Theory of laser-induced photoemission from a metal surface with nanoscale dielectric coating

Photoelectron emission is important to applications such as dielectric laser accelerators,^1–3 free electron lasers (FELs),^4,5 ultrashort x-ray sources,⁶ time-resolved electron microscopes,^7–10 ultrafast electron diffraction,¹¹ carrier-envelope detection,^12–16 and novel nano-electronics.^17–23 While enabling the exploration of matter at a temporal resolution of femtosecond and a spatial resolution of nanometer,^24–27 those applications rely on high-performance photocathodes or photoemitters, which are required to be of high efficiency and high stability.^28–30 Coatings, such as graphene, nano-diamond, silicon dioxides, and zinc dioxides, are proposed to be fabricated atop cathodes to protect them from degradation by ion and electron bombardment, or oxidization under poor vacuum conditions.^31–35 Coatings not only elongate the operational lifetime and the current stability of photocathodes but also enhance the quantum efficiency of photoemission by the lowering of the effective work function or the enhancement of the laser field.^{31,32,36–38} Analogous heterostructure photocathodes are prospective to optimize the quantum yield and emittance simultaneously for electron sources in x-ray free-electron lasers.^29,39

The development of theory for photoemission from coated cathodes facilitates the optimization of the design and performance of photocathodes. Commonly used Fowler–Nordheim-type equations, which assume photoelectron emission occurring in positive half cycles of the intense laser, are applicable only in the optical field tunneling regime but not in the multiphoton absorption regime.⁴⁰ Furthermore, it has been shown that negative half cycles also play a role in the photoemission process.¹⁶ Therefore, an exact model for photoemission from cathode’s with ultrathin coatings is desirable to uncover the interplay of various parameters on photoemission and provide insights into the development of photocathodes.

In this study, we construct an exact analytical quantum model for laser-driven photoemission from cathodes coated with a nanoscale-thick dielectric by solving the time-dependent Schrödinger equation (TDSE). The model is applicable to photoemission for arbitrary combinations of metal properties (i.e., work function and Fermi level), dielectric properties (i.e., thickness, relative permittivity, and electron affinity), laser field (i.e., wavelength, and field strength or intensity), and dc field. Based on the analytical solution, we investigate the effects of dielectric properties on photoemission. This analytical model is compared with the effective single-triangular barrier model³⁶ and modified Fowler–Nordheim equation^40,41 for photoemission from a dielectric-coated flat metal surface and a dielectric-coated pyramid-shaped nanoemitter.

In the one-dimensional (1D) model (see Fig. 1), electrons with initial longitudinal energy $ε$ are emitted from the flat metal surface coated with a nanoscale-thick dielectric, driven by a laser field and a dc bias. The laser field and dc bias field are perpendicular to the metal surface. For simplicity, the scattering effects of photoexcited electrons with other electrons and phonons in the metal and dielectric, the charge trapping effect in the dielectric, are ignored.^16,42,43 Therefore, the time-varying potential barrier in those three regions, i.e., metal, dielectric, and vacuum, reads

where $V0=W+EF$⁠, with $EF$ being the Fermi energy of the metal and $W=W0−ΔW$ being the effective work function including the potential barrier lowering by the Schottky effect due to the dc electric field $F0$⁠, $ΔW=2e3F0diel/16πε0εdiel$ when the maximum of the potential barrier including image charge potential is in the coating or $ΔW=2e3F0/16πε0$ when the potential maximum is in the vacuum; $χ$ is the electron affinity of the dielectric; e is the positive elementary charge; $F0$ and $F1$ are the dc electric field and laser electric field in the vacuum, respectively; $F0diel$ and $F1diel$ are the dc and laser electric field inside the dielectric, respectively; $ω$ is the angular frequency of the laser field; and d is the thickness of the dielectric. For a perfectly flat surface, $F0diel=F0/εdiel$ and $F1diel=F1/εdiel$ inside the dielectric with $εdiel$ being the relative permittivity of the dielectric.

