Theoretical analysis of 1D resonant tunneling behavior in ion-enhanced cold field and thermo-field emission

Over the past century, field electron emission sources have underpinned a number of technologies, including electron microscopes, radio frequency power amplifiers, space-based electric propulsion, and thermionic energy conversion devices.^1–3 In many of these applications, emission cathodes seek to maximize emission current density in order to reach high efficiencies; (cold) field emission cathodes take advantage of quantum tunneling under high applied electrical field, while thermo-field emission cathodes additionally apply high temperature to induce a thermionic component. The tunneling current for field emission was first described by Fowler and Nordheim⁴ and was further developed by Murphy and Good⁵ for large temperature ranges. These theoretical models, and the many that have followed, typically consider emission under ideal conditions—that is, emission from a clean surface into a vacuum.

However, there can be several circumstances where electron emission does not occur in a vacuum environment or from a clean surface. For example, electron emission can often occur in a gaseous environment in the presence of a plasma or gas discharge. Plasma environments have often been proposed for enhancing thermionic energy conversion devices,⁶ and in arcs and cathode spots, thermo-field emission is an essential electron generation mechanism.⁷ Additionally, a number of recent studies have shown that cold field emission becomes important in microscale plasma devices (∼1–10 μm), impacting gas breakdown as well as other properties.⁸ One natural consequence of the gaseous environment is that positive ions that formed in the interstitial gap by electron-impact ionization can enhance the emission current as they move towards the cathode; a phenomenon termed ion-enhanced emission that was studied at a fundamental level by Soviet physicists in the 1970s.^9–13 

An analogous behavior to ions enhancing field electron emission is how adsorbates (atoms or molecules that are adsorbed on the cathode) also impact emission current. The atom-metal surface interaction itself in the adsorption process has been investigated extensively since 1933,^14,15 and later, when field emission was used for analyzing the properties of adsorbed layers, a number of efforts focused on understanding the effect of adsorbates on field electron emission itself.^16–19 Enhanced emission due to surface contaminants and adsorbates can have adverse effects in applications where field emission needs to be suppressed. One example is particle accelerators, where high electric fields cause failure due to vacuum arcing. The injector part of accelerators is significant in length,²⁰ and as field emission sites form where gradients are large (as near irises or vanes), large and long structures have ample opportunity for dark current due to field emission to be injected where it is unwelcome or damaging. Recent studies have suggested that field emission in accelerators could occur at small spots where the work function is anomalously low,²¹ and it has long been suggested that surface contaminants and adsorbates are responsible for premature vacuum arcing such that great care is taken to ensure accelerator cavities are thoroughly cleaned.²² 

With both ion-enhanced and adsorbate-enhanced field electron emission, the enhancement occurs because the tunneling barrier is distorted. From a quantum perspective, the primary impact of the ion or the adsorbate core is that its own Coulombic potential distorts the potential barrier at the cathode surface. Consider the simplified one-dimensional (1-D) schematic shown in Fig. 1, where ε represents the energy normal to the surface. The Coulombic potential of the ion/adsorbate core (solid line) serves to both reduce the height of the potential barrier, much as in the Schottky effect, and also thin the first potential barrier, thereby enhancing tunneling emission in a non-resonant, direct tunneling process. Further, the potential well of the ion/adsorbate core can lead to resonant effects, where the electron effectively tunnels simultaneously through both potential barriers, which also enhances the emission current.

In many studies of ion-enhanced emission, the non-resonant, direct tunneling component has been approximated by incorporating the Coulomb potential of the ion into either the applied field term in the Fowler-Nordheim equation^23–26 or as a reduction in the work function in Schottky emission.²⁷ However, such a simplified approach discounts the potentially significant contribution of resonant tunneling to the current enhancement. Resonant tunneling, of course, is a well-understood and common phenomenon in semiconductor structures; in for example, resonant tunneling diodes.²⁸ Resonant tunneling occurs when incident electron energies are equivalent to the quasi-local levels within the barrier, where the transmission probability has a sharp maximum that can be—but generally is not—as large as unity. There have been extensive studies on these energy levels and the Lorenztian form of resonant transmission probability in heterostructures,²⁹ multimode nanostructures,³⁰ and tunneling diodes.³¹ However, the problem for field emission is notably more difficult as the effective mass is the rest mass of the electron, and the fields are appreciably higher.

