Detailed measurements of momentum distributions of emitted electrons have allowed the investigation of the thermal limit of the transverse emittance from metal photocathodes. Furthermore, recent developments in material design and growth have resulted in photocathodes that can deliver high quantum efficiency and are sufficiently robust to use in high electric field gradient photoinjectors and free electron lasers. The growth process usually produces photoemissive material layers with rough surface profiles that lead to transverse accelerating fields and possible work function variations, resulting in emittance growth. To better understand the effects of temperature, density of states, and surface roughness on the properties of emitted electrons, we have developed realistic three-dimensional models for photocathode materials with grated surface structures. They include general modeling of electron excitation due to photon absorption, charge transport, and emission from flat and rough metallic surfaces. The models also include image charge and field enhancement effects. We report results from simulations with flat and rough surfaces to investigate how electron scattering, controlled roughness, work function variation, and field enhancement affect emission properties. Comparison of simulation results with measurements of the quantum yield and transverse emittance from flat Sb emission surfaces shows the importance of including efficient modeling of photon absorption, temperature effects, and the material density of states to achieve agreement with the experimental data.

## I. INTRODUCTION

Effective operation of free electron lasers (FELs), linear accelerator facilities, and advanced X-ray light sources depends on providing reliable photocathodes^{1,2} for generation of low emittance, high-brightness, and high-current electron beams using conventional lasers. Modern developments in design and synthesis of materials have resulted in photocathodes that can have a high quantum efficiency and operate at visible wavelengths. These photocathodes are robust enough to work in high electric field gradient photoguns for application to FELs in dynamic electron microscopy and diffraction. Synthesis, however, often results in roughness, ranging from the nano- to the microscale. Thus, the effects of roughness on emittance are of significant importance to understand.

A new momentatron experiment concept was developed^{3} to measure transverse electron momentum and emittance. A successful application of the momentatron concept was demonstrated by Feng *et al.*^{4} to investigate the thermal limit of the transverse emittance from flat surfaces. Their experiments provide reference data on emittance from Sb cathodes. Effects of surface roughness can then be evaluated relative to the flat emission surface measurements.

Recently, advances in material science methods have been demonstrated (see, e.g., Chapter 5 in the study by Polyakov^{5} and references therein) to control the growth of photoemissive materials, e.g., Sb on a Si substrate, to create different types of rough layers^{6} with a variable thickness of the order of 10 nm.

Although analytical formulations of the effects of roughness have been developed, a full theoretical model and experimental verification are lacking. Our work aims to bridge this gap since coupled, high-fidelity theoretical and modeling treatment is of critical importance when investigating new cathode material concepts that include heterostructures, surface coatings, and modified barriers.^{7–9} Here, we report results on electron emission modeling and three-dimensional (3D) simulations from photocathodes with controlled surface roughness similar to grating surfaces that have been fabricated by nanolithography.

The rest of this paper is organized as follows. In Sec. II, we describe the models that we have developed to simulate electron excitation due to photon absorption, charge transport, and emission from metallic photocathodes with rough surfaces. In Sec. III, we discuss first simulation results from flat Sb emission surfaces and compare them with experimental data on the quantum yield (QY) and transverse emittance as a function of photon energy. Then, we describe simulations with controlled rough Sb surfaces and how the roughness affects emission properties. We provide a summary of the current work and some future directions that we are considering in Sec. IV.

## II. MODELING AND SIMULATIONS

In this section, we shortly cover the VSim^{10} computational framework and then describe the models that we have implemented to model metallic photocathodes with different piece-wise continuous emission surfaces.

The VSim software tool is based on the Vorpal computational physics kernel.^{11} It is a high performance, multi-dimensional code with extended capabilities for simulating electrostatic (ES) and electromagnetic problems, with or without the inclusion of charged particles.

Here, we use specifically its 3D mode of execution in parallel to simulate electron emission from metallic photocathodes with flat and controlled rough surfaces. Our approach includes electron excitation in response to absorption of photons with a given wavelength, charge dynamics due to drift and different types of scattering processes, representation of flat and rough interfaces, calculation of electron emission probabilities, which takes into account image charge and field enhancement effects across rough surfaces, particle reflection, or emission updates, and efficient 3D electrostatic (ES) solver for a simulation domain that has sub-domains with different dielectric properties.

### A. Three-step model for electron emission

The overall approach to model charge generation, transport, and emission that we have implemented corresponds to Spicer's 3-step^{12–14} emission model. A simplified diagram of the 3-step model for a metallic photocathode is shown in Fig. 1. It also includes specific temperature and energy threshold effects that we have implemented in the models.

