Model of photoemission and framework for relating quantum efficiency to stoichiometry Open Access

The recommendations of several extended reports and studies on photocathode research and development (R&D) have identified progress in higher performance photocathode materials for photoinjectors used in particle accelerators and x-Ray Free Electron Lasers (xFEL’s) as a critical priority. These reports strongly argue that “photocathode materials by design,”¹ also known as “Designer Photocathodes,”² intend to exploit engineered control of the stoichiometry and structure of photocathode materials, which may enable a stronger bonding of the cesium surface layer of high quantum efficiency (QE) photocathodes or even enable achieving an NEA surface through delta-doping techniques.³ The advancements are needed in order to optimally control laser absorption, electron transport, and photo-excited electron emission in the generation of high brightness electron beams needed to enable next-generation capabilities called for by the Department of Energy’s (DOE) roadmap.¹ Graded stoichiometries, in particular, are presented as a means to modify the cathode optical and transport properties. Specifically, the resulting very short wavelength light for extreme ultraviolet (EUV) lithography using a Free Electron Laser (FEL), in particular, will enable opportunities in semiconductor fabrication,⁴ a technological advancement with momentous consequences.

Alkali antimonides, such as Cs $3$Sb,⁵ K $2$CsSb,⁶ and Na $2$KSb,^7–9 have high QE, with the latter having reasonably good operation at elevated temperatures. Advances in understanding have moved the photoinjector community away from so-called “recipe-based” production to techniques¹⁰ that instead capitalize on materials and solid-state physics understanding of properties¹¹ and factors that affect performance¹² and allow for automated growth techniques.¹³ Doing so may allow for mitigating trade-offs between transverse emittance $ε n , r m s$ (alternately mean transverse energy or MTE) and the quantum efficiency (QE) of the material,^1,3,14–16 both of which increase in tandem, e.g., $ε n , r m s 4∝QE$ for metals, such as copper.^17,18 The processes responsible are also related to a correlation between response time and quantum efficiency^19–21 due to processes, such as scattering, during transport and the nature of the emission barrier,^22,23 considerations that also affect spin polarized sources for similar reasons.²⁴ Methods to quantify behavior of photoemissive materials that differ by small changes in stoichiometry are desirable for characterizing fabricated photocathodes and to enable reproducible and automated growth of them. The present study is to develop and vet a numerically agile means to characterize changes in the optical properties of stoichiometrically altered materials in a manner that will additionally be useful when simulations of their performance in photo-injectors performed with beam optics codes require flexible and computationally rapid methods to calculate critical quantities related to QE that govern the generation of high bunch charge electron beams.

Absorption bands in the experimental optical spectra of materials are associated with the resonance levels of atoms that can be associated with multiple oscillators of varying strength.^25–27 These resonances likewise appear in simulations using density functional theory.^28–30 The dielectric properties of the materials, in particular, are essential to calculate laser penetration depth and reflectivity¹⁷ in addition to accounting for complicating features, such as film thickness^24,31 or features in the density of states.³⁰ Another crucial component in the calculation of quantum efficiency is modeling the losses due to scattering processes as photoexcited electrons travel to the surface and are emitted, for which Monte Carlo methods are the preferred approach for semiconductors³² when scattering is not necessarily fatal to emission, especially for negative electron affinity (NEA) surfaces.^24,33,34 Such numerically intensive models are necessary even for otherwise simple materials, such as metals,³⁵ which exhibit complex wavelength-dependent properties (in addition to electron–electron scattering^34,36) that require fidelity to the optical and dielectric functions in predictions of QE. These high fidelity models, however, are better used to vet and justify emission simulation codes designed to provide computationally rapid methods of predicting quantum efficiency of photocathodes^2,34,37 in particle-in-cell codes in which millions of emission cells tiling a photocathode surface require QE evaluations over millions of time steps in a typical simulation of an electron bunch.^34,38 Such challenges are compounded if delayed emission occurs.²¹ Problems are further magnified if compound semiconductors and materials^3,39 have stoichiometries that vary, as it is expected that the optical constants will thereby vary and so exacerbate the difficulty in properly accounting for their properties in simulation codes that put an emphasis on rapid and flexible calculation methods to model beam formation and energy characteristics.

