A Thermodynamic Comparison of Nanotip and Nanoblade Geometries for Ultrafast Laser Field Emission via the Finite Element Method

Abstract

Strong laser field emission from metals is a growing area of study, owing to its applications in high-brightness cathodes and potentially as a high harmonic generation source. Nanopatterned plasmonic cathodes localize and enhance incident laser fields, reducing the spot size and increasing the current density. Experiments have demonstrated that the nanoblade structure outperforms nanotips in the peak fields achieved before damage is inflicted. With more intense surface fields come brighter emissions, and thus investigating the thermomechanical properties of these structures is crucial in their characterization. We study, using the finite element method, the electron and lattice temperatures for varying geometries, as well as the opening angles, peak surface fields, and apex radii of curvature. While we underestimate the energy deposited into the lattice here, a comparison of the geometries is still helpful for understanding why one structure performs better than the other. We find that the opening angle—not the structure dimensionality—is what primarily determines the thermal performance of these structures.

1. Introduction

Nanostructured cathodes have been utilized in electron microscopes [1], ultra-fast (low-energy) electron diffraction [2], and electron guns [3]. They are hailed for their high brightness and, in the case of metals, ultra-fast response times. Using a nanostructure such as a nanotip or a nanoblade enhances and confines the incident laser fields to the apex. This increases the current density, thereby improving the brightness. However, this process is limited by the structure’s ability to withstand intense illumination. Nanotips tend to break down in enhanced ultra-fast laser fields exceeding 10 V/nm [4,5], whereas nanoblades appear to attain surface fields of at least 40 V/nm [6] and likely up to 80 V/nm [7] according to the peak electron energies observed. A higher surface field leads to a larger current density and therefore a higher beam brightness [8]. Finding ways of improving thermomechanical survivability—for any structure—is therefore critical for improving the brightness in ultrafast cathodes.

Under laser illumination, energy is absorbed predominantly by the electron population. Electrons near the apex disperse energy on a timescale from a few to 10 s of femtoseconds, which is effectively instant. They then disperse energy throughout the bulk on a scale from 10 s to 100 s of femtoseconds [9]. However, a heated electron gas does not directly damage the material. Such damage would be caused by movements of the ions in the lattice, and thus the lattice temperature (the temperature of the phonon gas) is a better proxy for indicating damage. As electron-phonon scattering occurs on picosecond timescales, ultrafast illumination results in a thermal non-equilibrium, with the electrons attaining a higher temperature than the lattice. This known phenomenon has therefore been treated using two-temperature models [10]. Thus, if the laser pulse is short on the electron-phonon timescale, then the electrons disperse a substantial portion of the deposited energy before the lattice can absorb it.

In our previous analysis [9], we studied these timescales with simple geometric models to show that the dimensionality of the structure is the determining factor for how well it disperses energy; that is, the nanoblade effectively has two dimensions it may disperse energy through, while the tip only has one (for exceptionally small opening angles), and therefore the blade would more readily disperse deposited energy. As we find in the present analysis, this was actually a partially incorrect sentiment. To obtain a more accurate conclusion, we aim to accurately model the geometries using the finite element method (FEM). This removes the “fictitious apex” problem which had plagued the simple geometric analysis. Inclusion of both the electron and lattice temperatures is then straightforward as well, leading to a more complete model.

2. Materials and Methods

In this study, we solve the two-temperature heat equation for solid gold nanotips and nanoblades. We singled out gold because previous nanoblade studies have considered gold coated blades [7,8,9]. The heat equations follow:

$$\begin{array}{cc}\hfill \frac{C_e\left(T_e\right)}{2T_e}{\partial }_t{T}_e^2& =\frac{K_e\left(T_e\right)}{2T_e}{\nabla }^2{T}_e^2-g(T_e-T_i)+S(r,t),\hfill \\ \hfill C_l\left(T_l\right){\partial }_t{T}_l& =\nabla ·\left(K_l\left(T_l\right)\nabla T_l\right)+g(T_e-T_i).\hfill \end{array}$$

(1)

We use the free electron gas (FEG) model for the electronic thermal capacity and conductivity, $C_e=\gamma T_e$ and $K_e=K_{e0}{T}_e/T_{l0}$ (with $\gamma$ = 71.5 Jm⁻³K⁻² and $K_{e0}$ = 310 Wm⁻¹K⁻¹) [11,12]. Here, $T_e$ and $T_l$ denote the electron and lattice temperatures, respectively, $S(r,t)$ represents the surface heating function, g is the electron-phonon coupling constant, r is the spatial coordinate, t is the time, and ${\partial }_t\equiv \partial /\partial t.$ It is then natural to translate this to the provided temperature-squared electron heat equation.