The resonant processes in both ion-enhanced and adsorbate-enhanced field emission have been studied previously. Resonant tunneling in cold field emission was first discussed for atoms adsorbed on metal surfaces by Duke in 1967 and then by Gadzuk in 1970.^16,32 They approximated the potential barrier as a trapezoidal-triangle barrier with a square well in order to solve Schrödinger's equation and calculate the transmission probability. Around the same time, a theoretical and experimental study analyzed the current enhancement due to resonant tunneling through adsorbates, with results showing that the ground state level is broadened and shifted upward.¹² For studies on ion-enhanced emission, updated approaches to solving Schrödinger's equation to handle a more realistic Coulombic potential, such as the coupled angular mode method³³ and finite difference schemes, have been applied to both 1-D and 2-D axisymmetric models.^34,35 These studies, which focused on thermo-field emission in an electric arc, showed that at very high temperatures (>2000 K) and at high electric fields (1 V/nm), ion enhancement is primarily due to the non-resonant process, with only a minor contribution by resonant processes involving the n = 3 state (n is the quantum number). Resonant processes involving the n = 2 state, however, can be dominant when the field reaches 4 V/nm, but this diminishes as the field decreases, becoming negligible for weak fields (<0.1 V/nm). It has also been separately confirmed³⁶ that resonance effects contribute negligibly to cold field emission in the limit of low applied fields.

While these prior studies focused on specific conditions identifying if resonance occurs, we propose that if the conditions (electrical field, temperature) are well controlled, resonant emission can be intentionally designed into a system. One conceptual example is to use a semiconductor material as the cathode and to match the electron distribution in the cathode to the resonant states of an approaching gaseous ion or an absorbed ion. As recent studies have shown, manipulating electron emission from semiconductors into the gaseous phase gives rise to various interesting plasma phenomena,^37,38 and controlling resonant emission offers one such approach. However, resonant tunneling behavior must be first understood as a function of key parameters (i.e., applied field, ion position, etc.). In this paper, we focus on ion-enhanced emission and treat the incoming ion as single, positively charged, stationary ion, and solve the 1-D Schrödinger equation including the Coulombic potential of the ion and analyze resonant behavior. We use a combination of Airy functions as the outgoing wave instead of plane waves, which corresponds to a more realistic triangle barrier far away from the cathode without cutting the potential at zero, thereby allowing farther ion positions to be calculated. Resonant energies are analyzed by varying the applied field and ion position independently, and their relative impact on electron emission quantified. Within the context of perturbations to Schrödinger's equation, a simplified expression for resonant energies can be derived, and we show that under certain conditions, the asymmetry induced by the cathode surface causes unanticipated resonant conditions. This relatively simple equation can be used to both predict resonant energies for different electric field and ion position conditions and shows the scaling behavior for these two variables. While this work explores the effect of resonant enhancement from the perspective of positive ions in a plasma, this analysis also applies to adsorbates, where the Coulombic potential of the singly charge positive ion is equivalent to the adsorbate core.

The probability of an electron escaping from the cathode and being emitted is determined by the transmission probability, which describes the amplitude of the transmitted wave function relative to the wave function incident on the cathode surface. Under a 1-D approximation, the transmission probability D(ε) is determined by solving the time-independent, 1-D Schrödinger's equation

where Ψ(x) is the wave function and x is the distance normal to the cathode surface, measured from the cathode electrical surface. For ion-enhanced emission, we describe the electron potential energy V(x) as²⁷ 

where the first term is the vacuum potential energy defined relative to the bottom of conduction band (i.e., the Fermi energy plus the work function V₀ = ε_F + ϕ), the second term is the potential due to the applied electric field F, the third term is the Coulombic potential due to the image of an electron just escaping the cathode, and the final term is the Coulombic potential due to a singly charged ion at position L, which is normal to the cathode with a being the radial dimension along the surface. Here, we assume that since a is approximately the atomic radius r_a, and thus small, we can set it to zero and cut off the well at a fixed atomic radius to avoid the singularity. Eq. (2) then becomes

where x₀ is the position that V(x) = 0 (x < L) in Eq. (2) and r is the radius of the ion of interest. The bottom of the Coulombic ion well is then a fixed potential where Eq. (2) is evaluated at position V(x = L − r). For the perturbation analysis, we will discuss this in more detail.

