On modeling the induced charge in density-functional calculations for field emitters

Among many reasons for the continuing interest in field electron emitters is the developing use of “cold” emitters in cryo-electron microscopy. To minimize damage to biological specimens, the exposure to beam current must be kept to the minimum that is consistent with obtaining contrast with a defocused objective lens. Under these conditions, the energy spread of the beam influences the minimum spot size at the specimen that can be achieved. Room-temperature (or “cold”) emitters have a lower energy spread than Schottky emitters and have contributed to recent advances in resolution for biological specimens.¹ 

Field emitters of practical interest often consist of dense metals, whose large numbers of mobile electrons per atom make heavy demands on computer time. An indication of their behavior can be found more swiftly by studying simpler structures of less dense material. This study reports the results obtained with density-functional theory (DFT) as defined by Hohenberg, Kohn, and Sham^2,3 and reviewed by Jones.⁴ Since 1990, DFT has been used for many calculations on carbon nanotubes (CNTs), where its ability to give results directly in physical coordinates enables it to solve the end states of finite structures more easily than by working in k-space. Here, we report some detailed results on the electronic structure of a single-wall (5,5) CNT terminated with a hemispherical cap.

Our initial motive for study of this well-tried subject was to confirm that the linear-scaling DFT code ONETEP,⁵ which uses adaptive orbitals, would be suitable for modeling an emitter of very small apex radius. In 2005, at a talk in Cambridge, Csányi et al. reported⁶ a trial of ONETEP to model the apex region of a (5,5) CNT. A short section of the capped end was represented in a single (cluster) DFT cell and the potential on the cell boundary was set by a separate macroscopic calculation.

It is well known that when a CNT is placed in an externally applied electric field, a net imbalance of charge is induced at the apex^7,8 and the field there is enhanced to many times the applied value.^7–17 Calculations made on a short section of tube with a range of the enhanced field have been successful in enabling the emitted current to be calculated.^12–14 The major obstacle to obtaining results that are consistent between applied field, net charge, and the resulting potential distribution through the emitter surface has been that of calculating the induced charge for use in the DFT solution.

In 2014, it became possible to extend our earlier work as a topic in the Simdalee2 project of the EU-FP7 programme. In developing our analysis, we realized that inclusion of net charge in the cell is incompatible with the infinite repetition of cells that most DFT codes assume by default: if every cell in an infinitely periodic sheet contains unbalanced charge, then the potential will diverge without limit.

The first step in avoiding this problem is to specify, in the input data for the code, that the calculation should be for a single cell (a cluster calculation). It is then also necessary to supply the net charge in the cell, but not its distribution. We found that this magnitude of charge can be obtained simply from the flux theorem attributed to Gauss Maxwell and others,¹⁸ by integrating the normal component of electric flux over all the boundary surface of the DFT box. Details are given in Sec. II. The resulting solution shows the charge density distribution both inside and outside the atomic framework, the movement of the Local Density of States (LDoS) with an externally applied field f, the distributions of individual orbitals and their occupancies, and the potential distribution near the CNT surface that is consistent with the charge in the DFT box caused by the full length of the CNT. Solutions were found for many values of f, allowing the induced-field enhancement factor (IFEF) to be deduced. The range of f used is believed to start below that causing current to flow. The results suggest that some values of f may have exceeded the threshold for current flow, but this is not certain since this calculation did not allow charge to leave the DFT cell. No estimates of current have been made, as we wanted first to examine details of the pre-emission state. We reported earlier^19,20 some first results of this work and a test of the influence of exchange and correlation on the value calculated for the work function of the CNT cap.

In this paper, V represents the “effective potential” (energy) as defined by Kohn and Sham,³ differing from the classical potential by a density-dependent term that vanishes as the density tends to zero. Energy and potential values are shown relative to the Fermi level in the CNT. The magnitude of electronic charge is denoted by (positive) e and electric fields are denoted (as usual) by the negatives of their formal definitions. The externally applied field (as defined in Sec. II) is written as f. In this paper, the chemical potential and Fermi level E_F are regarded as identical, enabling this theory to be applied to “cold” (that is, room temperature) emitters. Kohn and Mattsson²¹ (KM) suggested that a logical choice for “the surface” of a curved emitter would be the (infinitely thin) equipotential surface at E_F. They also considered the tapering of charge density through this equipotential (see the  Appendix). The location at which the axis of the CNT intersects the Fermi equipotential in zero field is denoted here by z_F. The origin of z was set at the intersection of the hemisphere of carbon nuclei with the axis, so the axial position of the pentagonal end ring of nuclei was −0.024 nm. The terms “Fermi equipotential” and “Fermi level” refer to the same energy, but “Fermi equipotential“ implies also the surface extending in 3D physical space. There are two spin states per energy level; in this study, they remained paired when the electric field was applied without magnetic field, and here no distinction is made between them.