The electron wave functions $ψ(x,t)$ in the metal, dielectric, and vacuum are obtained by solving the time-dependent Schrödinger equation (see  Appendix A for solutions),

where $ℏ$ is the reduced Planck's constant; m is the electron effective mass, set to the electron rest mass in all three regions for simplicity; and $ϕ(x,t)$ is the potential given in Eq. (1).

The electron transmission probability, $w(ε,x,t)=Jv(ε,x,t)/Ji(ε)$⁠, is defined as the ratio of transmitted electron probability current density in the vacuum $Jv$ to the incident electron probability current density in the metal $Ji$⁠, both of which are calculated from the electron probability current density $J=iℏ/2m(ψ∇ψ∗−ψ∗∇ψ)$⁠. It is easy to show the time-averaged transmission probability as

or

with $k0=2mεℏ2$⁠; $E3n=ε+nℏω−Up3−V30$ is the drift kinetic energy in the vacuum, with the ponderomotive energy $Up3=e2F124mω2$ and $V30=W+EF+ed(F0−F0diel)$ for $F0≠0$ or $V30=W0+EF$ for $F0=0$⁠; and $κ3=(2meF0ℏ2)1/3$⁠; $T3n$ is the transmission coefficient of the wave in the vacuum (see  Appendix A for more details).

The electron emission current density is obtained from

where $D(ε)$ is given in Eq. (3) and $N(ε)=mkBT2π2ℏ3ln[1+exp(EF−εkBT)]$ is the flux of electrons impinging normal to the metal–dielectric interface, which is calculated from the three-dimensional (3D) free electron theory of metal,^43–45 with $kB$ being the Boltzmann's constant and T being the temperature.

Based on the theory developed in Sec. II, we provide an analysis of the photoemission from metallic cathodes coated with dielectric. The metal is assumed to be gold, with nominal work function $W0=5.1eV$ and Fermi energy $EF=5.53eV$⁠. The laser has a wavelength of 800 nm, corresponding to the photon energy of 1.55 eV. These are the default properties of the metal and laser respectively, unless prescribed otherwise.

Figure 2 shows the effects of coating dielectric properties (i.e., thickness d, relative permittivity $εdiel$⁠, and electron affinity $χ$⁠) on the electron transmission probability from a flat metal surface. Since most of the photoemission occurs with electron initial energies near the Fermi level at ambient temperature, the free electrons inside the metal are assumed to have an initial energy $ε=EF$⁠. The electron transmission probability through the nth channel $wn(ε=EF)$ calculated from Eq. (3), for the dielectrics of a different thickness d, relative permittivity $εdiel$⁠, and electron affinity $χ$⁠, is plotted in Figs. 2(a)–2(c), respectively. The laser has a field strength of F₁ = 5 V/nm. The dc electric field is $F0=0$⁠. It is found that the dominant photoemission is through four-photon absorption (n = 4) under the fields provided, regardless of the dielectric properties. By checking the Keldysh parameter $γ$⁠, it is found that $γ≥3.59$ for all the cases in Figs. 2(a)–2(c) (see Fig. S8 in the supplementary material), indicating multiphoton absorption process. The dominant channel, n = 4, is consistent with the ratio of the work function to the photon energy $⟨W/ℏω⟩$⁠, where $⟨⟩$ represents the next nearest integer to the value inside the bracket. $wn(ε=EF)$ increases with decreasing $εdiel$ or increasing $χ$⁠, due to the narrowed or reduced barrier in the dielectric, which is similar to dc field emission from thin dielectric-coated surfaces.³⁸ However, $wn(ε=EF)$ has no clear monotonic dependence on d, as shown in Fig. 2(a).

The electron transmission probability $D(ε=EF)$⁠, which is the sum of $wn$ over all channels, is presented as a function of dielectric thickness, relative permittivity, and electron affinity in Figs. 2(d)–2(f), respectively, under various laser field strengths $F1$ [see Fig. S1 in the supplementary material $D(ε=EF)$ vs $F1$]. It is obvious that the transmission probability increases when the laser field strength increases. In Fig. 2(d), the transmission probability shows approximately periodic peaks with respect to the dielectric thickness. These peaks are due to resonance in the quantum interference⁴⁶ between electron waves transmitted to and reflected from the dielectric–vacuum interface, which forms constructive interference when the dielectric is of a particular thickness (see Figs. S2 and S4 in the supplementary material). As the laser field strength increases, peaks on the curves shift toward larger thicknesses, as indicated by the gray dotted line in Fig. 2(d). The physics behind this shift lies in the fact that the wavelength of the electron waves inside the dielectric increases with the laser field (see Fig. S3 in the supplementary material), which can also be indicated by the wavenumber $2mE2n/ℏ2$⁠, with $E2n=ε+nℏω−Up2−(EF+W−χ)$ and $Up2=e2(F1diel)2/4mω2$⁠.