The present work is the first of a two-part effort to relate changes in stoichiometry to experimentally measurable quantities that characterize photocathode performance, but to do so in a theoretically controllable manner. The first part is the development of a methodology to characterize emitters when their stoichiometry can result in a change of quantum efficiency while at the same time provide a computationally agile means to calculate the optical parameters for advanced simulation codes^{34,37,38,40,41} modeling photocathode performance and predicting beam properties in photoinjectors. The second part is to apply these methods to the predictions of DFT applied to multialkali antimonide photocathode materials with variations in their stoichiometry so as to demonstrate the efficacy of the method for characterization efforts measuring quantum efficiency. The present study is, therefore, structured as follows. First, a description of various photoemission theory models is given to identify how the optical constants are utilized. Next, the complexity of the optical constants and their relation to reflectivity and laser penetration depth are applied to a common photocathode material, copper, even though copper photocathodes are not the intended material on which follow-on studies will be based. To be clear, copper is not the target material for the development of present methods. Rather, copper is a well-understood and characterized material for which optical data^42–45 and theoretical treatments^46,47 are available over an extensive range, and it, therefore, serves as an ideal physical material on which to demonstrate the application of the method developed in the present part of the study, in contrast to the ad hoc optical model on which the developed procedures are vetted. The intent is to incrementally show the value of the methodology by first applying it to a more familiar and easier (metallic) material. We reiterate that the target materials for our studies are the multi-alkali antimonides, as it is their stoichiometric composition that motivates the present study. Last, a rapid and simple model of the dielectric function, on which reflectivity and laser penetration depth, in particular, depend upon, is given and shown to provide a means to relate variations in the latter to changes in the parameters used to model the former. The model is designed to be computationally rapid and simple so as to meet the needs of both photoemission simulation and experimental characterization.

When Eq. (12) is used to characterize experimental photoemission data, then a single set of parameters matches reported data even though the authors of different data sets may not agree on particular parameters (e.g., $E g=1.6$ and $E a=0.4$ eV characterize the Cs $3$Sb data using the present theory of both Spicer⁵⁷ and Taft and Philipp,⁵⁸ even though those authors suggest different values). Furthermore, the separation of $h(s)$ from the parameters making up the coefficient of $QE$ means that finding the impact on the optical constants on $QE$ has now become simply a requirement to find out how $R(ω)$ and $δ(ω)$ are affected, as $h(s)$ remains unchanged. Consequently, to examine the presumptive impact of stoichiometry on $QE$⁠, it is enough to examine the variations that affect $[1−R(ω)]/[1+p(E)]$ in Eq. (12).

To prevent notational difficulties in what follows, the energy $E$ will be taken as the parameter of interest describing the photoexcited electron, and so, the wave number will be designated $k(E)$ when it appears. When by itself, $k$ will be the imaginary part of the complex index of refraction, or $ℑ( n ^)=ℑ(n+ik)=k$ below.

Using the well-known case of copper, the effect of subtracting off the Drude and Lorentz parts is readily visible in Fig. 3(a) based on $μ=7$ eV. Representative estimates of the other parameters from Table I result in the Resonant terms of Fig. 3(b). It is seen first that the Drude and Lorentz contributions are fairly well specified by the behavior of $K i$ near the origin and the plasma frequency $ℏ ω p$ and second that the peaks in the cumulative resonant terms are individually well modeled even by ad hoc values determined by trial and error. Such an ad hoc prescription to determine all of the Lorentzian parameters of Eq. (25) is inadequate for the needs of simulation: the damping terms $Γ j$ vary considerably and so affect the size of $f j$ and the location of $ω j$⁠. Methods to determine “best fit” parameters for a collection of curves of a given formula to a curve based on their sum exist as part of high performance packages (e.g., MATLAB), but first, such methods do not lend themselves to ready manipulation in a manner that addresses the needs of rapid characterization and alteration of parameters to reveal sensitivities to particular resonances, as will be needed when attributing changes to—and possibly subtle variations in—stoichiometry; and second, they tend to produce a large collection of Lorentzians (with both positive and negative weight factors $f j$⁠) that are antithetical to generating a characterization/simulation protocol. The presence of negative weighting factors, in particular, reported in an earlier analysis,³¹ is a subtle indication that the numerically determined “best fit” damping factors may be too broad (or narrow) and so have to be corrected by ad hoc offsets that result in a comparatively unwieldily number of resonant terms, a consequence of addressing resonances sequentially rather than en masse.