In the first step, part of the photons from the incident radiation are absorbed. This leads to excitation of electrons from filled levels in the conduction band to higher energy empty ones. A level in the conduction band with energy $E$ is occupied with probability given by the Fermi function

where *T* is the temperature, *μ* is the chemical potential, and *k _{B}* is the Boltzmann constant. An example of an electron excited due to an absorbed photon with energy $\u210f\omega $ is shown in Fig. 1 with the transition from energy levels $E0$ to $E1$.

The second step consists of electron transport of excited electrons. Since the electric field in metals and semimetals is approximately zero, the electron transport is mostly diffusive. For metallic materials, electron-electron (el-el) scattering is the dominant process that affects emission.

In the third step, electrons that attempt to cross the interface with vacuum are emitted with probability that depends on their energy, momentum, and surface potential. If an electron is not emitted, it is reflected back into the photocathode material. A constant surface potential, shown in Fig. 1, is given by the vacuum level at *μ* + *ϕ*, where *ϕ* is the work function. For this case, electrons with energy $E<\mu +\varphi $ have zero probability of emission. Thus, when modeling electron transport to investigate electron emission, we can define an energy threshold $Eth$ set to a value that depends on the surface potential energy such that electrons with $E<Eth$ will have zero or exponentially small probability of emission. These electrons can be removed from the simulation since they will effectively not affect emission properties. For the constant vacuum level, we have $Eth=\mu +\varphi .$

In the presence of an applied electric field and when the image charge contribution is included, the surface potential has the shape given by *V*(*x*) in Fig. 1. In this case, the top of the potential energy is lowered due to the Schottky effect. Moreover, there is a non-zero probability that electrons with energies lower than the top of the potential energy barrier will tunnel and be emitted. However, even in this case, if we lower the electron energy sufficiently below the top of the barrier, the probability of emission decreases exponentially and becomes sufficiently small to effectively prevent emission through tunneling. Thus, we can also define a threshold energy $Eth$, as shown in Fig. 1, and keep in simulations only electrons with energies $E>Eth$. This approach can significantly increase the speed of the simulations since we do not have to push throughout a run all excited electrons but only the ones that could potentially be emitted.

Furthermore, when photons are absorbed, we can load in a simulation only the electrons that are excited to energies $E>Eth$ since electrons excited to energies lower than $Eth$ will effectively be not emitted into vacuum.

Once electrons with energies $E>Eth$ are loaded, they are evolved using the Monte Carlo approach to model electron transport. Our implementation^{15} is based on the method described by Jacoboni and Reggiani.^{16} In this approach, electrons are pushed ballistically until a scattering time, sampled from the scattering rates of the processes involved, is reached. Then, the specific scattering process is determined and executed.

In metallic materials, the most important process for modeling emission is el-el scattering. The reason is that one such scattering event is effectively sufficient to lower the energy of an excited electron with $E>Eth$ to a value below $Eth$, thus removing it from the list of electrons that could be emitted.

We show one such el-el scattering process in Fig. 1. A second electron with initial energy $E2$ sampled from the filled levels in the conduction band scatters with the excited electron with $E1>Eth$. Due to the Pauli exclusion principle, only empty final states are allowed for both electrons. The probability for an empty energy level is $1\u2212f(E)$. At room temperature, most of the levels with $E<\mu $ are filled. Thus, it is most likely that the final energies of the two electrons will be higher than *μ* as shown in Fig. 1. Since $E2<\mu $, the final energy $E4$ of the second electron will then be higher than its initial value $E2$.

Due to the conservation of energy in the scattering process, the decrease in the first electron's energy will be $E4\u2212E2$ (note that the largest energy the first electron can lose in the scattering is approximately $E1\u2212\mu $). If the excess energy $\u210f\omega \u2212\varphi $ is of the order of 1 eV (which is often the case when operating photocathodes) and since the highest photoexcited electron energy is $E1max\u2248\mu +\u210f\omega +O(kBT)$, a single el-el scattering process is often enough to bring even an electron on $E1max$ to a final energy lower than $Eth$. The process shown in Fig. 1 indicates such a transition to a final energy $E3<Eth$ when a photoexcited electron with initial energy $E1>Eth$ scatters with the electron initially at $E2$.

### B. Electron excitation due to photon absorption

We model electron excitation with an exponential decay of absorbed laser light intensity relative to positions on the photocathode surface. Electrons can be excited due to normal or oblique light incidence (relative to a reference plane). For modeling emission from antimony, electrons are selected for excitation from occupied states at a given temperature *T* in the conduction band from the distribution function $p(E)\u223cg(E)f(E)$, where $g(E)$ is the density of states (DOS) obtained from the study by Bullett.^{17} Electrons are created only if their final state with energy $E+\u210f\omega $ is not occupied [with probability $1\u2212f(E+\u210f\omega )$].