In the current analysis, lattice heating is entirely attributed to heat transfer from the thermalized electron distribution via electron-phonon scattering. The electron-phonon coupling constant g varies from material to material and for differing temperatures [13]. According to Figure 4d in Ref. [13], and by extrapolation, for electron temperatures up to 30,000 K, the coupling constant in gold is less than g = 3 × 10¹⁷ Wm⁻³K⁻¹, and therefore we use this approximate constant value. On the whole, this leads to an overestimation of the lattice heating for cool electron temperatures but is somewhat accurate at high temperatures. We additionally run calculations for g = 0.2 × 10¹⁷ Wm⁻³K⁻¹ (a low temperature limit; see Appendix C). This alteration does not affect our conclusions. Similarly, the electron thermal capacity increases beyond being linear (and one expects a similar increase for the electron thermal conductivity), but to retain the simpler temperature-squared heat equation, we continue with the simpler linear approximation.

The lattice thermal conductivity and capacity follow from the Debye model [14,15]. For computational purposes, we integrate by parts to find the final forms which are efficiently implemented in the code:

$$\begin{array}{cc}\hfill C_l\left(T_l\right)& =9k_{b}n\left[-\frac{x}{e^x-1}+\frac{4}{3}{D}_3\left(x\right)\right]\hfill \\ \hfill K_l\left(T_l\right)& =k_b{\left(\frac{\pi n^2}{6}\right)}^{1/3}\sum _i{v}_i\left[-\frac{x_i}{e^{x_i}-1}+\frac{3}{2}{D}_2\left(x_i\right)\right]\hfill \end{array}$$

(2)

with $i\in \{L,T\}$ the longitudinal and transverse sound modes respectively (the transverse mode is doubly counted), the relative Debye temperature, $x_i={\mathsf{\Theta }}_{Di}/T_l$, the number density of atoms, n = 5.91 × 10²⁸ m⁻³, the transverse and longitudinal sound velocities, $v_T=1200$ m/s and $v_L=3240$ m/s [16], respectively, and the Debye functions $D_N\left(x\right)=\frac{n}{x^N}{\int }_0^{x}dt\frac{t^N}{e^t-1}$. In the conductivity formula, the modal Debye temperatures are calculated via the sound velocity [15] and are ${\mathsf{\Theta }}_{DT}$ = 139.16 K and ${\mathsf{\Theta }}_{DL}$ = 375.74 K. For the heat capacity, we use the Debye temperature for gold from the literature: ${\mathsf{\Theta }}_D$ = 170 K.

2.1. Linear Dielectric Heating

For the nanoblade structure, part of the field enhancement may be attributed to traveling surface plasmon polaritons (SPPs) which trail the laser as it traverses the length of the blade. This invites a simpler linear analysis for low fields. For a given frequency, each material has a complex dielectric constant, with the imaginary component indicating loss of energy into the material. The SPP wavevector, k, and longitudinal decay constant, $\kappa$, may be found from the dielectric constant:

$$k=\frac{\omega }{c}{\left(\frac{{\epsilon }_r}{{\epsilon }_r+1}\right)}^{1/2},\kappa =\frac{\omega }{c}{\left(\frac{{\epsilon }_r}{{\epsilon }_r+1}\right)}^{3/2}\frac{{ϵ}_i}{2{ϵ}_r^2},$$

(3)

where $\omega$ denotes the wave frequency, c denotes the speed of light, and ${\epsilon }_r=-25$ and ${\epsilon }_i=1.25$ are the approximate real and imaginary relative permittivities at the laser wavelength 800 nm, respectively [17]. The electron temperatures in our analysis exceed 30,000 K, necessitating the inclusion of a variable dielectric constant. Such considerations increase the heating and peak lattice temperatures [18]. The nonlinear $U_p\approx$ 50 eV-scale dynamics of the constituent electrons within the SPPs at 80 V/nm further complicate this consideration (with $U_p$ being the ponderomotive energy of the field). Here, we continue with constant dielectric constants as we only compare the performance of these structures. Some of the nonlinear contributions are accounted for by vacuum heating in Section 2.2 below.