Figure 2(e) shows that the transmission probability $D(ε=EF)$ decreases with the relative permittivity of the dielectric $εdiel$ for a given laser field, due to the smaller field $F1diel=F1/εdiel$ inside the dielectric. When $εdiel$ is large, e.g., $εdiel>2$⁠, $F1diel$ inside the dielectric would be relatively small for a given $F1$⁠; thus, the incident electron would see a double-triangular potential barrier before emission (see Fig. 1), yielding a rapidly decreasing slope in $D(ε=EF)$ vs $εdiel$⁠. When $εdiel$ is small (<2), $F1diel$ would be larger, such that the incident electron would see only a single-triangular barrier inside the dielectric (since the barrier at the dielectric–vacuum interface would be below the electron initial energy level), yielding a smaller slope for $F1=6,8,and10V/nm$⁠. The trends of these curves are found to follow closely those of the “area under the curve” in the potential barrier (i.e., WKBJ approximation, see Fig. S5 in the supplementary material).

Figure 2(f) shows the effect of the dielectric electron affinity on photoelectron transmission probability, with $εdiel=2$⁠, and $d=1nm$⁠. $D(ε=EF)$ increases with $χ$⁠, due to the lowering of the potential barrier. There appears distinct resonance peaks on each curve for a given $F1$ due to quantum interference. The peaks shift toward a larger $χ$ as the laser field strength increases, which is indicated by the gray dotted lines in Fig. 2(f). Note that the model recovers photoemission from bare metal surfaces^42,43 when $d=0$ or when $χ=0$ and $εdiel=1$ (i.e., vacuum).

Figure 3 shows the effects of dielectric properties on photoemission current density. The photoemission current density J, calculated from Eq. (4), is plotted as a function of dielectric thickness d in Fig. 3(a), for laser field strengths from 2 to 10 V/nm. J decreases with d for $0<d<1nm$⁠. The current density is almost constant for $d>1nm$⁠, while there appears some slight resonance peaks, such as the orange dots on the curve for $F1=2V/nm$⁠. These features are similar to those of dc field emission from dielectric-coated surfaces (cf. Fig. 5 of Ref. 38). The resonances are not as strong as those in Fig. 2(d) for $D(ε=EF)$ vs d. This is because photoemission current includes the electron emission from all incident energy levels in the metal, which, in combination, smoothens the curve and reduces the strong emission peaks from a single initial energy level. Note that in our model, electron–phonon scattering and electron–electron scattering effects in dielectrics are not considered. According to the experiment in Ref. 47, a mean free path of a few nanometers has been observed for photoexcited electrons in dielectrics.^47,48 Therefore, for a dielectric thickness smaller than the mean free path, the scattering effect inside the coating may not be important. Figure 3(b) shows J as a function of relative permittivity $εdiel$⁠. The curves exhibit two distinct slopes in the semilog scale plot, which can be inferred from the transmission probability in Fig. 2(e). Figure 3(c) presents the effect of dielectric electron affinity on photoemission current density. Generally, emission current density increases with electron affinity for a given field strength, due to the lowering of the surface potential barrier. Mild resonant peaks in J are also observed as $χ$ changes.