In Fig. 2, we show the effective distribution function $p(E)$, normalized to represent a probability distribution, over the interval of energies relevant for electron excitation in Sb that we have considered here. This range is determined by the highest photon energy of 6.2 eV used in the experiments. The figure also shows that there are some filled energy states above the chemical potential. This is due to the non-zero temperature allowing for electrons to be excited above the vacuum level even for some $\u210f\omega \u2212\varphi <0$ excess energies.

We developed an algorithm to access these energies in order to obtain results for the quantum yield and transverse emittance for energies from the tail of the $p(E)$ distribution when $\u210f\omega \u2212\varphi <0$. We provide some details of this algorithm when we discuss the simulation results.

For all simulations we did for emission from antimony, over the range of photon energies $4.4\u2264\u210f\omega \u22646.25\u2009eV$, we used an absorption length of 10 nm, derived from optical data.^{18}

### C. Electron scattering

In VSim, we have previously implemented the ensemble Monte Carlo method for charge transport for two semiconductors: diamond^{15} and GaAs.^{19} These models take into account different types of electron and hole scattering processes.

For metallic materials, it is important to include electron-electron scattering in the modeling since it is the main process that affects electron emission. Here, we have implemented the unified model for el-el scattering in metals proposed by Ziaja *et al.*^{20} It is applicable over a wide range of energies and is efficient for use in Monte Carlo transport simulations. Moreover, it is relatively straightforward to implement for different metals. However, this unified model is a simple description of el-el scattering and does not include temperature effects. A more detailed theory, as the one given in the study by Jensen *et al.*,^{21} should be used when these effects become important. For our work here, the energies of photoexcited electrons that are emitted are several electron-volts above the Fermi level and temperature effects arising from el-el scattering are negligible.

In this model, the electron mean free path (MFP), $\lambda (E)$, is given by the simple formula

where the initial electron impact energy $E$ is in eV, $Etheh$ is the effective energy threshold for production of an electron-hole pair as a result of the scattering, $E0=1$ eV, and *a*, *b*, *A*, and *B* are the fitting parameters with units that give the MFP in Å. For metals, $Etheh=\mu $ (or the Fermi energy $EF$ when $\mu \u2248EF$). In semiconductors, the threshold is given by the gap energy $Etheh=EG$.

The fitting parameters are determined using experimental data and/or full band structure calculations. The values of these parameters, determined using this approach, are provided^{20} for a number of different metal and semiconductor materials.

The parameters *a* and *b* determine the MFP in the low energy regime $EF/G<E<EP$, where $EP$ is the plasmon energy. This is the regime of interest for electron emission from metallic materials since the energy of photons used is of the order of 1 eV, leading to excited electrons with $E<EP$. In this regime and assuming that the speed of an electron is given by $v(E)=2E/m*$, where *m** is the electron effective mass, the el-el scattering rate is given by $\Gamma (E)\u2248a\u0303(E\u2212Etheh)b$, where $a\u0303\u2261a2/m*$. Both the free electron gas model and the Fermi-liquid theory lead to *b* = 2. However, using Eq. (2) to fit data for a number of metals, the values of the model parameter *b* were obtained^{20} in the range of 1.5 ≤ *b* ≤ 5.0.

The top plot in Fig. 3 shows the MFPs for three metals calculated with such parameters. We selected these three metals to show the range over which the MFPs vary when $E\u2212\mu $ is of the order of 1 eV. Over this range of energies, the MFPs of Cu and Ag are similar, while the MFP of Be is over two orders of magnitude higher when $E\u2212\mu $ approaches 1 eV from above. In the bottom plot of Fig. 3, we show the corresponding scattering rates for these three metals considering only the range of interest for photocathodes. For electron energies in the interval $3<E\u2212\mu <7\u2009eV$, the rates vary from approximately 10^{12} 1/s to 10^{15} 1/s.

Since we currently do not have data, similar to the types of data used by Ziaja *et al.*^{20} in order to obtain the fitting parameters of the unified el-el scattering model given by Eq. (2), we bracketed the Sb rates over the energy range $3<E\u2212\mu <7\u2009eV$ by using a low and high rate to span the range determined for other metals. These two rates are also shown in the bottom plot in Fig. 3. The parameter *a* was fixed for both rates. The higher el-el scattering rate is calculated with *b* = 3.8, while for the lower one, we used *b* = 1.5.