We considered a macrostructure containing a (5,5) CNT of total length 50 nm, standing perpendicular to and in contact with a planar cathode of diameter 100 nm, which was spaced by 100 nm from a parallel similar anode [Fig. 1(a)]. The free end of the CNT was closed by 30 carbon atoms on a hemispherical surface, forming half a fullerene molecule. With a general anode-cathode spacing d, and with a lateral extent of the electrodes much greater than d, the magnitude of field resulting from applying voltage V_a between these electrodes in the absence of the CNT would be V_a/d. This field is denoted here as f and is described as the “applied field.” To simulate the CNT in the macroscopic system, we replaced it with a cylindrical rod and hemispherical cap, both conducting. The rod, connected to the cathode plane, was 49.7 nm long and had a diameter of 0.978 nm, as found by iteration for the Fermi equipotential outside the carbon nuclei in zero field [Figs. 1(b) and 2]. We then set an anode-cathode voltage and used the classical 3D Poisson solver FlexPDE²² to find a solution for voltage throughout the macrostructure [Fig. 1(a)]. The potential between the cathode and the anode on the system's boundary was set by linear interpolation to provide a Dirichlet boundary.

Within the macrostructure, a “DFT box” was defined as a cube of side 5 nm [shown small central square (red online) in Fig. 1(a)]. The box must be sufficiently large that the classical potential and the effective potential as defined by Kohn and Sham³ (KS) can be expected to match at the box boundary. Its location was chosen to enclose 1.83 nm of the CNT with its apex at the center of the box, leaving space to model also the barrier region. From the solution for the macrosystem with the substitute conductor occupying the same length as the whole CNT, we extracted values of voltage V_b and normal voltage gradient (∇V)_n over the surface of the DFT box. Since these were obtained with the whole conductor present, they result from the charge induced on its full length when V_a is applied. Then, by integrating (∇V)_n over the complete surface area of the box, we found the charge q_b contained within it. This is expected to be a close approximation to the charge induced by the same f on the same section of the CNT as of the conductor. Although the section of tip in the box appears disconnected, the continuity of the CNT ensures that the Fermi level in the tip equals that of the cathode plane.

A regular grid of 410 × 410 × 410 mesh points was defined in the DFT box. The CNT section contained 30 carbon nuclei in the hemisphere and 12 layers of 10 carbons each in the cylinder, terminated with 10 hydrogens. With four valence electrons per carbon nucleus, the total number of valence electrons used for calculation was 610 and with two electrons per energy level, the highest occupied orbital in zero field was the 305th (#305). The KS number density is the sum of the squared magnitudes of occupied KS orbitals. The boundary conditions supplied to ONETEP included the potential V_b (x,y,z), the total charge within the box (without specifying its distribution), and an approximation to the expected structure. The DFT calculation had no knowledge of the substitute conductor and assumed only that the Fermi level in zero field was constant throughout the CNT structure. For each value of f, the structure, orbitals, and their occupancies were found iteratively. Applied fields f up to 0.189 V nm⁻¹ induced up to 3.5 excess electronic charges into the box.

The resulting solution for potential by DFT can be compared with the macroscopic solution. The intersection of the axis with the Fermi equipotential in zero field was found to be at z_F = 0.08 nm. An offset to V_b is needed to match the solutions on the boundary because the work function is ignored by the classical solver but is calculated by the DFT solution. Thus, for the boundary specification, we added 4.43 eV to the classical potential solution, producing the agreement shown in Fig. 1(b).

The program ONETEP⁵ (Order-N Electronic Total Energy Package) iterates alternately between adapting the shapes of orbitals (in physical space) and optimizing the density kernel. Minimization of the total energy determines the one-particle density matrix for the Kohn–Sham (KS) system. Since forces at the atomic scale have a limited range,²³ the range of the basis for the localized orbitals is conveniently controlled by defining a cutoff energy. The ONETEP code was modified by members of the Theory of Condensed Matter group at Cambridge University Department of Physics to solve a nonperiodic cell with a given boundary voltage and enclosed charge. Initial optimization of the CNT structure for each value of the applied field reduced the maximum force component to 0.044 eV/Å. The results shown here were found by using a functional based on the local density approximation (LDA), despite possible inaccuracy, to gain a wide range of information within the computer time available; thus, we used the PW92 functional,²⁴ with cutoff at 1000 eV. A few runs have also been made with the functional PBE (Ref. 25), which implements the generalized gradient approximation (GGA). The maximum radius of the adaptive basis functions was 12 Bohr radii, and the projector augmented wave method²⁶ was used for the ion core potentials. Further details are given in Refs. 19 and 20.