It is interesting to find that a flat cathode surface with dielectric coating can emit a larger current density than an uncoated case, when the dielectric has a sufficiently small relative permittivity $εdiel$ or a large electron affinity $χ$⁠, which would result in a narrowed or lowered potential barrier. Figures 4(a) and 4(b) provide examples of such cases, as shown in the region above the gray dashed lines. A similar electron emission enhancement phenomenon is also demonstrated for field emission from dielectric-coated surfaces.³⁸ It should be pointed out that the scattering of electrons with phonons, impurities, and even with other electrons inside the dielectric is not considered in the model. Photoelectron emission from ultrathin oxide covered devices shows an exponential attenuation behavior for the relatively thick oxide layer (2.5 ∼ 15.3 nm), with the dominant mean-free-path of the photoexcited electrons inside SiO₂ ∼ 1.2 nm.⁴⁷ Therefore, for a dielectric thicker than the mean-free-path, electron scattering effects cannot be ignored.

Figure 5 shows the effects of the dc electric field on photoemission from dielectric-coated metal surfaces. The transmission probability through different n channels from initial energy $ε=EF$ under a different $F0$ is shown in Fig. 5(a). As $F0$ increases, the dominant emission channel shifts to a smaller n. The narrowed and lowered (due to the Schottky effect) surface potential barrier by the static field enables more photoemission mechanisms, such as photo-assisted field emission $(1<n<4)$⁠, multiphoton emission $(n<0)$⁠, and direct tunneling $(n=0)$⁠. The electron transmission probability is greatly enhanced by the dc field.

The electron transmission probability $D(ε=EF)$ is plotted as a function of the dc field $F0$ in Fig. 5(b) for different laser field strengths. The combined dc field and laser field results in an emission probability significantly larger than that from either field alone. When $F1=0$⁠, the electron transmission probability recovers the dc field emission from dielectric-coated surfaces.³⁸ 

Figure 5(c) shows the emission current density as a function of the dc field under various laser field strengths. The shape of the emission current density is similar to the transmission probability from $ε=EF$ in Fig. 5(b). For $F0≲12V/nm$⁠, the slope of the curves varies with the laser field strength $F1$⁠, which indicates that n-photon-assisted field tunneling dominates in this range, where n becomes smaller as $F0$ increases. When $F0≳12V/nm$⁠, the slopes of the curves for all four cases are almost the same, because the dominant emission becomes dc field tunneling.

In this section, we compare the results of our exact analytical model with the effective single-barrier quantum model (ESQM)³⁶ and modified Fowler–Nordheim (FN) equation^40,41 for photoemission from dielectric-coated metal surfaces. The description of the ESQM is found in Ref. 36. A short account of the modified FN equation is provided in  Appendix B.

Figure 6 shows a comparison of the three models for photoemission from 1D flat cathodes. The electron emission probability $D(ε=EF)$ as a function of the laser field strength $F1$ is plotted in Fig. 6(a). It can be seen that, for flat surfaces with coating, the ESQM overestimates the photoemission probability, but the modified FN equation underestimates the photoemission probability, as compared with the exact double-triangular barrier model. The ESQM models the double barrier using an effective single-triangular barrier based on the WKBJ approximation,³⁶ where the effective barrier height is chosen at $Weff=W0−χ$⁠. When $F1<6V/nm$⁠, the dominant emission process is multi-photon absorption, since the curve scales as $D∝F12n$ from both the ESQM and the exact double-barrier quantum model. Here, $n=3$ for the ESQM and $n=4$ for the exact double-barrier model, which is determined by the ratio of the barrier height $Weff$ to the photon energy (⁠$ℏω=1.55eV$ for a 800 nm laser) in each model, with $Weff=W0−χ=4.1eV$ for the ESQM and $Weff=W0=5.1eV$ for the exact double-barrier model. The smaller effective potential barrier used in the ESQM also explains the greater transmission probability by the ESQM. The regime of multiphoton absorption is further confirmed by the Keldysh parameter, $γ=4.75$ (Exact) and $γ=5.68$ (ESQM) for $F1=6V/nm$ (see Fig. S9 in the supplementary material). The abrupt slope change at $F1≈12V/nm$ from the exact model is due to the channel closing effect.^42,49 The underestimation of $D(ε=EF)$ from the FN equation is expected, because the FN equation takes account of optical field tunneling only, which is significantly smaller than other n-photon processes in the multiphoton absorption regime. For $13<F1<15V/nm$⁠, the curve for the FN equation overlaps with that for the exact model, as the emission enters the optical field tunneling regime.

