The recently computed result for charge density penetrating into the barrier region of a capped (5,5) carbon nanotube in zero applied field is compared with the analysis of Kohn and Mattsson for a narrow region near the Fermi equipotential. The decrease of charge density with distance into the barrier is found to be faster for the computed result, possibly resulting from a limitation in the density functional used.
The steady-state charge density and potential distribution around the hemispherical cap on a (5,5) carbon nanotube (CNT) in an applied external field were recently computed1 by the use of density functional theory (DFT) with boundary conditions set by macroscopic first principles. Individual orbitals and their occupancies, with the distributions of potential and total charge density, were calculated self-consistently for a range of externally applied fields by the use of the adaptive and order-N DFT code ONETEP.2 The method showed that as the external field f was increased, the maximum potential in the barrier region decreased (Fig. 1). The computation also showed that charge density extended outside the Fermi equipotential, penetrating evanescently into the barrier region, even in zero applied field (Fig. 2).
The DFT functionals used for this calculation were the readily available local density (LD) and generalized gradient (GG) approximations. However, it is known that these functionals are not ideally accurate for large and rapid changes of density as at a metal-vacuum interface. In particular, they do not model well the long-range or asymptotic variations of potential and density. The physical density and potential distributions were thus not modeled exactly, so the accuracy of the computation of charge penetration into the barrier is not clear. An analytic estimate would obviously be helpful for comparison with the computation.
In 1998, Kohn and Mattsson (identified hereafter as KM) discussed3 many topics relevant to field emission. These include the nonuniformity of charge density at an emitter surface, the near-sightedness of atomic forces, and the inverse radius and density of the exchange hole. They also found that, where the difference of the potential from the Fermi level is proportional to the axial distance from the Fermi equipotential (in physical space), the distribution of the amplitude of an orbital through the surface can be described simply by a known function. The problem of the accuracy of functionals was avoided by returning to a more basic form of the Kohn–Sham (KS) statement of DFT, expressing the density as a sum of densities of individual orbitals. In this paper, we outline the analysis given by KM. A numerical estimate from the analytic result predicts charge penetration into the barrier in zero external field with greater density than was shown by our computation. The discrepancy may be due to the behavior of the DFT functionals used.
This brief paper is not intended to relate to any particular model of current emission or comparison of models, such as those4,5 in the established literature. It discusses only a recent computation of unassisted charge penetration into the barrier region of a field emitter.
A. Kohn–Sham DFT
According to the theory given by KS,6 the total (number) density of orbitals at a point r, n(r), can be written as the sum of the intensities of separate orbitals ϕj at that point,
Minimization of the total ground-state energy shows that each orbital obeys an equation that resembles Schrödinger's,
where is the reduced form of Planck's constant, m is the mass of the electron, veff and ɛj have dimensions of energy, and ɛj is the lowest eigenvalue for orbital j. This equation differs from Schrödinger's in that the effective potential veff includes the classical potential due to Coulomb interactions and also a contribution from exchange and correlation effects. The particles forming the orbitals are “noninteracting electrons” in the sense that their Coulomb and exchange interactions are taken into account in forming veff and are not further included explicitly. A functional is needed to approximate the exchange contribution to veff.
B. Kohn–Mattsson theory
KM defined the “edge surface” S of an emitter as the surface at which the effective potential veff is equal to the chemical potential, μ. For any point r, they defined a z-axis through r and perpendicular to the nearest part of S, with z increasing from the barrier into the emitter. This direction is opposite to the transport of charge in field electron emission. KM also defined possible variations transverse to z, but they are not considered here so we express v(r) [ = v(0,0,z)] as v(z).
The F used by KM is the negative of the gradient of potential energy for their direction of z.
C. Notation in this paper
Comparison of our computed results with the analytic result of KM in the two coordinate systems requires frequent changes of sign, in particular, when gradients of quantities are considered. This comparison is eased by using the same direction for the z-coordinate in both calculations. As the z-direction used in our computed results was fixed, the analysis of KM is restated here with the following changes of notation:
Our axial coordinate z increases from the emitter toward the barrier, with z = zF at S, the surface at the chemical potential.
- We denote the electric field (enhanced if an external field is applied) by a variable FV equal to the negative of the electric field as defined using our z coordinate. Then, positive values of FV accelerate an electron in the +z direction. At the emitter-barrier interface, the strong field due to dipole and exchange effects confines electrons in or near the emitter. Here, the gradient of the effective potential, dveff /dz, is positive, so Fv is negative. Also, we express FV in units of voltage/distance, as is usual in the literature of the field emission. We consider behavior in a small range of z near zF, where veff can be approximated by(2)
Here, e is the magnitude of the charge on an electron and Fv0 is the field at zF in zero applied field.