Eq. (1) can be re-arranged in a more concise form

with $k2=2m[ε−V(x)]/ℏ2$⁠. This second order ordinary equation can be integrated numerically using a 6th order Numerov discretization,^39,40 and while other analytical techniques could be used (e.g., the transfer matrix approach), we found that using a high-accuracy numerical approach gave us the benefit of not approximating any portion of the potential with little impact on computational speed. However, we did benchmark against transfer matrix solutions to confirm the accuracy of our numerical scheme.

Starting from the region far away from the cathode (x → ∞), where V(x) ≈V₀−eFx and the wave function is a pure outgoing wave, we assume the outgoing wave to be a combination of Airy functions^3,41

where z = f^−2/3k², and $f=2meF/ℏ2$⁠. The differential equation is integrated backwards until $x<0$⁠, where $V(x)=0$ and the form of the wave function is known to be

Finally, the transmission probability D(ε) can be described as

where $k02=2mV0/ℏ2$⁠, and t(k) is the ratio of the amplitude of the outgoing wave to that of the incident wave. Note that we use the unnormalized wave functions since the ratios are not dependent on the normalization.

It should be noted that this 1D approximation assumes tunneling directly along the axis connecting the external ion nucleus to the cathode and that the transmission probability for tunneling along other directions that avoid the nucleus will presumably be less than the values calculated here. Notwithstanding, it is useful to calculate a presumably maximum limit on the whole electron transmission probability. Another item of note is that the system considered here only contains only one external ion. If there are other ions nearby, their presence will enhance the field F that is experienced by the electron and are expected to enhance the transmission probability for emission through a single external ion. The single ion, 1D assumption we give here is useful for capturing important physical effects although the absolute magnitude for emission will be influenced by both of these consideration as well as other realistic effects (such as non-ideal cathode surface).

The differential emission current density is the product of the transmission function and the supply function at a given energy level

The supply function N(ε,T) physically describes the number of electrons per second per unit area having normal energy within the small range dε, and we assume the well-known Fermi-Dirac metal formulation such that

where m is the electron mass, k_B is Boltzmann's constant, h is the Plank's constant, and T is the cathode temperature. To calculate the total emission, we can integrate the differential emission current over all energies

In the present work, we consider an argon ion (Ar⁺), with an atomic radius r = 0.71 Å, moving slowly toward a copper cathode with Fermi energy $εF=7.0$ eV and work function $ϕ=4.65$ eV. Because the time scale of the ion motion is slow relative to the timescale of tunneling,³⁶ we treat the ion as stationary at a distance L perpendicular from the cathode. We vary the applied field F from 0.1 V/nm to 5 V/nm, under which the resonant process is important. The cathode temperature is set to be either T = 300 K or 3000 K to analyze relative extremes of cold field emission and thermo-field emission, respectively. The grid size is Δx = 0.1 Å and the electron energy ε (measured with respect to the bottom of the conduction band) range is from 0 to 12 eV with an increment of 0.0012 eV. We conducted verification studies to confirm that the mesh was converged and benchmarked our computational results against well-known analytical models. Under this discretization, the numerical scheme is of adequate accuracy compared to more exact approaches such as transfer matrices.