In Sec. III, we compare the results from simulations on emission from flat Sb surfaces with experimental data on the quantum yield and transverse emittance. As we find out, the results with the higher el-el scattering rate show better agreement with these datasets. Thus, we can use the QY and emittance experimental data to obtain the values of the parameters *a* and ** b** of the el-el scattering model, which leads to the best level of agreement. This optimization shall be explored in a future investigation.

### D. Modeling electron emission

In the Monte Carlo method to simulate electron transport and emission, when an electron in a photocathode material attempts to cross its emission surface to move into vacuum, a probability of emission is calculated using the surface potential at the location where the electron hits the surface. Once this probability is known, the electron is emitted or reflected back according to its value [the electron is emitted if the emission probability is higher than the value of a (pseudo)random number generated from a uniform distribution].

The calculation of the emission probability simplifies significantly if the potential is translationally invariant in the emission surface plane. For this case, the potential is a function of only the coordinate along a direction normal to the emission surface. The translational invariance also requires that the transverse electron momentum is conserved in the emission process (see, e.g., the study by Bell^{23}), This is the approach that we had previously implemented for modeling emission from diamond^{24} and GaAs cathodes.^{19}

We have implemented^{24} several models for the calculation of the probability of emission (for a detailed exposition of electron emission physics, see, e.g., the study by Jensen^{25} and references therein). Of these models, the one based on the transfer matrix method allows us to readily include arbitrary piece-wise continuous surface potentials and effects due to anisotropic effective masses (of importance when modeling emission from diamond^{24}) and changes to the electron effective mass when it transitions from the photocathode material into vacuum. The latter effect has important restrictions when transverse momentum conservation is taken into account.^{19,24} For these reasons, all simulation results presented here were investigated using the transfer matrix method to evaluate electron emission probabilities.

For the simulations on flat Sb surfaces, we used uniform electron affinity *χ* = *μ* + Φ, where the work function of Sb was Φ = 4.5 eV as in the analysis of the experimental data.^{4} The surface potential energy also includes terms due to the applied field *F* and the image charge

where

with *K _{s}* being the static dielectric constant of the emitter and $Q0=q2/(16\pi \epsilon 0)$, with

*q*being the fundamental charge and

*ε*

_{0}the permittivity of vacuum. The

*x*axis is considered to be normal to the flat emission surface. The fundamental charge is included in the vacuum applied field

*F*such that all terms in Eq. (3) are in eV units. In the simulations of Sb photocathodes, we used

*K*= 80, a value determined from infrared absorption measurements

_{s}^{26}at low temperatures. A value of

*K*= 100 has also been used in a previous study

_{s}^{27}of antimony and is derived from the experimental data in the study by Cardona and Greenaway.

^{18}

For modeling of emission from rough photocathode surfaces, we implemented an approach that is based on the approximation that the surface potential can be treated locally (in a small surface area centered at any point where an electron attempts to cross into vacuum) as effectively one dimensional and represented by Eq. (3). In the case of a rough surface, *x* in Eq. (3) is a coordinate along the direction of the local normal at the crossing point on the surface. Examples of such local normal directions are shown in Fig. 4 for three locations on a ridge rough surface similar to the grated surfaces being grown experimentally. The figure shows the longitudinal electric field calculated using VSim's ES solver. It is from one of the 3D simulations on emission from controlled rough Sb surfaces, discussed in detail in Sec. III.

In Fig. 4, normal directions at three different locations are shown with line segments that extend into the vacuum sub-domain. The ES potential is evaluated along such local normal directions when emission probabilities are calculated. This allows us to include the field enhancement effect on electron emission from rough surfaces. As expected and shown in Fig. 4, the electric field magnitude is enhanced near the tips of the ridges, leading to higher Schottky barrier lowering there and increased probability of emission.

For rough surfaces, treating the surface potential as a one-dimensional function with its argument representing the distance from the surface along the local normal direction is an approximation since the potential is no longer translationally invariant (as in the case of a planar surface with constant electron affinity). This approach allows us to efficiently calculate emission probabilities and to also include the field enhancement effect in the modeling. Moreover, for length scales of the order of 10 nm over which the potential in vacuum at the interface is evaluated for use in the transfer matrix calculations, the curvature effects on the field lines near the surface are small. Thus, the evaluation of the transmission probability is along the classical trajectory^{28} and can be taken along a straight path in the direction of the local normal to the surface.^{29}

In addition to modeling emission/reflection when an electron in the photocathode attempts to cross the interface surface with vacuum, we have implemented the capability to always reflect vacuum electrons that attempt to cross the interface in the reverse direction. This could occur due to rough interface surfaces with different curvatures or to specific space-charge distributions near the emission surface. Generally, we can treat such vacuum electrons in a similar way by calculating a transmission probability given the surface potential. However, in this study, we implemented an algorithm that always reflects them (setting the probability for their transmission into the photocathode material to zero).