The orbitals nearest the Fermi level are usually described as the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO). As the (5,5) CNT behaves as a conductor, Ensemble DFT (Ref. 5) for a given temperature is used and the states are allowed to have fractional occupancy according to the Fermi–Dirac distribution (FDD), so the standard acronyms for orbitals are somewhat inaccurate (possible alternatives might be HOBFL and LOAFL). However, the energy range of the FDD is a few kT, and at 300 K, kT is about 26 meV. The range of the FDD is, thus, much less than the typical separation between levels near E_F found with the limited number of atoms in our CNT section.²⁷ So here we continue to identify the orbitals nearest the Fermi level as HOMO and LUMO, but their occupancy may differ from 1 or 0. When both states of the HOMO are occupied, induction of one excess electron is enough to occupy one state of the next higher level, which becomes the new HOMO.

A check can be made for consistency of the DFT solution. The data provided to ONETEP include the charge within the cell and the potential distribution over the box boundary, but not the normal potential gradient there. After calculation, the results from ONETEP can be used to estimate the potential gradient at the boundary, for comparison with that found by the Poisson solver. It is straightforward to compare the slopes of the equipotentials on the two sides of the boundary. Figure 1(b) shows typical equipotentials found by the two methods.

Figure 2(a) shows that most of the electronic charge is concentrated close to the atomic nuclei, as expected. However, the surface of very low density shown in Fig. 2(c) is continuous, suggesting that the charge at that density is mobile and outside the framework defined by the nuclei. The skin of charge for this density appears to correspond to the π* delocalized charge known for graphene. It lies slightly inside the surface at the Fermi equipotential, which forms a cylinder of radius about 0.489 nm from the axis of the (5,5) CNT and a hemisphere of similar radius over the cap, while the radius of the cylinder of nuclei is 0.315 nm.

The Fermi equipotential was proposed by KM²¹ as a convenient definition for “the surface” of the CNT. Classically, an electron with the energy of the Fermi level would not travel beyond z_F. However, the potential gradient found at z_F (Fig. 3), though large, is not infinite. When a potential proportional to a locally defined z is inserted in Eq. (2.8) of KS³ for an orbital, the equation can be solved with an Airy function^28,29 that changes from near-sinusoidal (in the emitter) to rapidly decaying at z_F and for a short distance beyond it, into the barrier (see the  Appendix). Thus, the charge is predicted to extend into the barrier beyond the (first) Fermi equipotential by a distance of the order of 0.2 nm, as was found by Han and Ihm.³⁰ Our calculation of the charge density (Fig. 3) also shows this penetration beyond z_F.

Many reports on CNTs note that as the applied field is increased, identifiable peaks in the LDoS decrease in energy relative to the Fermi level.^{7,9,12,30–33} We reported previously¹⁹ the LDoS found for the CNT described here, for four values of the applied field. Many peaks in the LDoS, including a prominent one with energy E near E_F, decreased in energy as the applied field increased, at a rate of 4.25 eV/V nm for fields of 0.054–0.162 V nm⁻¹. We attribute this high rate to the change in net charge induced at the tip by the change in the applied field. As the field is increased, the population of states up to the Fermi level is maintained by the further occupation of orbitals, successively above the one which was the HOMO in zero field. On the initial field increase from 0 to 0.054 V nm⁻¹, the peak near E_F moved as if at the same rate plus an additional shift of −0.33 eV. Figure 4 shows the LDoS for the same hemispherical cap but as part of a longer section of 1050 electrons and orbitals in total in zero field. The twin peaks close to E − E_F = 0 shown here did not appear in the result for 610 electrons. The LDoS shown here is more populated than the earlier result, as expected for the greater number of orbitals in the whole CNT. For the full length of the nanotube, the number of electrons and KS states would be greater by a factor of 15.6. Some indicative charge density surfaces for orbitals near the Fermi level are shown in Fig. 5. With 610 electrons and 2 states per orbital, the HOMO and LUMO in zero field are numbered #305 and #306. The axial wavelength of charge distribution in these orbitals exceeds that of the atomic structure by a factor of 3. In the next higher orbitals, #307 and #308, charge is concentrated at the apex. This pair appears to correspond to localized orbitals, and #305 and #306 to extended orbitals, as reported earlier.^14,30,33,34 Applying a field of 0.054 V/nm induces one unbalanced electron into the DFT box and causes one state in level #306 to be occupied and become the HOMO. Raising the field to 0.104 V/nm fills the second state in level #306. A further increase in field to 0.162 V/nm fills one state in #307, which becomes the HOMO for that field. The change of HOMO with applied field suggests that variation of the field will vary the charge distribution over the apex and so may change the pattern on a distant screen.³³ 