The SPP group velocity, $v_g=\partial \omega /\partial k=c{\left(1+1/{ϵ}_r\right)}^{1/2}$, associated with this dispersion relation is the speed at which bound electromagnetic energy traverses the surface. Therefore, the dielectric loss power density reads

$$S_d(r,t)=\frac{du}{dt}=\kappa v_{g}u,$$

(4)

where u is the electromagnetic energy surface density. For a surface plasmon polariton, this is given by

$$u=\frac{{\epsilon }_0}{2}\frac{{\kappa }_{z1}^2}{k^2}{E}^2\left[\frac{1}{2{\kappa }_{z1}}\left(\left(1+{\left(\frac{k}{{\kappa }_{z1}}\right)}^2\right)+{\left(\frac{\omega }{{\kappa }_{z2}c}\right)}^2\right)+\frac{1}{2{\kappa }_{z2}}\left({\epsilon }_r\left(1+{\left(\frac{k}{{\kappa }_{z2}}\right)}^2\right)+{\left(\frac{\omega }{{\kappa }_{z2}c}\right)}^2\right)\right],$$

(5)

where ${\kappa }_{z1}=\sqrt{k^2-{\omega }^2/c^2}$ and ${\kappa }_{z2}=\sqrt{k^2-{\epsilon }_r{\omega }^2/c^2}$ are the spatial extent of the plasmon into a vacuum and into the material, respectively. Note that the dependence on the field is squared, and thus, as expected, the energy deposition rate is directly proportional to the intensity of the field.

2.2. Vacuum Heating

The above prescription accounts for low field perturbations. However, we are interested in reaching significantly high fields. We expect nonthermal carriers to account for the majority of lost energy present in the system by naïvely extrapolating Figure 4 in Ref. [10] to V/nm fields. This is a point of contention, and the extension of work of Ref. [10] to fields relevant for this analysis may serve as future work.

The nonthermal electrons in the system considered here arise from the vacuum dynamics of the quasi-free surface plasma. This plasma undergoes ponderomotive motion in the enhanced laser field, with most of the charge returning to the material and transmitting into bulk. Therefore, one may just sum up much of the deposited energy at a high field by studying the near-field dynamics. This is precisely the work in Ref. [19], where a heating intensity was found:

$$S_v(r,t)=\frac{du}{dt}=\frac{\eta {\epsilon }_0}{4\pi }{v}_{\mathrm{osc}}{E}^2,$$

(6)

with $\eta \approx 1.57$ being a numerically attained constant and ${\epsilon }_0$ denoting the vacuum electric permittivity. The ponderomotive velocity is given by $v_{\mathrm{osc}}=\frac{e}{m_e\omega }E$, where $m_e$ denotes the electron mass. Thus, the energy deposition rate is proportional to the laser intensity to the power of $3/2$. For the chosen values, vacuum heating locally surpasses linear (temperature-independent) dielectric heating at $E\approx$ 40 V/nm.

The energy in this model is carried initially by nonthermal electrons. However, electrons in a good conductor (aluminum) and within the energy range from 2 eV to 500 eV (which encompasses any electron undergoing such ponderomotive motion for the fields we consider) have a range of less than 10 nm [20]. Therefore, the energy in this analysis is deposited essentially throughout the apex. The thermalized electron distribution effectively approaches equilibrium instantly at this length scale and on all other timescales [9]. Thus, only direct heating of the lattice by nonthermal electrons and plasma heating beyond that which we cover here remain as contenders for damaging effects, which we leave for a future investigation.

2.3. Finite Element Method Set-Up

We create meshes modeling the nanoblade and nanotip structures using Gmsh [21] and solve for the electron and lattice temperature over time with the FEM using SfePy [22].

Figure 1 shows a 2-dimensional (2D) cross-section of the blade and tip structures. The nanoblade geometry is created with a circle of a radius R embedded in a 2D wedge with a half-opening angle $\theta$ such that the wedge and circle are tangent to each other, and any wedge above the tangent point is removed. The center of the circle is $10R$ above the bottom of the wedge. To test for accuracy, we double this height and observe a peak lattice temperature difference of only up to 33.5 K (see Appendix A). By symmetry, only the right-hand side of the 2D slice is needed. We use nearly the same recipe for the nanotip, with the difference being that the 2D slice is revolved about the central axis by 90 deg to create a tip quadrant.