## III. RESULTS

Here, we first explain the simulation setup to model the quantum yield and transverse emittance. In Subsections III A and III B, we describe the results from the simulations and compare them with the experimental data when using flat Sb emission surfaces. We report the results from simulations with a controlled rough emission surface in Subsection III C. We discuss how this type of roughness affects the quantum yield and emittance when compared to the simulations with the flat emission surface.

We set up and ran simulations to investigate how two different el-el scattering rates and controlled surface roughness affect the spectral response of the quantum yield and transverse emittance. For modeling emission from flat and three-ridge rough Sb surfaces, we ran most of the simulations with a uniform work function of *ϕ* = 4.5 eV. We used periodic boundary conditions along the transverse directions (*y* and *z*). There is a constant potential difference maintained across the *x* length of the simulation domain, leading to an applied field magnitude in the vacuum region of the order of 1 MV/m. The field magnitude varies on the rough emission surface from around 0.2 MV/m near the valley bottoms to around 3 MV/m near the tips of the ridges as shown in Fig. 4.

The controlled rough surface has a ridge period of 394 nm, a ridge height of 194 nm, and a width of the ridge flat top of 79 nm. An approximate *x*–*y* cross-section of the simulation domain with the ridge rough surface is shown in Fig. 4 (the Sb sub-region is below the contour of the rough surface, while the vacuum sub-region is above it). A 3D visualization of excited and emitted electrons from the rough surface simulations is provided in Subsection III C.

The simulation domain size for both the flat and the 3-ridge emission surfaces is 0.4268 × 1.182 × 0.394 all in *μ*m along *x*, *y*, and *z*, respectively, with a sample resolution of 90 × 264 × 16 number of cells. A typical time step of *dt* = 2.5 × 10^{−16 }s was used in the simulations.

### A. Quantum yield

QY is defined as the ratio of the number of electrons emitted to the number of absorbed photons. In the simulations, we have direct access to the data of emitted electrons as a function of time. We also control the number of photo-excited electrons that are loaded in response to absorption of photons. Thus, a simple approach to obtain QY from simulations is to load a given number of photo-excited electrons due to absorption of photons with specific energy and then determine the number of emitted electrons until the excited electrons relax their energies (mainly due to el-el scattering) and emission effectively stops. However, this approach is inefficient particularly when the photon energy approaches the effective work function (the work function decreased by the Schottky barrier lowering). The reason is that initial levels from which electrons are excited are sampled from the distribution function proportional to the Fermi function and the density of states, as we discussed in Subsection II B. This leads to a significant number of photo-excited electrons that do not have sufficiently high energy to be emitted.

To address this problem, we implemented an efficient algorithm that loads a specific number of electrons above a given energy threshold while also recording how many electrons were excited to reach this goal. We also need the total number of photo-excited electrons in order to calculate the QY. This algorithm makes it possible to explore the emission regime when the photon energy decreases below the work function. Then, only electrons excited from the tail of the distribution function with energies above the chemical potential could potentially be emitted.

Using this algorithm, we started all runs by loading 3 × 10^{5} excited electrons with energies $E>\mu +4.4\u2009eV$. For the highest photon energy $\u210f\omega =6.25\u2009eV$, this goal was reached after we obtained approximately 7.8 × 10^{5} samples from the distribution function for initial energy levels from which electrons can be excited to unoccupied levels with probability $1\u2212f(E+\u210f\omega )$. This number of samples increases when decreasing the photon energy. For the lowest photon energy $\u210f\omega =4.4\u2009eV$ used in the simulations, approximately 1.7 × 10^{8} electrons had to be excited to reach the goal of 3 × 10^{5} electrons with energies above the given threshold. Note that the energy threshold for loading excited electrons is 100 meV lower than *μ* + *ϕ* with *ϕ* = 4.5 eV for Sb and approximately 35 meV lower than $\mu +\varphi effmin$ where the minimum effective work function is reached at the highest Schottky barrier, lowering near the tips of the ridges. The loaded excited electrons are transported inside the Sb sub-domain until they are emitted or their energy drops below the threshold of *μ* + 3 eV (in which case they are removed). Electrons emitted in vacuum are pushed until they exit the simulation domain due to ballistic motion in the applied field.