Figure 2 shows characteristic transmission probabilities for varying applied electric field F (Fig. 2(a)) and ion position L (Fig. 2(b)). In Fig. 2(a), the ion position is fixed at L = 1 nm and the applied field is varied from 1 V/nm to 3 V/nm. As the applied field grows, the transmission probability increases universally at energies $ε<V′0$⁠, where $V′0$ is the reduced Schottky barrier (defined as the maximum of the first potential barrier in Fig. 1). This is not unexpected as higher fields result in a greater tunneling effect. Further, we can clearly observe the resonance peaks along the transmission probability curves. While the overall transmission probability increases with electric field, we observe that the resonance peaks move from higher to lower energies as the applied field increases, which is typical band bending behavior observed in quantum tunneling systems.

In Fig. 2(b), we fix the applied field at F = 1 V/nm and vary the ion position L from 5 nm to 1 nm. Here, we find that as the ion position decreases the overall transmission probability increases, and resonant peaks move from lower energies to higher energies—in contrast to increasing electric fields. This indicates that as an ion approaches the cathode, there will be a greater enhancement in the transmission probability due to the positioning of the resonant level and the lowering of the barrier height due to the ion.

From Fig. 2, it is clear that under sufficiently high fields, as the ion approaches the cathode it can induce resonant emission with a transmission probability that can approach unity. We evaluated the impact of these resonant effects on field and thermo-field emission by calculating the ion-enhanced emission current density using Eq. (10) at cathode temperatures of 300 K (cold field emission) and 3000 K (thermo-field emission).

Figure 3 shows the enhancement ratio (j_ion/j_no-ion), defined as the ratio of the ion-enhanced current density (j_ion) to “straight” emission (j_no–ion), as a function of the ion position L for an applied field of F = 1 V/nm (solid line). Here, we define straight emission as emission with no ion present, which would correspond to the theoretical cases considered in the classic Fowler-Nordheim⁴ and Murphy-Good⁵ analyses. For both cold field and thermo-field emission, the current density of straight emission is obtained using our same numerical calculation of Schrödinger's equation but without any ion, and we benchmarked our analysis against a transfer matrix-based analysis.⁴² For F = 1 V/nm, the straight cold field emission current density is relatively low (⁠$jno−ion=1.025×10−15$ A/m²), while the straight thermo-field emission current density is very high (⁠$jno−ion=1.66×107$ A/m²). In order to isolate the impact of the enhanced tunneling process, we artificially removed the effect of resonance by effectively deleting the peak in the transmission probability (Fig. 2) and using a smoothing technique for the remaining curve, which we show as the dashed line in Fig. 3. The dashed line is the contribution to enhancement due to non-resonant processes, such that the difference between the solid line and dashed line is the enhancement due to resonant processes alone.

As shown in Fig. 3(a), for cold field emission (300 K), the current begins to be enhanced once the ion is less than 10 nm from the cathode surface, and the enhancement ratio can be several orders of magnitude. Most of this enhancement comes from direct, non-resonant tunneling, but once the ion reaches 3 nm from the cathode, resonant effects begin to play a major role. At this distance, the current density can increase from 10⁻⁵ A/m² to hundreds of A/m², as compared to $jno−ion=1.025×10−15$ A/m². This enhancement is primarily due to strong resonance at two ion positions, 0.95 nm and 2.1 nm, indicating that two resonance levels have been sequentially brought into range of the supply function electrons. On the other hand, in Fig. 3(b) for thermo-field emission (3000 K), the enhancement is not nearly as significant, and the enhancement ratio approaches only 300. Further, the resonant contribution is relatively small as compared to the non-resonant effect, which is consistent with Refs. 38 and 39. This is because the thermal contribution to the emission current is extremely strong at this temperature and is always present by comparison, so that the non-resonant case is not exponentially small as for pure field emission. In this case, therefore, the ion enhancement is primarily due to the ion lowering the potential barrier, rather than resonant effects. Only when the highest resonant energy moves very close to the barrier maximum does the contribution from resonant processes become distinguishable. This happens when L is small (<1 nm) such that the heights of the two barriers become almost equal.