The results for KS effective potential as a function of physical coordinates and applied field give further information. The potential on a section through the cap is shown in relief for an applied field of 0.162 V nm⁻¹ in Fig. 6, and the potential on axis for a range of applied field in Fig. 7.

In zero applied field, (V − E_F) changes monotonically from strongly negative, through −10 eV at z ∼ −0.02 nm to the vacuum level at z > ∼0.5 nm. The work function is a defined part of that change and there is no obvious change of slope or other glitch at V = E_F. Realistic values of work function are obtained (4.45 eV by LDA, 4.22 eV by GGA) only when the exchange-and-correlation calculation is included, as reported by Masur et al.²⁰ These values are slightly smaller than those of Zhao et al.³⁵ who found 4.68 eV and Chen and Lee³⁶ who obtained 4.78 eV, for a capped (5,5) CNT. The Pauli principle causes the density to have an “exchange hole” (modeled classically by the image effect), which implies that the potential in zero field varies asymptotically as 1/(z − z₀) (for an undefined z₀). This variation is not well modeled by the LDA or GGA functionals, implying that the potential rises more slowly for z > z_F than is shown in Fig. 7.

For our CNT of length 50 nm, the maxima of potential in the region z > 0 and 0.026 ≤ f ≤ 0.189 V/nm (Fig. 7) are found to vary more linearly with the applied field than in Fig. 4(c) of Peng et al.,¹⁰ where the CNT length was 1 μm.

The change in field on the axis induced by the applied field, at a point within the CNT cap or in the barrier region, can be found from the difference

between the potentials at z with and without field f. Figure 8 shows DV plotted against z with f as a parameter, and in Fig. 9 DV is plotted against f with z as a parameter. In Fig. 8, the behavior resembles that shown for low fields in Fig. S1 of the supplementary material published for Ref. 16 and also shows the effect of greater applied and induced fields. Our calculation also shows a lowering of the mean potential within the CNT as f is increased, believed to be due to the increase in inserted charge, and a substantial increase in mean field when f is raised to 0.162 V/nm, enough to occupy an end orbital.

Where z < 0 and at low fields (f ≤ 0.108 V/nm), DV varies little with z (Fig. 8), showing that the large variation of V with z near the atomic cores is little changed when a typical field is applied. The initial drop of about 0.33 eV between f = 0 and f = 0.026 V nm⁻¹ for all z is discussed below. Figure 8 also shows that, in the same range of z, when f rises to 0.162 V/nm the average slope of DV(z) increases to about 1.7 V/nm, suggesting penetration of a field much greater than f. This tilt appears to happen when three extra electrons are induced into the DFT box, causing orbital #307 (an end state) to be occupied. However, this deduction is valid only if no current flows with this applied field.

Where z > z_F, |DV| is generally larger than it is in the range z < 0. Figure 9 shows that in the vacuum region, DV varies linearly with f. Linear fitting produces the relation

with p = 0.0031 V/nm and r = −0.346 eV for z > z_F. The function s(z) can be constructed from the slopes in Fig. 9 and is shown in Fig. 10. From (1), the applied field f produces in the vacuum region an induced local field F(z), which is (with the same polarity as f)

The rate of change of F(z) with f is (−ds/dz), which is a function of z, shown in Fig. 11. Clearly (−ds/dz) is a measure of the enhancement of the applied field f.

Within the accuracy of the linear fitting, (−ds/dz) is independent of f at any z > z_F. This result is found without reference to FN theory or the FN plot. It parallels those of De Castro et al.,^15,16 which are discussed below.

The residual step r = −0.346 eV found in (1) appears as f changes from zero to 0.026 V nm⁻¹ and is otherwise independent of f and z. This initial step in potential V is similar to the initial step of −0.33 V found in the energy E of the LDoS (Ref. 19) as an external field was applied. It is obviously desirable to investigate these similar changes in E and V, which could have arisen from some systematic cause such as a change in the calculation of the Fermi level.