Figure 6(b) shows emission current density as a function of the laser field strength $F1$⁠, showing similar trends as the emission probability in Fig. 6(a). The curves from the three models converge as the laser field increases to 15 V/nm, when the strong optical field tunneling becomes dominant.

A pyramid-shaped plasmonic resonant photoemitter coated with an atomically thick dielectric has been numerically demonstrated to provide an emission current of orders of magnitude larger than a bare photoemitter.³⁶ The substantially improved photoemission is ascribed to the increased field enhancement by confining the plasmonic field inside the dielectric waveguide along the metal surface. Although the ESQM was used to estimate the photoemission current and has been verified by comparing it with the modified FN equation, it is desirable to calculate photoemission from the exact analytical model. Figure 7 presents emission current density J, calculated from the exact double-barrier quantum model, ESQM, and a modified FN equation for photoemission from the pyramid-shaped gold emitter with SiO₂ coating, as a function of the externally applied laser field strength $Fext$⁠. Full wave optical simulation shows an approximately linearly decaying laser field inside the dielectric at a resonance wavelength of 608 nm, with the maximum field enhancement factor $β$ at the metal–dielectric interface (⁠$x=0$ in Fig. 1). To accommodate the double-triangular barrier quantum model and the modified FN equation, the laser field inside the coating is assumed uniform, with the field strength being the one at the metal–dielectric interface, and the laser field in the vacuum is assumed to be that at the dielectric–vacuum interface, i.e., $F1diel=β(x=0)Fext$⁠, and $F1=β(x=d)Fext$⁠. Those three models manifest a quantitatively good agreement for $Fext>0.05V/nm$⁠, where the emission enters the strong field tunneling regime.³⁶ This transition is also indicated from the value of the Keldysh parameter approaching unity, $γ=1.89$ (Exact) and $γ=2.46$ (ESQM) for $Fext=0.05V/nm$ (see Fig. S10 in the supplementary material). For $0.01<Fext<0.05V/nm$⁠, emission current densities calculated from both the ESQM and the FN equation are smaller than that from the exact double-barrier model.

One may expect that the assumption of the uniform laser field inside the coating being the maximum field at the metal–dielectric interface would result in an overestimation of the emission current in the small field regime. However, calculation from the exact double-barrier model with $F1diel$ determined by the slope of the line connecting the peaks of the potential barrier at the two interfaces (see Fig. S6 in the supplementary material) shows that emission current densities from the ESQM and the FN equation are still smaller for $Fext<0.03V/nm$⁠, despite the fact that such an approximation of $F1diel$ would overestimate the potential barrier, thus underestimating photoemission from the exact model. Therefore, applying the ESQM to a dielectric-coated plasmonic resonant Au photoemitter³⁶ may underestimate the photoemission current for $Fext<0.03V/nm$⁠, and the exact double-triangular barrier model may be used to give a more accurate estimation of photoemission from coated photoemitters.

In summary, we have constructed an analytical model for photoemission from metal surfaces coated with an ultrathin dielectric, by exactly solving the one-dimensional (1D) time-dependent Schrödinger equation subject to a double-triangular barrier. The model manifests various electron emission mechanisms, i.e., multiphoton emission, static field tunneling, photon-assisted field emission, optical field tunneling, and thermionic emission, depending on the applied fields (laser field and dc field) and temperature. The effects of dielectric properties on photoemission are investigated. It is found that a flat metal surface coated with a dielectric of smaller relative permittivity and larger electron affinity can photoemit a current density larger than the uncoated metal due to the lowered surface barrier. For dielectric-coated nanoemitters, photoemission can be greatly enhanced due to the nonlinear field enhancements near the coating.