Although optical data for photoemitter materials, such as copper, the alkali and multi-alkali antimonides, or specific perovskites³¹ are available either through measurement or density functional theory (DFT) simulations, such data cannot suffice to vet the theoretical methodology because the actual ${ f j,ℏ ω j, Γ j}$ parameters for measured/simulated data are unknown, and so data fitting may fall short for assessing the accuracy of algorithms. To justify the methods to be presented, a theoretical model with specified parameters that result in a theoretical $K i$ behavior is preferable. Its utility for parameter determination is then judged by how well known values are returned: an analytical test case quantifies its accuracy and limitations. A particular difficulty to focus on is accounting for or resolving closely spaced resonant terms with differing features. To that end, a model is formed from ad hoc ${ f j,ℏ ω j, Γ j}$ to create a $ΔK$ obtained from a sum of Lorentzians over a range of $ℏω$⁠.

A methodology is sought that is rapid, flexible, easily administered without requiring specialized “best fit” procedures, and so amenable to adjustments that offset noisy data, particularly to achieve a manageable number of parameter sets that are flexibly manipulated. To vet that model, a means to assess accuracy in order to quantify the impact of changing a resonant term (the hypothetical consequence of altering stoichiometry) is made possible by creating an analytic model against which the predicted parameters can be compared. Six resonances with the parameters given in Table I and represented in Fig. 4 are used to create $ΔK= K i− K d− K l$⁠. Particular features are purposely introduced, such as narrow and broad peaks or “problematic” closely spaced resonant terms $ℏ ω 2$ and $ℏ ω 3$⁠.

The model parameters are found as follows. The pseudo-experimental data, called Actual, is mapped over to uniformly spaced values $[ℏ ω l,Δ K l]$⁠, e.g., $ℏ ω l=lℏ ω max/N$⁠, where $N$ is chosen to be suitably large and $l∈(0,1,…N)$⁠. To indicate “data,” the $ℏω$ values will use a $l$-subscript to reference one of the uniformly spaced values, but a $j$-subscript to reference one of the $N r$ resonances, with $N r=6$ for the present example. When examining with actual data, the values of $ℏ ω l$ can be spline-interpolated from data that are non-uniformly spaced to fulfill this condition if the data are, in fact, measured and, therefore, non-uniform. Uniformity enables employing finite difference methods to approximate derivatives.^18,67 The peaks of the $l$-data set are found and presumed to lie near a resonance $ℏ ω j$⁠. An estimate of $Γ j$ is made from the curvature near that peak. The Prediction stage is an automated hunting for peaks in $ΔK$ and their parameterization; the Correction stage is to examine those peaks interactively and determine modifications to the damping width term $Γ j$ and the location of $ω j$ so as to better model the behavior of $K i(ω)$⁠, with the intent of finding reasonable weighting factors $f j$⁠.

The designation of the weighting factor $f j ∗$ with an asterisk is to emphasize that these estimates are only provisional. The presence of other resonances alters them, thereby necessitating a correction to the parameters. The means to do so are taken up next.

Having now developed a method of reasonable accuracy to characterize a material by its optical constants, attention is now turned to examining how variations in those constants reveal themselves in the reflectivity $R ( ω )$ and the penetration depth parameter $δ ( ω )$⁠, which are obtained as follows. The complex index of refraction $n ^=n+ik$ is found by Eq. (17). From $n ( ω )$ and $k ( ω )$⁠, $δ ( ω )$ and $R ( ω )$ are subsequently evaluated from Eqs. (15) and (16).