We plot the QY results from the simulations on emission from flat Sb surfaces and the two el-el scattering rates in Fig. 5. As expected, the QY for the runs with the higher el-el scattering rate, plotted in Fig. 3 with *b* = 3.8, is lower than the QY obtained from the runs with the lower rate. This is due to the longer electron lifetime. For the higher scattering rate, the el-el scattering reduces the energy of excited electrons faster, thus preventing more of them from reaching the emission surface with sufficiently high energy to be emitted.

In Fig. 5, we also plot the QY from experiments^{30} on emission from an Sb photocathode with a flat surface. The experimental data are in agreement with the simulations using the higher scattering rate over the range of photon energies considered, even for $\u210f\omega <\varphi $. This indicates that the implemented algorithm to probe the tail of the distribution function near the Fermi energy is effective to study temperature effects on emission.

### B. Intrinsic emittance

The transverse emittance^{1,2,25} in photoinjectors is limited by the intrinsic transverse emittance (or simply called intrinsic emittance here) of electron beams emitted from photocathodes.^{31–33} The intrinsic emittance originates from the photocathode material and its surface geometry and depends on the type of emission process.

If there is no correlation between the transverse position and momentum distributions of emitted electrons,^{34} the intrinsic emittance per mm of rms laser spot size is given by^{35}

where *σ _{y}* is the laser spot size in mm,

*m*is the electron mass in vacuum,

_{e}*c*is the speed of light, and

*p*is one of the transverse momentum components of an emitted electron. A theory

_{y}^{35}based on the approximation that the electron density of states is constant and that the zero temperature representation of the Fermi function (given by the Heaviside function) can be used for metals leads to

This theory predicts that the intrinsic emittance is defined only for excess energies $\u210f\omega \u2212\varphi \u22650$ and becomes zero when the photon energy is equal to the work function (or the effective work function when the Schottky effect is taken into account in the applied field).

If the assumption that the zero temperature representation of the Fermi function is lifted and Eq. (1) is used but still assuming a constant DOS, the intrinsic emittance can be expressed^{36} via the polylogarithm function.

A more accurate treatment that includes the DOS of electrons over the energy range from which they are excited and temperature effects^{4} leads to

$where\u2009h(E,\varphi ,\u210f\omega )=23\u2212\mu +\varphi E+\u210f\omega +13(\mu +\varphi E+\u210f\omega )23$.

In the simulations, we calculate the mean transverse energy (MTE) of vacuum electrons at emission and when they cross a diagnostic surface in a given location, with *x *=* Const*, near the exit of the simulation domain. The MTE is then used to obtain the intrinsic emittance since under equipartition, $MTE=\u3008py2\u3009/me$, and compare with the models given by Eqs. (6) and (7).

The intrinsic emittance per mm of rms laser spot size for Sb evaluated from these three theories as a function of excess energy is shown in the top plot of Fig. 6. Note that for *T *≠ 0 with $g(E)=Const$, the integration in Eq. (7) reduces to the expression^{36} based on the polylogarithm function.

As expected, the theory for the emittance, Eq. (6), based on the assumption that the *T* = 0 K representation of the Fermi function is applicable, cannot be used to obtain the intrinsic emittance for photon energies close to the (effective) work function and is not defined for $\u210f\omega \u2212\varphi <0$. The other two theories are good approximations for the intrinsic emittance in this region and approach its thermal limit $\u03f5y,\u200ath/\sigma y=kBT/mc2$ when $\u210f\omega $ decreases below the (effective) work function.

The theory based on the constant DOS leads to a good approximation of the intrinsic emittance since only a small interval of energies with $E>\mu $ contributes in the evaluation of the integrals in Eq. (7), and over such a small interval, the DOS remains approximately constant. Once the photon energy exceeds the work function and increases, a larger interval of energies with $E<\mu $ starts to contribute in the integrals in Eq. (7). If the DOS deviates from a constant over this interval, then we would expect the intrinsic emittance to deviate from the predictions of the models using a constant DOS. This is indeed the behavior that we see in the top plot in Fig. 6 since the effective distribution function $p(E)\u223cg(E)f(E)$ shown in Fig. 2 increases quickly when $E$ decreases below *μ* and deviates from a constant.

The DOS effect on the intrinsic emittance of antimony is also confirmed by the experimental data.^{4} The results from the simulations with the implemented models using a flat Sb emission surface and the two scattering rates are shown in the bottom plot of Fig. 6. They are compared with the experimental data^{4} and the three theoretical models.

Similar to the results on QY, the simulations with the higher el-el scattering rate are again in agreement with the experimental data. This level of agreement is only possible if both the finite temperature and the accurate DOS effects are included in the modeling.