The results quoted above appear to be self-consistent but have been obtained using a functional of LDA type, which is known not to model well the asymptotic variation of potential as 1/(z−z₀). For any development of this calculation, it will be desirable to use a more suitable functional, as well as to extend the method as feasible to calculate steady-state current flow.

In some earlier modeling of CNTs,^9,10,17,31 values for field enhancement factors (FEFs) were obtained by dividing the calculated total field at (or near) the emitter surface by the externally applied field f, as was done earlier for similar geometries by classical methods.³⁷ Classically, the field in the vacuum just outside a conducting hemisphere-on-a-cylinder is zero when the applied field is zero. However, when the effective potential is calculated at an atomic scale (Figs. 3 and 7), the field at the interface is found to be large even when the applied field is zero. This occurs because the Fermi level is an intermediate value on the continuous change of potential from strongly negative (relative to E_F) at the nearby carbon cores to positive at the vacuum level. At the Fermi level, the strong retarding field is only slightly reduced by f. Thus, the ratio of total field at the surface to f is sure to vary with f and to be infinite when f = 0. A more useful measure of enhancement is the ratio of the change in surface field (the induced field) to f, as was found by Han and Ihm⁷ and as deduced here as (−ds/dz).

More recently, this IFEF was considered by De Castro et al.^15,16 for single- and double-ended CNT and similar structures, of half-lengths 0.64–4.04 nm, using periodic cells and the SIESTA code. In both these papers, the IFEF was found to be independent of the applied field over a range of axial distance, with good agreement between values calculated both classically and by DFT. The maximum values for (5,5) CNTs were between 1.9 and 3, which at first sight is unexpected since classical calculations^37,38 for this geometry suggest a minimum value of 3 at the classical surface of the tip. However, the largest values shown appear to be for points further than z_F from the end ring of carbon cores. The plot in Fig. 11 suggests that for a (5,5) CNT of length 50 nm the IFEF would be largest at z_F, about 0.1 nm from the end ring of carbon cores, in accordance with the maximum slopes in Fig. 8.

The maximum (−ds/dz) of 50 found at z_F for the 50 nm (5,5) CNT can be compared with classical estimates denoted by γ, using a relation such as γ ≈ 1.2 (L/a + 2.15)^0.90 (Ref. 37). If we first take the radius a of the CNT cap as that of the carbon cores, 0.315 nm, then L/a = 159 and γ = 116. However, the sheath of mobile electrons spreads at least to the Fermi equipotential, with a radius of 0.49 nm. Using this value in the ratio L/a, γ becomes 79. When we note that the mobile charge penetrates further into the barrier by at least 0.12 nm (Fig. 3), γ falls to 65. Thus, the spread of charge outside the cylinder of carbon cores has a strong influence on the size of the IFEF.

For the 50 nm CNT, an applied field of 0.1 V/nm induces an additional field on axis at z_F of 5 V/nm, which agrees with the values said to be typical for the onset of field emission.⁷ Just outside z_F, the field that is implied by the steep rise in potential that forms the work function (Fig. 7) is about 54 V/nm and retarding, so it is not clear that an induced accelerating field of 5 V/nm is enough to control current flow. As mentioned above, earlier work and Fig. 3 show that charge can penetrate the barrier to z > z_F even in the zero field, so it does not seem necessary to assume that added field at z_F is needed to pull charge into the barrier.

The use of a single nonperiodic cell for DFT calculation allows solution with unbalanced charge in the cell. This enables modeling of the field induced at the apex of a CNT immersed in an applied field. The charge induced in a DFT cell that is part of a larger system was found by using classical modeling to define (∇V)_n over the boundary surface of the DFT cell, with the CNT replaced by a conductor of similar geometry. The DFT code ONETEP used for this work provided (1) single-cell solution with inserted charge; (2) details of LDoS of KS orbitals; (3) details of individual orbitals including end states; (4) adaptive iteration for orbitals and occupancies; (5) order-N solution.

The use of these methods for a capped (5,5) CNT shows a low-density but continuous sheath of mobile charge outside the framework of atomic nuclei and close to the surface of the Fermi equipotential. The Fermi equipotential may for convenience be used to define the surface of the CNT, as suggested by Kohn and Mattson (KM). However, both the theory of KM and the DFT solution reported here predict that a small fraction of the maximum charge density penetrates beyond the Fermi equipotential, into the potential barrier, even in zero applied field. Further results for a steady state with no current flow show an induced enhancement factor that is a function of position but is independent of the applied field. These results for a single CNT were obtained using only Kohn–Sham modeling at an atomic scale, as implemented by the ONETEP DFT software.