It is important to note that the ion is not stationary at position L in plasma systems. Rather, it is drifting toward the cathode with some speed that can be roughly approximated by the applied electric field multiplied by the ion mobility. The total enhancement, therefore, is simply the integral under the curves in Fig. 3, accounting for this approximate ion speed.^26,36 In doing this calculation, we again see that the impact of the resonant states is extremely significant, especially when the emitter is cold, as this integrated enhancement is $7.37×1015$ at 300 K and 2.44 at 3000 K. While non-resonant processes are important, i.e., the lowering and effective thinning of the potential barrier, resonant effects still account for much of this enhancement, especially as the ion comes <3 nm from the cathode.

We can isolate the effect of resonance by quantifying the percentage contribution that the resonant processes make to the enhanced current as shown in Figure 4. As anticipated, for cold field emission as the ion comes close to the cathode (3 nm to 0.5 nm) and under moderately high fields (0.5 to 3 V/nm), the contribution of resonance to ion enhancement is significant and sometimes can be almost 100%, while at large L or extremely high fields, the resonance contribution is negligible. For thermo-field emission there is a resonance contribution up to 50% near L = 1 nm and F = 1.2 V/nm, but overall it is much less significant than cold field emission.

As noted above, for cold field emission under an applied field of 1 V/nm there are two local maxima in the enhancement ratio, which are obvious in both Figs. 3(a) and 4(a). At room temperature, the supply function has a sharp decrease just above the Fermi level as shown by the black curve in Fig. 5, which makes the differential current density (defined in Eq. (8)) peak near the Fermi energy. As the ion moves closer to the cathode, the resonant states shift to high energies (Fig. 2(b)), and both the n = 3 and n = 4 eigenstates approach the Fermi energy. As shown in Figure 5, when the energies of these eigenstates align with the Fermi energy, at ion positions of L = 2.1 and 0.95 nm, respectively, the differential current density is maximized. This indicates that it may be an avenue toward tailored ion-enhanced emission if we deliberately match the energy of the cathode electrons to the ion resonant states, so that when the ion is at a particular distance the effect is maximized.

It is clear that under favorable conditions, an ion will significantly enhance field emission current density through resonant tunneling. Further, this enhancement can be particularly strong when the resonant energies are in the range of the Fermi energy of the cathode (Fig. 5). It can be shown through a perturbation analysis (see supplementary material) that the applied electric field, ion image, and electron image act as perturbations to the wave function of a single ion. If we treat the ion's Coloumb potential as a 1D hydrogen atom, the wavefunctions take the form of Whitaker functions with eigenenergies $εn0$⁠, as defined by Eq. (S6) in the supplementary material. The perturbed eigenenergies are then functionally dependent on the positions of the ion and the applied electric field and take the form

We note that a similar expression was also proposed in Ref. 33. It is important to note that the Whitaker solutions to the single Coulomb well, under the necessary coordinate transformation (see supplementary material), are orthonormal, odd functions, which means that their first and second order perturbations are also odd functions. As such, the second order perturbations are generally zero.

Equation (11) gives us important scaling behavior for the resonant energy levels of the approaching ion as a function of L and F. When L is large, the electrical field plays an important role (as L → ∞, $εn≈εn0−eFL$⁠), and the energy shift is like band bending. When L is small, the image charge induced by the cathode surface is dominant (as L → 0, $εn≈εn0+e216πε0L$⁠), and each energy state will have a sharp rise as the ion moves closer. These are qualitatively consistent with our numerical findings in Fig. 2 as well as others.³³ 

To assess the accuracy of the simple analytical expression in Eq. (11), we compare it with the resonant energies obtained from our numerical calculation of the transmission probability from Fig. 2. However, in our numerical calculation of the transmission probability, we cut off the ion potential well at a well width corresponding to the atomic radius of Ar⁺ (Eq. (3)). When the infinite well is cut off, the energy levels $εn0$ are no longer accurately described by Eq. (S6) in the supplementary material; instead, they will move to higher energy states that are more similar to a 1-D finite box. Additionally, using a cut-off, finite well yields even and odd wave functions that lead to non-zero second order corrections for each perturbation, whereas the second order perturbations are exactly zero in the derivation of Eq. (11). But after closer examination we found that when compared with the first order corrections, those second order terms are small. In order to be consistent in the comparison, we replace $εn0$ in Eq. (11) by numerically solving the eigenenergy $εn0′$ of a cut-off, finite well.

