (FE) applications have been extensively studied experimentally and theoretically [1–5]. Various improvements to fabricate well-aligned CNT arrays have been achieved, but non-uniformities are always present. To build precise arrays is expensive and difficult in extending to large areas. Simulation of CNT arrays is cost effective; however, simulation of these structures including non-uniformity is rare in the literature. To model non-uniformities in FE, it is necessary to understand their effects on the emission current. The simulation of FE in large domains is notoriously difficult especially in three dimensions, which is necessary in this analysis. The difficulties include long simulation times, large computer memory requirements, and computational instability. The first analysis of this kind is the recent work of Shimoi and Tanaka . They managed to perform three-dimensional (3D) simulations based on boundary elements that avoided meshing the volume of the 3D domain. They simulated carbon nanofibers with random position and height to match the emission pattern that they obtained experimentally. In this work, we perform simulations of non-uniform CNTs with dispersions in height, radius, and position in limited ranges and with small CNT aspect ratios aiming to correlate the current from non-uniform arrays with the current expected from perfect arrays. We restrict our analysis to a hemisphere-on-a-post model [4, 6–8], in which the CNTs are regarded as perfect conductors, with a smooth surface and oriented normal to the substrate. In this report, we shall refer to these idealized tubes as CNTs.
The simulations are performed using software COMSOL® v.4.2a, which is based on the finite elements method. The CNT array, as shown in Figure 1, is regarded as purely electrostatic system. A macroscopic vertical electric field of 10 GV/m is applied on the domain. The side boundaries have symmetry boundary condition, which states that there is no electric field perpendicular to these boundaries (E.n = 0) making them act as mirrors. These conditions determine the norm of the electric field in the domain.
Each simulation run, identified with the number of the run, k, has a particular set of randomized parameters that yield the normalized current, I k . The I k values from a 3 × 3 domain present large variations, but after averaging 25 simulation runs, we obtain a smoother behavior, which is the expected values of the stochastic I k . The error in I k decreases by a factor of 1/√k. In FE experiments, the observed current is the average over a large number of CNTs. We did 25 simulation runs of 3 × 3 CNTs, which is physically similar to simulate 225 CNTs in one run. However, the latter calculation is impossible due to memory and numerical instability. Even a 3 × 3 array takes a rather long time to simulate, and some of our results were not reliable at large spacing. We simulated arrays with 1 × 1, 2 × 2, 3 × 3, and 4 × 4 randomized CNTs. The average current depends on the size of the domain, but the convergence is fast. The normalized currents as a function of the spacing for 3 × 3 and 4 × 4 arrays are exactly the same within the error. Hence, a 3 × 3 domain is already large enough to represent a random field of CNTs.
Results and discussion
Equations (5) to (7) have no physical meaning; they are mere interpolating functions only to provide numerical values between the simulated points. These interpolating functions were chosen for representing the shape of the curves by taking the logarithmic scale of the x-axis into account.
where I phr is given in Eq. (11).
Let us give an example: consider a non-uniform array with α p = 0.4, α r = 0.5, α h =0.8 observed microscopically and s = 2 h yielding an average emission of 1 μA. From Eqs. (14), (15), and (16), we calculate a normalized current of I = 1.28, which corresponds to the 1 μA; Ihigh = 4.94 (3.86 μA) and σIhigh = 1.90 (1.48 μA). Now, suppose Imax is 10 (7.81 μA), then the fraction ξ of emitters that will burn out at 1 μA is smaller than 0.04% according to Eq. (17). In this example, Imax is constant: otherwise, the calculation of ξ will be more elaborate. If Imax is a known function, then ξ must be integrated over Imax for a refined estimative. However, we shall not deepen our analysis on ξ in this paper.
We simulated the behavior of the field emission current from non-uniform arrays of CNTs and obtained correction factors to multiply the current from a perfect CNT array toward the currents of non-uniform arrays. These correction functions are valid if the allowed dispersion in height and radius is kept inside the limits of 50% and 150% of their average values and if the randomization of the CNT position is done inside the designated unit cell. The uneven screening effect in non-uniform arrays causes many CNTs to become idle emitters while few may become overloaded and burn out. To avoid this, uniformity is desired: however, non-uniformities are always present in some degree, and our model describes how to treat them. This model can also be used in estimating how many CNTs are expected to burn given their tolerance and the total current extracted from the array.
We like to point out that in a previous work , we showed that the emission from 3D CNT arrays can be simulated in a two-dimensional (2D) rotationally symmetric system with proper boundary conditions. The currents from the 2D and 3D arrays are also related by a factor that is a function of the aspect ratio and spacing of the actual array. The combined correction factor from Eq. (14) and the procedure in  can considerably ease the modeling of FE from non-uniform CNT arrays, as they can be reduced to perfectly uniform arrays, which may be treated in a 2D model.
This work was supported by the National Council of Technological and Scientific Development (CNPq) of Brazil.
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