The bottom plot in Fig. 6 shows simulation results for the emittance calculated at the emission surface and also at another surface parallel to it but near the end of the simulation domain. All vacuum electrons eventually cross the second diagnostic surface due to their drift in the applied electric field. The transverse emittance is effectively the same at the two diagnostic surfaces. This confirms the expected behavior since for the simulations with a flat emission surface, the transverse electric field in vacuum is practically zero.

### C. Surface roughness effects

In this Subsection, we describe the results from the simulations on electron emission from the ridge controlled rough surface. We consider how this type of surface roughness, given the assumptions of the implemented models, affects the quantum yield and transverse emittance.

All the results presented here are from simulations with the laser light impacting the rough surface in a direction normal to the flat top sub-regions of the ridges. This is the same direction (along the negative *x* axis) as in the simulations with flat Sb emission surfaces. We have not taken into account light interference effects on the rough surface. Thus, the light intensity profile on the rough Sb surface is considered uniform.

Under these assumptions, excited electrons are loaded at positions sampled from an exponential decay distribution along the *x* axis and with local origins at positions where photons impact the rough surface. The initial distribution of the 3 × 10^{5} electrons loaded is shown in the bottom plot in Fig. 7. Electrons in Sb are shown with red spheres.

This initial distribution of excited electrons already shows an important difference due to the rough surface. In the case of a flat emission surface, all electrons loaded at a given distance *x _{s}* relative to the surface have to traverse it before they can attempt to cross the interface and possibly be emitted. For the rough surface, however, electrons loaded on the sides of the ridges at the same distance

*x*relative to the position on the rough surface along

_{s}*x*are effectively closer to the vacuum interface since the closest distance for any of these electrons is along a local normal direction towards the surface and not along the

*x*direction. Effectively, electrons loaded under the sides of the ridges have shorter distances to travel to reach the emission surface. Under the ridge tops and the valley bottoms, the distribution of loaded electrons is similar to the distribution under the flat emission surface.

The other three plots in Fig. 7 show electron position distributions at 12.5 fs intervals since the start of the simulation. Electrons emitted in vacuum are shown with green spheres. The sequence of plots confirms the diffusive expansion dynamics in Sb, the fast emission response time characteristic of metallic materials, and the quick reduction of excited energetic electrons due to el-el scattering. Electrons in Sb with energies that drop below the *μ* + 3 eV threshold are removed during the run by the implemented algorithm.

The QY results from the simulations with the rough surface are shown in Fig. 8. Similar to the results with the flat emission surface, the QY with the higher el-el scattering rate is lower compared to the results from the runs with the lower scattering rate. In the latter case, again electrons in Sb survive over a longer period of time. This allows more of them to reach the emission surface with sufficiently high energy for emission.

The QY from the rough surface simulations shows an increase relative to the corresponding results from the flat surface. While the probability of emission near the tips of the ridges has increased due to field enhancement and the Schottky barrier lowering, we consider the small QY increase to be due to the increase in the surface area and emission specifically from the sides of the ridges.

Since photons absorbed on the sides of the ridges lead to excited electrons that are closer to the vacuum interface compared to electrons due to photon absorption at flat surfaces, we expect an increase in emission from the sides of the ridges. To test this, we ran simulations with a variable work function. It was set to a sufficiently high value on the sides of the ridges to effectively prevent electron emission there, while on the ridge tops and the valley bottom, the work function remained unchanged (kept at *ϕ* = 4.5 eV). The QY spectral response from the variable work function runs is also plotted in Fig. 8. Comparing the simulation results from a flat surface with the same el-el scattering rate confirms that preventing emission from the sides of the ridges lowers the QY below the values from the flat emission surface.

Apart from the surface-dependent work function effect on QY, the spectral response could also be affected by light intensity variation on the rough surface. Given the periodicity of the ridges and the range of photon wavelengths used, light interference could lead to a non-uniform intensity profile on the surface. This will affect the distribution of excited electrons and, thus, the QY. We have not considered this effect here. However, we are planning to investigate it in a future study by using another code to calculate the light intensity profile on the rough surface. Then, we will implement a new electron loader that can read-in the intensity profile and load electrons in the photocathode according to it.

The dependence of the transverse emittance on excess energy from the rough surface simulations is shown in Fig. 9. The results from the runs with the uniform work function and the two scattering rates are shown in the top plot of the figure. The emittance has increased when compared to the results from the simulations with the flat surface shown in Fig. 6.