Our model is compared with the effective single-barrier quantum model and modified Fowler–Nordheim equation, for both flat cathodes and three-dimensional (3D) nanoemitters. It is found that both the effective single-barrier model and the modified Fowler–Nordheim equation may underestimate photoemission from the dielectric-coated nanoemitters in the multiphoton absorption regime, and that our exact model may give a more accurate estimation. In the strong-field optical tunneling regime, the three models show quantitatively good agreement. The results further confirm that plasmonic resonant tip photoemitters with thin dielectric coating may be promising for higher yield electron sources.³⁶ 

While a wide range of dc and optical fields are used in this theoretical study of photoemission from 1D flat cathodes, the operation of real-world cathodes poses limitations on the dc field and laser field (intensity, pulse duration, and repetition rate). The breakdown strength for dielectrics shows strong dependence on their relative permittivity and thickness. A theoretical ultimate breakdown strength of 1.5 V/nm for SiO₂ is reported⁵⁰ and an optical breakdown laser pulse with a duration of 25 fs and field strength of ∼20 V/nm is experimentally observed.⁵¹ Laser pulses can also induce heating and even ablation of the metal, which depend strongly on the laser wavelength, pulse duration, and intensity. A laser pulse of 450 fs duration at 248 nm, with a peak intensity of up to 13.5 × 10⁹ W/cm², is experimentally used for photoemission from a polycrystalline copper flat surface, and a nonlinear increase of photocurrent due to the laser heating effects is observed.⁵² As the laser duration decreases, the applied intensity can be increased. In Ref. 14, a laser of 14 fs duration, 1.65 μm wavelength, and up to 7.4 × 10¹³ W/cm² intensity (after geometrical field enhancement) is used to study photoemission from single-crystalline gold tips. For realistic nanostructured cathodes or sharp nanoscale emitters, because of the strong field enhancement factor^53,54 and possible further enhancement due to plasmonic resonance,³⁶ the externally applied laser intensity and dc bias field are expected to be orders of magnitude smaller than those shown in this work for 1D flat surfaces that are considered for simplicity. These important factors should be taken into consideration for the design of practical photocathodes.

In this model, the dielectric is assumed ideal, with only electron affinity and relative permittivity taken into consideration. The metal, the dielectric, and the interfaces between them are all assumed to be perfectly aligned and with no defects or impurities. However, localized traps may exist in the bulk dielectric and at the metal–dielectric interface, especially when the dielectric layer becomes relatively thick,⁵⁵ which may result in a trapping of photoexcited electrons from the metal. Trapped charges can have strong effects on photoemission. Experiments show that the reduction of trapped charges by annealing lead to a red shift of the maximum peak of photoemission energy spectrum and the formation of a higher energy plateau.⁴⁰ Additionally, the mean free path or the scattering length of electrons within the dielectric is on the order of a few nanometers.^40,41,47,48 As the coating thickness increases, the scatterings (i.e., electron–phonon, electron–electron, electron–impurity) inside the dielectric become important. Future work may consider charge trapping effects as well as scattering effects inside the dielectric in the photoemission model. The effects of space charge^19,21 and pulsed excitation¹⁶ may also be studied in the future.

See the supplementary material for the effects of dielectric properties on photoemission mechanisms, quantum interference inside the dielectric, enclosed area by the potential barrier profile, comparison with an effective single-barrier quantum model and modified double-barrier Fowler–Nordheim equation, and the calculation of the Keldysh parameter.

This work was supported by the Air Force Office of Scientific Research (AFOSR) YIP (Grant No. FA9550-18-1-0061), the Office of Naval Research (ONR) YIP (Grant No. N00014-20-1-2681), and the Air Force Office of Scientific Research (AFOSR) (Grant No. FA9550-20-1-0409).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The exact solution to Eq. (2) in the metal $(x<0)$ is

which represents the superposition of the incident plane wave with an electron incident longitudinal energy $ε$⁠, and a set of reflected waves with an energy of $ε+nℏω$ after photon absorption $(n>0)$ or emission $(n<0)$ processes. $R1n$ is the reflection coefficient.

In the dielectric $(0≤x<d)$⁠, the solution to Eq. (2) is obtained by the following Truscott transformation and separation of variables:^{16,42,43,56–59}

for $F0=0$ or

for $F0≠0$⁠, where $T2n$ and $R2n$ are the transmission coefficient and reflection coefficient of electron waves through the nth channel in the dielectric, respectively; $E2n=ε+nℏω−Up2−V20$ is the drift kinetic energy in the dielectric, with the ponderomotive energy $Up2=e2(F1diel)24mω2$ and $V20=EF+W−χ$⁠; $Θ(x,t)=exp[−iε+nℏωℏt+ieF1dielsinωtℏωx+ie2(F1diel)2sin2ωt8ℏmω3]$⁠; $Γ(x,t)=exp[−ie2F0dielF1dielsinωtℏmω3]Θ(x,t)$⁠; $ζn=(x+eF1dielcosωtmω2+E2neF0diel)(2meF0dielℏ2)1/3$⁠. $ψII$ represents the superposition of the forward traveling waves and the reflected waves in the dielectric.