- We use the scaling length given by KM but define it with changes to account for the sign and units of Fv. We also write the length as λ to reduce confusion with other symbols,(3)
We approximate by regarding μ as equal to the Fermi level, and we refer to the 3D physical surface at the Fermi level as the Fermi equipotential.
D. Axial variation of orbitals near the Fermi equipotential
where ɛj is the eigenvalue satisfying (1) for a self-consistent distribution of charge and potential. We also define a dimensionless ratio xj by
whose real solutions, Ai(xj) and Bi(xj), have properties that are known.7,8 We expect the wave amplitudes to decrease on transit through the barrier (x > 0), so we use only the solution
The function Ai(x) is shown without ambiguity of sign in Fig. 9.3.1 of Ref. 8. This result agrees with Fig. 4 of KM on reversal of the z-coordinate; however, in Fig. 4 and Eq. (20) of KM, their z is relative to the scaling length, like our x above. Approximate solutions of (6) can be deduced by the JWKB method, but they diverge as |x| → 0. In contrast, the Airy functions are exact solutions of (6), finite at x = 0.
III. ANALYTIC AND COMPUTED RESULTS FOR CHARGE DENSITY
For any orbital j, we can find values of ϕ (dropping the subscript) in the small range of z for which the approximation (2) is valid. The range of potential over which this approximation can be made is certainly less than the work function; here, we take 0 < veff < 2 eV as an order-of-magnitude assumption for the linear range. The axial distance corresponding to this potential range is 2/e|Fv0|. Fv0 is the field at the Fermi level in zero applied field; for the (5,5) CNT reported earlier,1 we find it to be about −54 V nm−1. Hence, the axial distance over which the field can be assumed constant is of the order of 2/54 ≈ 0.037 nm. For the same value of Fv0, the scaling length λ is about 89 pm. In this case, the range of x over which the Airy function is a reasonable approximation is no more than about 0.42. The values of Ai(x) for x = 0 and x = 0.42 are about 0.355 and 0.250. Thus, the amplitude of a single orbital decreases by a factor of the order of 0.70, and its intensity by 0.50, over the axial range outside z = zF in which the local field can be assumed constant.
The result of our computation can be compared with this analytic result. From Fig. 2, the electron density at z = zF is approximately 0.013 eÅ−3. When x = 0.42, then (z − zF) ≈ 0.037 nm, as above. In Fig. 1, the Fermi level is at zF = 0.08 nm, so in the figure, x = 0.42 corresponds to z ≈ 0.12 nm. At this z, the density is about 0.0037 eÅ−3. The computed density at x = 0.42 relative to that at x = 0 is thus.0037/.013 = 0.28, which is appreciably less than the 0.5 estimated analytically. Thus, the analytic solution, though limited to the linear range of the barrier, predicts penetration of greater charge density over this range than we have observed thus far by computation.
A possible reason for the more rapid decrease of computed charge density in the barrier is that the functional used for computation is known to produce potentials which at large distances (∼1 nm) from the cores fall off exponentially, faster than varying inversely as is expected from the exchange/image effect.
Over the narrow range of z where veff is proportional to (z − zF), the function Ai(x) for the amplitudes of orbitals can be approximated by
The analytic solution and our computation thus affirm that some density of charge occurs outside EF even in zero applied field.
At the outer interface between barrier and vacuum, similar principles may apply but the details will differ. Only the potential barrier at the emitter surface is considered here.
It was reported earlier9 that as the externally applied field is increased, peaks identifiable in the local density of states (LDoS) for the hemispherical cap move monotonically to more negative energy, by about 1 eV for the range of the external field used. It can also be seen from (4) that the distance by which zj is offset from (z − zF) is proportional to the eigenvalue ɛj. One might speculate that if (a) a peak in the LDoS continues to identify a single orbital as it is moved by the external field and if (b) there is a monotonic relation between orbital energy and ɛj, then varying the external field may move the charge density for each orbital along the axis. The question of the relation between the energy of an orbital and the corresponding KS eigenvalue has been widely discussed. Here, we note a report10 that calculations made with suitable long-range corrections produce accurate values of orbital energies.
Our DFT computation of the charge density outside the cap on a (5,5) CNT showed some charge penetration into the potential barrier over a short range beyond the Fermi equipotential. This computed result has been compared with the analytic result of Kohn and Mattsson for the axial variation of orbital amplitude over the same range. The computation predicts a more rapid decrease of total density with distance into the barrier than the analytic result for the density of a single orbital. The density functionals used for the computation are known to be inaccurate for long-range effects such as the exchange (or image) interaction. However, both methods clearly agree qualitatively that charge penetrates the barrier beyond the Fermi equipotential, even in zero field.
Conflict of Interest
The author has no conflicts to disclose.
C. J. Edgcombe: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).