Figure 6 shows a comparison of the simplified expression in Eq. (11) with our numerically calculated resonant energies from Fig. 2. From a practical aspect, the relevant resonant energies only exist above zero energy and below the lower maximum of the double barrier; therefore, the corresponding energy states start from n = 2. In Fig. 6(a) we vary the ion position, and it is clear that the formula accurately captures the resonant energy variation with $L$⁠, including the linear behavior at large L when the electric field is the dominant perturbation and the non-linear behavior at small L when the electron image is the dominant perturbation. In Fig. 6(b), on the other hand, there is greater discrepancy between our model and the numerical results when the applied electric field is varied, especially for very high applied fields (F > 2 V/nm). This difference is due to the cut-off, finite ion potential well in Eq. (3) that we used for the numerical calculations. Specifically, this artificial cut-off creates a second order dependence on F that becomes large as F increases to large values. As the expression in Eq. (11) treated the ion well as a true Coulombic potential, no second order behavior is included and is thus not captured in Fig. 6(b).

The comparison in Fig. 6 shows that the resonant energies can be predicted fairly accurately by a simple perturbation analysis of Schrödinger's equation. While there is some issue with how we handle the ion potential in both the expression and our numerics, we note that currently there is no consensus on how to treat potential wells for real atoms.^43,44 Therefore, within the limits of the model that we use here, where we approximate a = 0, we believe our results capture physical effects in a reasonable and representative manner, especially for modestly high fields. While this model could be improved by using a more realistic value for a or a true 2-D axisymmetric analysis, our 1-D approximation leads to a closed form expression in Eq. (11) that accurately captures the dominant resonant behavior.

Perhaps more importantly, Eq. (11) provides a conceptually simple approach to designing high performance cathodes when ions are present in the system. As noted above, in a plasma environment, the ions are not stationary, but ostensibly the total enhanced emission is simply the integral over the position L as the ion approaches the cathode. In this case, one could simply match the range of ion resonant energies to the Fermi energy of a metallic cathode or to the specific bands of a semiconductor cathode in order to achieve high resonant emission. Another application is adsorbate-enhanced emission, where an atom is adsorbed on the cathode. Since the core position L is fixed in this case, we can match the specific resonant energy to the Fermi energy or conduction bands of the cathode to promote emission. Typically, an electropositive adsorbate is sought that significantly reduces the work function in order to enhance “direct emission.” We are proposing an alternative perspective, where both the adsorbate and cathode are matched to promote “resonant emission,” which we have shown dominates the local emission current density for cold field emission. It is clear that either application that takes advantage of resonant tunneling could be an avenue to high emission current technologies.

In the present work, a 1-D model for cold field and thermo-field emission has been used to analyze the impact of an ion at a given distance from the cathode surface, which occurs in microscale plasma conditions for cold field emission and during arcs and plasma-based thermionic emission systems for thermo-field emission. The numerical calculations show that the emission current is significantly enhanced by an ion near the cathode surface due to both resonant and non-resonant effects. When the applied field is greater than a few V/nm and the ion is a few nm away from the cathode, the resonant process dominates the enhancement and significant local emission current densities can be achieved. It has been shown that as the resonant energies shift along with the applied field and ion position, there is a maximum augmentation in the emission current density when the resonant energy coincides with the Fermi energy of the cathode.

In order to design cathodes to capitalize on this enhancement, the simple expression derived here based on a 1-D perturbation analysis can be used for the ion resonant energies as a function of applied field and ion position. Going forward, this analysis provides a basis for optimizing (or suppressing) ion-enhanced or adsorbate-enhanced field emission current by matching (or mismatching) the resonant energies to the shape of the supply function determined by the cathode material.

See supplementary material for the detailed perturbation analysis to derive the analytical relationship for resonance energies in Eq. (11).

This material is based upon work supported by the National Science Foundation under Award No. FA9550-11-1-0020.