The local grid size (set when creating the points used to define the above geometries) is set to $0.1\times R$ for the blade and $0.5\times R$ for the tip. For the blade only, at the tangent point and at the apex, this value is refined to $0.05\times R$ and $0.01\times R$, respectively. The tip apex mesh is quite coarse, but this turns out to not be an issue (see Appendix A). The time step, $dt$, for the first 80 fs is 0.4 fs, and for the remaining 3920 fs (for a total simulation duration of 4000 fs), it is 19.6 fs.

We reduce the heat equations into their weak forms for the FEM calculation:

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}d\mathsf{\Omega }\left[\frac{\gamma }{2}{s}_e\frac{d{T}_e^2}{dt}+\frac{K_{e0}}{2T_{l0}}\nabla s_e·\nabla T_e^2+g(T_e-T_l)s_e\right]& ={\int }_{{\mathsf{\Gamma }}_{\mathrm{surf}}}d\mathsf{\Gamma }S(r,t)s_e,\hfill \\ \hfill {\int }_{\mathsf{\Omega }}d\mathsf{\Omega }\left[C_l{s}_l\frac{d{T}_l}{dt}+K_l\nabla s_l·\nabla T_l-g(T_e-T_l)s_l\right]& =0\hfill \end{array}$$

(7)

with $s_e$ and $s_l$ as the dummy variables associated with $T_e^2$ and $T_l$, respectively, $\mathsf{\Omega }$ as the area/volume within the structure and ${\mathsf{\Gamma }}_{\mathrm{surf}}$ as the facets in contact with vacuum.

The electron and lattice temperatures are initialized to 300 K. We assume that the structures are adhered by their bases to a heat bath at 300 K, necessitating an essential Dirichlet boundary condition on this facet for both distributions. The electron heating at the surface is accounted for within the weak form with a surface heating function $S(r,t)$. All other boundary conditions are of the Neumann type with zero flux (no heat traverses the metal–vacuum boundary or the symmetry planes), and the weak form naturally defaults to this condition when not otherwise specified.

The heating term includes a combination of the linear plasmonic loss and the nonlinear vacuum heating, where $S(r,t)=S_d(r,t)+S_v(r,t)$. However, both of these heating models are dependent on the local field strength $E(r,t)$. We choose the local field as a function of the position such that it mimics the fields we expect near these structures, akin to a conducting cylinder or sphere embedded in an otherwise uniform electric field. For points above the circle center and for $t<4\tau$, where $\tau$ = 8 fs is the full-width half-maximum of the field strength, one has:

$$E(r,t)=E_0{sin}^2\left(\frac{\pi }{4}\frac{t}{\tau }\right)cos\varphi \left(r\right)e^{-{\kappa }_{z1}\left(\right|r|-R)},$$

(8)

where $E_0$ is the peak surface field, ${sin}^2(·)$ is the pulse envelope, $cos\varphi$ is the field profile about the apex ($\varphi =\measuredangle \mathrm{A}Or$ is the angle created from the vertical axis by a ray from the circle’s center to the point r on the surface) and the exponential term diminishes the field intensity away from the apex. We choose ${\kappa }_{z1}$ as a reasonable scale for the field drop off.

3. Results

3.1. Timescale Analysis

The timescales previously described are further illuminated by the temperature time plots in Figure 2 and the spacetime plots in Figure 3. These are for the nanoblade structure with R = 20 nm, E₀ = 80 V/nm and $\theta$ = 30 deg, though all the cases we study here are qualitatively the same. Heating of the structure proceeds as follows: (I) On a scale of 10 s of femtoseconds, the illuminating laser heats up the electron distribution. As the laser turns off, the apex quickly comes to equilibrium, resulting in a slight dip in temperature. (II) On a scale of 100 s of femtoseconds, the electronic distribution continues to disperse energy throughout the bulk of the structure and deposits some energy into the lattice. (III) On a scale of picoseconds, the electrons have dispersed much of the thermal energy initially deposited, and the lattice has reached its peak temperature. The electron temperature is less than the lattice temperature, indicating that the electronic distribution is beginning to cool the apex. This sequence confirms that the electron heating and cooling processes are on quite different timescales from lattice heating and cooling.

3.2. Comparison of Structures

The primary result of this study is the comparison of the tip and blade structures. The apex lattice temperatures as a function of time for the blade and tip structures with $\theta =3$ and 30 degrees are shown in Figure 4. The curves follow a qualitatively similar path by first heating, reaching a peak temperature and then slowly cooling. The structures corresponding most closely to real structures are the 3-degree tip and 30-degree blade. Here, one can immediately see that the realistic blade outperforms the realistic tip, albeit only by about 200 K.