The authors are grateful to M. C. Payne for initially suggesting the use of ONETEP and for providing access to the code and facilities of the TCM group at Cambridge. They also thank G. Csányi, A. Nevidomskii, and C. K. Skylaris for their earlier demonstration of ONETEP, and members of the ONETEP group including J. Dziedzic, E. Linscott, and G. Constantinescu for more recent advice and assistance in running it. The research leading to these results has received funding from the People Programme (MarieCurie Actions) of the European Union's Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement No. 606988. S.M.M. and C.J.E. also thank Thermo Fisher Scientific for financial support during completion of her thesis and for further discussions with G. Schwind and A. Bahm.

The authors have no conflicts to disclose.

S. M. Masur: Data curation (lead); Investigation (equal); Software (lead); Visualization (lead). C. J. Edgcombe: Conceptualization (lead); Formal analysis (lead); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (lead); Writing – review & editing (lead). C. H. W. Barnes: Funding acquisition (equal); Project administration (equal); Resources (lead); Supervision (equal).

The results for potential used for Figs. 7 and 8 can be made available by arrangement.

Kohn and Mattson²¹ (KM) noted that at the junction between a conductor and vacuum the density of charge must be nonuniform. They replaced the local density approximation by considering instead Eq. (2.8) of KS³ for an orbital, in limited ranges of z near where the effective potential V equals the chemical potential. There (with a small shift in z-coordinate to account for the eigenvalue) the amplitude of the orbital can be defined by Airy's equation, whose solutions are known.^28,29 In obtaining this equation, a scaling length l is defined, equivalent here to l = (ℏ²/2eFm)^1/3 where ℏ is the reduced Planck constant, e is the magnitude of electronic charge, F is the field in volts/length, and m is the electronic mass. The general solution for orbital amplitude is then a linear combination of Airy functions of (z/l), valid over a small range of z near V = E_F.

KM set the origin of local coordinates for each boundary at a point (such as z_F) where the effective potential V equals the chemical potential. When a background field is applied, the physical directions of the total field are opposite at the two boundaries of the barrier. KM first defined F (written here as F_KM) in their Eqs. (4) and (5) and then in their (6) chose to make F_KM positive at both boundaries,

where V as defined in previous papers has SI units of energy, F_KM has units of energy/length, and their z-coordinate is written here as z_KM. These definitions require F_KM to be >0, and V′ (0) to be <0. At the inner boundary, V increases steeply with distance outward from the emitter, that is, as z (as used here) increases. These definitions are consistent for the inner boundary when z_KM there is taken to increase in the opposite direction to z, that is, if the “Interior region” in Fig. 1 of KM is taken as the emitter. Then, Fig. 4 of KM describes the amplitude of one orbital at the inner boundary, provided that z_KM for a charge is understood as becoming more negative as the charge moves into the barrier. With F_KM > 0, the scaling length l_KM is also >0, and in the barrier (z_KM/l_KM) is <0. So Fig. 5 in KM also applies near the inner boundary if ζ is taken to increase in the opposite direction to charge movement.

The same result can be obtained by specifying instead that the +z-axes maintain the same physical direction at the two boundaries. Then, the field values at the two boundaries, as derivatives of V, can have opposite signs, but for consistency the scaling length l (proportional to F^−1/3) becomes negative at one boundary. We continue to use the convention that the negative of the field defined formally from potential is represented by F (but here in volts/length). With this convention,

So at the inner barrier F is negative, and consequently so are l and ζ = z/l. Also at the inner boundary, our z increases in the opposite physical direction to z_KM, so transit into the barrier corresponds to going to more negative z_KM in Fig. 4 of KM. Thus, the behavior predicted at the first barrier is the same by both methods.

For a local (retarding) field of −5.4 × 10¹⁰ V m⁻¹ at z_F as in Fig. 7, l ≈ 89 pm. This solution is valid in the range over which V can be assumed proportional to z; here we consider a range of ±2 eV. This change occurs in δz = ±37 pm, with δ(z/l) = ±0.42 and 0.46 > Ai(z/l) > 0.25. Then, in zero field, the charge density is not zero at z_F, but falls continuously as z increases, as is expected from the curvature of V(z). KM named the resulting density distribution an “edge gas” or “Airy gas.”