In the vacuum $(x≥d)$⁠, the exact solution to Eq. (2) is

for $F0=0$ or

for $F0≠0$⁠, where $T3n$ is the transmission coefficient in the vacuum; $E3n=ε+nℏω−Up3−V30$ is the drift kinetic energy in the vacuum, with the ponderomotive energy $Up3=e2F124mω2$ and $V30=W+EF+ed(F0−F0diel)$ for $F0≠0$ or $V30=W+EF$ for $F0=0$⁠; $Ξ(x,t)=exp[−ied(F1−F1diel)sinωtℏω−iε+nℏωℏt+ieF1sinωtℏωx+ie2F12sin2ωt8ℏmω3]$⁠; $ηn=(x+eF1cosωtmω2+E3neF0)(2meF0ℏ2)1/3$⁠. $ψIII$ denotes the outgoing waves traveling to the vacuum.

Continuity of the wave function and its derivative at both the metal–dielectric interface $(x=0)$ and the dielectric–vacuum interface $(x=d)$ and Fourier transform yields the solutions for $R1n$⁠, $T2n$⁠, $R2n$⁠, and $T3n$⁠,

where $δ$ is the Dirac delta function; $P1nl=12π∫02πP1n(ωt)e−ilωtd(ωt)$⁠, $Q1nl=12π∫02πQ1n(ωt)e−ilωtd(ωt)$⁠, $P2nl=12π∫02πP2n(ωt)e−ilωtd(ωt)$⁠, $Q2nl=12π∫02πQ2n(ωt)e−ilωtd(ωt)$⁠, $P3nl=12π∫02πP3n(ωt)e−ilωtd(ωt)$⁠, $Q3nl=12π∫02πQ3n(ωt)e−ilωtd(ωt)$⁠, $Z3nl=12π∫02πZ3n(ωt)e−ilωtd(ωt)$⁠, $P4nl=12π∫02πP4n(ωt)e−ilωtd(ωt)$⁠, $Q4nl=12π∫02πQ4n(ωt)e−ilωtd(ωt)$⁠, and $Z4nl=12π∫02πZ4n(ωt)e−ilωtd(ωt)$ are the Fourier transform coefficients. For $F0=0$⁠, we have

with $ρ(ωt)=exp[ie2(F1diel)2sin2ωt8ℏmω3]$ and $ϱ(ωt)=exp[ie2F12sin2ωt8ℏmω3]$⁠.

For $F0≠0$⁠, we have

with $Υ(ωt)=exp[−ie2F0dielF1dielsinωtℏmω3+ie2(F1diel)2sin2ωt8ℏmω3]$⁠, $s(ζn)=Ai(−ζn)−iBi(−ζn)$⁠, $r(ζn)=Ai(−ζn)+iBi(−ζn)$⁠, $t(ζn)=iAi′(−ζn)+Bi′(−ζn)$⁠, $u(ζn)=iAi′(−ζn)−Bi′(−ζn)$⁠, $κ2=(2meF0dielℏ2)1/3$⁠, $Λ(ωt)=exp[−ie2F0F1sinωtℏmω3+ie2F12sin2ωt8ℏmω3]$⁠, $v(ηn)=Ai(−ηn)−iBi(−ηn)$⁠, $w(ηn)=iAi′(−ηn)+Bi′(−ηn)$⁠, and $κ3=(2meF0ℏ2)1/3$⁠.

Following the procedure in Ref. 41, a modified Fowler–Nordheim equation is formulated. For the scenario of an ultrathin dielectric on a metal surface, a double step barrier is formed. If the zero of energy is taken at the Fermi level, the lowermost barrier profile bent by the laser field reads