The peak achieved lattice temperatures as a function of the peak surface field $E_0$ is shown in Figure 5 for the same structures shown above. As expected, the peak temperature increases monotonically with increasing field strength. Again, the realistic blade outperforms the realistic tip for the same field strengths. The “fictional” three-degree blade is the only structure which appears to reach the melting point here.

When sweeping over the opening angles instead, Figure 6 is quite illuminating. Here, we again used a peak field of 80 V/nm. It is now understandable that the blade structure does not outperform the tip in general. Actually, the tip appears to outperform the blade. At about $\theta$ = 4 deg, the blade apex reaches the melting point of gold, while the tip does not reach it at all. For an increasing opening angle, the peak temperature drops drastically, indicating that the opening angle is the stronger independent variable.

Finally, by sweeping over the apex radius of curvature for the 30 deg blade at E₀ = 80 V/nm, we obtain Figure 7. The increasing apex size increases the total energy deposited into the structure, thereby increasing the peak lattice temperatures achieved. Even though the electron distribution within the apex still equilibrates rapidly, there is more thermal energy for essentially the same bulk structure below.

4. Discussion

In our previous study [9], we posited that the blade structure would outperform the tip structure thermomechanically as a consequence of its higher dimensionality. The blade may disperse energy effectively in two dimensions, while the tip only has one effective dimension, considering their typical small opening angles. Here, however, we found a different reason for their performance gap. Actually, for the same opening angles, the tip outperforms the blade.

This leads us to the conclusion that using a larger opening angle is a better avenue for improving thermals than using a different structure altogether. A pyramid structure [23], which is tip-like but with large enough opening angles, may achieve higher surface fields and therefore higher brightness emissions with its larger opening angle. However, the pyramid structure retains quite a small interaction area, which may necessitate an array.

We were not able to accurately predict the damage thresholds here. As a matter of fact, we only predicted melting at the apex for quite narrow blades below 4 deg and for large apexes. We also predicted tip survivability at fields up to 80 V/nm, which is certainly not realistic. Let us now discuss several ways in which this analysis may be built upon.

Firstly, the blade structure likely does not attain a bulk geometric enhancement that leads to 80 V/nm fields. Rather, these fields are likely obtained via hot spots [6]. Inclusion of these hot spots is of interest for this study. One would additionally need to model the electromagnetic and plasmonic fields to properly implement the hot spots. However, their inclusion ultimately reduces the heating of the structure compared with this smooth analysis. Any localized electronic heating at the hot spots quickly dissipates into the bulk of the apex according to the effectively instant relaxation time at these small scales [9], and thus one would not predict direct damage to these hot spots through this additional consideration alone.

We additionally used the FEG approximation for the electronic heat capacities and conductivities. These properties, as well as the electron-phonon coupling factor, are calculated by integrating over the density of states [13]. At energies about 2 eV $\approx$ 20,000 K below the Fermi level, d-shell electrons begin to contribute to both these quantities. We achieved temperatures that extended beyond this range, and therefore inclusion of the density of states is necessary for a more accurate electronic thermomechanical model at ultra-fast timescales. This may perhaps be more impactful when studying the effects on emissions as opposed to survivability.

Additionally, the electron-phonon coupling was weak compared to the value we used for low electron temperatures and increased for increasing electron temperatures. However, the high electron temperatures were so short-lived that much of the lattice heating occurred when coupling was weak. Improving this would also decrease the observed lattice temperatures (see Appendix C), indicating that there must be more ways to heat the lattice near the apex.

One process which would lead to higher surface temperatures would be direct deposition of energy into the lattice by the nonthermal electrons. If this is a strong enough process, then the lattice temperature may increase much more quickly before the electrons may disperse the energy via electron-phonon coupling, followed by electronic cooling. As the lattice is less capable of dispersing energy, this may lead to damage.

Finally, we would like to improve upon the electron heating mechanisms. The dominant heating mechanisms vary depending on the configuration of the system. Modeling the surface as a solid density plasma, which is appropriate at such high surface fields, would invoke the normal skin effect, anomalous skin effect, high-frequency skin effect and sheath inverse bremsstrahlung [24]. All these listed improvements will serve to more accurately model these structures, their emissions and their viability as long-lifetime cathodes.