On the hypothesis that a change in stoichometry would change one (or more) of the parameters $( ω j , Γ j , f j )$⁠, one Lorentzian in the optical region was varied, with the remaining parameters held to the Actual parameters of Table II. The varied Lorentzian was chosen to be (⁠ $n=2$⁠), and only one of the three parameters was varied at a time. The consequences are shown in Fig. 7. The magnitude of variation depends on the magnitude of the parameter and its difference with Table II values for the $j=2$ terms. We emphasize that the variations chosen are to make the consequences readily visible in the figure and do not correspond to changes anticipated for variations in lattice constants resulting from changes in stoichiometry. Rather, what is of importance here is in what manner the variation occurs. Changes in $ω 2$ cause a local shift of the $( R , δ )$ lines; changes in $Γ 2$ affect the sharpness of the peaks nearest to the $j=2$ resonance; and changes in $f j$ cause an overall shift (but not a constant one) in the magnitude of the $( R , δ )$ curves. Consequently, the extraction of the $( ω j , Γ j , f j )$ parameter from two different samples may provide, by way of variations in how a subset of their parameters behaves, an indication of a dependence on stoichiometry. Such an analysis shall be presented separately. In anticipation of it, observe the addition of cesium to Na $2$KSb to create (Cs)Na $2$KSb changes the lattice constant from 0.7727 to 0.7745 nm⁶⁸ with similar changes in related materials, such as the cubic K $2$CsSb, which is stable under alkali excess;⁶⁹ changes in stoichometry result in bandgap changes, such as $E g=0.57$ eV for Cs $2$KSb but 1.62 eV for Cs $2$KSb;⁷⁰ and DFT is a useful means of discerning the lattice parameters of such materials, e.g., predicting a lattice parameter of 0.875 87 nm in good agreement with the experiment for KCsSb photocathodes.⁷¹ Therefore, DTF already reveals a discernible change in parameters characterizing crystal lattices when stoichometry is varied, supporting the expectation that it will likewise reveal changes in optical constants.

The optical terms $R ( ω )$⁠, $δ ( ω )$⁠, coupled with a model for the relaxation time $τ ( E )$ based on electron–electron and electron–phonon scattering,^{18,27,34,52,72–74} the coefficient of $QE ( ω )$ [Eqs. (7) and (8)] can be evaluated straightforwardly. The optimized set of extracted Resonant parameters then enables a refined estimate of quantum efficiency using theoretical means. More complex cases, such as photoemission from, e.g., alkali antimonides^5–9 or perovskites,³⁹ may be complicated by the demand for more resonance terms, but the procedure is expected to be similar,³¹ as shall be undertaken in a subsequent study. In that study, the $K i ( ω )$ of Cs $3$Sb for differences in atomic ratios of Cs to Sb (close to 3:1) will be related to the changes in the optical parameters, which will then be related to changes in QE through the alterations of $R ( ω )$ and $δ ( ω )$ in Eq. (12) as anticipated by Fig. 7. An example of the DFT results for Cs $3$Sb evaluated using DFT is shown in comparison with copper in Fig. 8. It shows a more nuanced structure over a greater photon energy range than copper, a feature that is believed will facilitate the relation of QE variation to stoichiometric changes.

The dielectric functions of photoemissive materials are a key component of theoretical models to predict their quantum efficiency as they give rise to the reflectivity $R ( ω )$ and laser penetration depth $δ ( ω )$ parameters needed by moments-based³¹ or three-step models.¹⁷ Using copper as a motivational example, a Lorentz–Drude–Resonant (LDR) model was shown to be compatible with a description of the imaginary part of the dielectric function $K i$⁠. A method of modeling the optical properties with a small number of LDR parameters (weight $f j$⁠, resonant frequency $ω j$⁠, and damping factor $Γ j$⁠) was vetted on an exact and ad hoc model that replicated challenges such parameter extraction models face and shown to meet the needs of modeling, simulation, and characterization. It was then shown how small changes in those parameters give rise to variations in $R ( ω )$ and $δ ( ω )$ in a manner that, it is argued, will be of use in the characterization of measured sample data that may differ by stoichiometry, or in the parameterization of similar data resulting from a density functional theory (DFT) analysis of the optical terms, studies that shall be taken up separately: DFT is a promising method to ascertain how changes of atoms in the unit cells are reflected to changes in the $( ω j , Γ j , f j )$ terms.

K.L.J. and P.G.O. gratefully acknowledge support provided by the Los Alamos National Laboratory (LANL). This work is supported by the U.S. Department of Energy, Office of Science, Office of Accelerator Research & Development and Production (ARDAP) under Triad National Security, LLC (“Triad”) Contract Grant No. 89233218CNA000001, FWP: LANLE6E7.

The authors have no conflicts to disclose.

Kevin L. Jensen: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead). Dimitre Dimitrov: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Funding acquisition (equal); Investigation (supporting); Methodology (supporting); Validation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Vitaly Pavlenko: Conceptualization (supporting); Funding acquisition (lead); Investigation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Patrick G. O’Shea: Conceptualization (supporting); Funding acquisition (supporting); Investigation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available within the article.