This is mainly due to emission from the sides of the ridges. Electrons are emitted there with similar transverse momenta relative to local normals but large ones relative to the longitudinal direction since local normals on the ridge sides deviate by a large angle from the *x* axis. Moreover, there are transverse electric fields in the vacuum regions between the ridges that also modify electron transverse momenta. Due to the presence of these transverse electric fields, the emittance is different when determined at the emission surface compared to that when evaluated at the diagnostic surface near the simulation exit. This effect is not present in the simulations with the flat Sb surface as shown in Fig. 6 since in those simulations, the transverse electric field in vacuum is approximately zero.

The results from the rough surface simulations with the same variable work function set up to prevent emission from the sides of the ridges are shown in the bottom plot in Fig. 9. These results are now similar to the ones from the flat emission surface since the momentum distribution of electrons emitted from the ridge flat tops and the valley bottoms does not deviate significantly from the one observed in the flat surface simulations. Nevertheless, the emittance shows an increase that is due to the transverse electric fields, mainly in the vacuum regions between the ridges, emission at the ridge tips, and the corners of the valley bottoms. The transverse fields also cause the difference in emittance when calculated at the emission surface and at the diagnostic surface near the end of the vacuum sub-domain.

## IV. SUMMARY

We implemented models to simulate electron emission from metallic photocathodes with controlled rough surfaces in the 3D VSim code. The implementation includes accurate and efficient treatment of DOS and Fermi-Dirac distribution effects both in the excitation of electrons due to photon absorption and in handling el-el scattering. This allows modeling of emission even when the photon energy decreases below the work function, and only electrons excited from the tail of the Fermi-Dirac distribution with energies $E>\mu $ could be emitted.

Simulation results on emission from flat Sb surfaces with the higher el-el scattering rate considered show that the spectral response of the quantum yield and the transverse emittance is in agreement with available experimental data^{4} and the theory that takes into account the Fermi-Dirac distribution and density of states effects. Thus, it is of interest to use experimental data from emission experiments to determine the parameters of the unified model for el-el scattering, which optimize the agreement with the data. Emission from rough surfaces leads to an increase in the transverse emittance.

We expect that simulation results with the developed models will allow us to further improve our understanding of experimental data on electron emission from other metal photocathodes. We are also considering to investigate how the unified el-el scattering model performs for modeling of emission from metals for which the parameters of the model have been determined from other kinds of experiments^{20} and to implement a model^{21} for el-el scattering that includes temperature effects.

Currently, we do not include light interference effects and how the light intensity varies on rough surfaces due to this. Adding this effect will lead to a different distribution of loaded electrons, effectively modifying the emission properties. We are planning to include this effect in the future. Another extension of the models is to include emission from layered photocathodes, e.g., Sb on Silicon where the Sb layer is of the order of 10 nm and effects due to the Sb-Si interface are considered as well. We are also investigating how to introduce a time-varying light absorption when the light intensity profile varies and the direction of incidence on the surface is arbitrary.

There is a theoretical model for the quantum efficiency^{35} (directly related to the QY) that treats the el-el MFP within specific approximations and also uses a constant DOS and the *T* = 0 K representation of the Fermi function. We are considering to compare our simulation results with this theoretical model and then to extend it to include temperature and DOS effects, similar to the comparison that we did here with the models for the intrinsic emittance. It will be interesting to check if the accurate values of the fitting parameters *a* and *b* of the unified el-el MFP model can be determined using this approach. Accurate modeling of el-el scattering will be of increasing importance when modeling QY at higher photon energies and from photo-emissive layers. In these cases, the QY could increase due to more electrons near the emission surface with sufficient energy to be emitted, including the ones that survive over a longer period of time even after they have undergone one or more el-el scattering events.^{21}

We are also planning to do simulations with rough surface shapes that have been investigated previously.^{37,38}

The currently implemented emission models do not include quantum surface effects^{39} due to rough structures comparable in linear size to the electron De Broglie wavelength. We are investigating how to include such quantum effects and types of surface roughnesses in the modeling as well.

## ACKNOWLEDGMENTS

D. A. Dimitrov would like to thank Kevin L. Jensen, David Smithe, Erdong Wang, Triveni Rao, and John Cary for helpful discussions. This work was supported by the Office of Science, Office of Basic Energy Sciences of the U.S. Department of Energy under Contract Nos. DE-SC0013190, DE-AC02-05CH11231, and KC0407-ALSJNT-I0013, and by the National Science Foundation under award PHY-1549132, the Center for Bright Beams.

## References

*An Engineering Guide to Photoinjectors, CreateSpace Independent Publishing Platform*, edited by T. Rao, D. H. Dowell (2013), ISBN-10: 1481943227, ISBN-13: 978-1481943222.