Simulating the quantum efficiency (QE) from photocathodes used in accelerators and photoinjectors requires accounting for the properties of the photoemissive material, the optical properties, scattering coefficients, and doping concentrations of which are dependent upon the stoichiometry. We present a rapid and flexible optical model that can be used to investigate the consequences of changes in the dielectric properties and their impact on the QE through such factors as reflectivity and laser penetration depth. Differences in materials can then be characterized by changes to parameters used to evaluate the dielectric function in a Lorentz–Drude–Resonant model. A method to characterize data is motivated by the example of copper and vetted by application to an exact analytical model. The effects of changes in parameters describing the resonant terms, which aggregate in the visible and UV regions of the spectrum, are shown.

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 3Sb,5 K 2CsSb,6 and Na 2KSb,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 Q E for metals, such as copper.17,18 The processes responsible are also related to a correlation between response time and quantum efficiency19–21 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 thickness24,31 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 scattering34,36) 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 photocathodes2,34,37 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 materials3,39 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 codes34,37,38,40,41 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 data42–45 and theoretical treatments46,47 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.

Theoretical models of quantum efficiency often focus on near threshold conditions, for which the Fowler–DuBridge relation48 is the best known one. Its predictions (made using a Sommerfeld model of an electron gas in a metal) utilize similar models behind both thermal and field emission problems, and in fact, all three can be limiting cases of the general Thermal-Field-Photoemission (GTFP) equation18,49–52 for current density. Introduce the following parameters: T is the temperature, β T = 1 / k B T, β F is the slope of the Gamow tunneling factor, ϕ is the effective barrier height above the Fermi level (Schottky lowered work function), ζ ( 2 ) is the Riemann zeta-function, and other terms have their usual meanings. In terms of them,
(1)
which accounts for field and temperature dependent modifications contained in the second term. Equation (1) gives a good account of Q E from metals away from threshold,53 but other effects influenced by the physics below cause departures at higher photon energies ( ω).
The Three-Step Model (TSM) of photoemission10,46,54–56 was instrumental in developing a predictive model applicable to both metals and semiconductors, taking into account optical properties, transport issues, and surface conditions. It separated the processes behind quantum efficiency evaluation into
  • absorption of the incident photon,

  • transport of the photoexcited electron to the surface, and

  • emission of the photoexcited electron past the surface barrier.

A similar framework that cleaves to the same three-part structure, but one based on an integration over momentum k z rather than normal energy E, and so in keeping with quantum mechanical models of emission through and over a barrier, was based on integrands that were products of powers of k z and k with the putative distribution function of emitted electrons to describe Q E and ε n , r m s. It is, therefore, called the Moments Model (MM),18,31,50 the predictions of which largely match the TSM approaches even if details differ depending on models behind parameters, such as electron mean free path, scattering rates, optical constants, and transmission probabilities. The treatment herein will adhere to the conventions established in the MM approaches. The simple moments model31 integral for the current density is
(2)
where the symbols refer to
  1. Absorption ( A), accounting for reflection R ( ω ) and attenuation into the photoemissive material governed by δ ( ω ),

  2. Transport ( T) of an electron having absorbed a photon of energy ω and undergoing scattering events characterized by a scattering time τ and losses factor f λ, and

  3. Emission ( E) over a barrier or surface structure governed by a transmission probability D ( k ).

The number of photo-excited electrons is J Δ t A o / q, where Δ t is the duration of the pulse, A o is the illumination area, and q is the magnitude of the electron charge. The number of photons is I ω Δ t A o / ω, where I ω is the laser intensity. Quantum efficiency is, therefore, the ratio of the first to the second and is simplified if emission is prompt and only tracks the laser pulse, in which case J / q Q E × ( I ω / ω ), or
(3)
where the moments M n ( k j ) are specified by
(4)
where d 3 k is the wave number volume element in its usual representation, d 3 k = 2 π k 2 sin θ d k d θ, k j is a Cartesian component of k, e.g., k z = k cos θ, and a factor of 2 in the numerator of the coefficient accounts for electron spin. The usage of k will pose difficulties due to its potential confusion with the optical attenuation parameter k associated with the imaginary part of the index of refraction n ^ = n + i k. Observe the convention here: complex quantities, such as n ^, are so indicated by the presence of the hat, or circumflex above the letter. To forestall this confusion, what k represents will be described in each section it is used.
In the present section, k is taken to be a wave number so that k is a momentum and E = 2 k 2 / 2 m is energy parabolic in momentum. To leading order, the transmission probability D ( k z ) behaves as a step function such that 2 k z 2 / 2 m > μ + ϕ for emission from metals where μ is the Fermi level and ϕ = Φ 4 Q F is the Schottky lowered work function: that approximation leads to the Simple Moments Model (SMM). Semiconductors follow a similar equation but use an electron affinity E a and bandgap E g. In polar coordinates where θ is measured from the normal to the surface, then k z / m = ( k / m ) cos θ. The maximum angle θ m for which emission occurs is then
(5)
(6)
For NEA photocathodes, E a < 0, and so, θ m = π / 2. When the SMM is restructured in terms of E and x m = cos θ m, it becomes
(7)
(8)
(9)
where Δ m = μ + ϕ ω for metals and Δ s = ω E g for semiconductors, y is a dummy integration variable, and x m = cos θ m. Let the laser penetration depth be governed by the depth parameter δ ( ω ) such that the intensity decays into the material as exp ( z / δ ), where z is the distance from the surface. The loss factor for thick materials is then given by
(10)
where p ( E , ω ) = δ ( ω ) / [ v ( E ) τ ( E ) ], where v ( E ) = k / m is the velocity, E ( k ) = 2 k 2 / 2 m is the energy, and τ ( E ) is the mean time between scattering events at that energy: v τ is, therefore, the mean free path, and p is the ratio between the depth parameter and the mean free path (modifications are required if the material is thin). It can be shown31 that approximating the transmission probability by unity within the limits of the energy integral (that is, considering only above barrier emission for the Schottky–Nordheim barrier and setting its transmission probability to unity) gives
(11)
where the second line (approximation) is in the Fowler–DuBridge limit of ( ω ϕ ) 0, commensurate with Eq. (1) in the low temperature limit. In characterizing quantum efficiency, it is common to treat R ( ω ) and p ( μ ) as approximately constant (particularly in simulation), given that as a practical matter, photocathodes are operated at the longest permissible laser-generated wavelength and so often threshold conditions are sought, and/or high photon frequency optical data are not always available. The consequence of ignoring variation in the optical terms contributes to the departures visible at high photon energy, as in Fig. 1 for copper, where two forms of Eq. (11) are shown and compared to experimental data.55 
FIG. 1.

Performance of Eq. (11) theory compared to copper photoemission data55 (courtesy of D. Dowell, SLAC). The reference point is ( ω o , Q E ( ω o ) for ω o = 4.88 eV.

FIG. 1.

Performance of Eq. (11) theory compared to copper photoemission data55 (courtesy of D. Dowell, SLAC). The reference point is ( ω o , Q E ( ω o ) for ω o = 4.88 eV.

Close modal
Performing a similar analysis for semiconductors ultimately result in53 
(12)
where s ( ω E a E g ) / E a and p = p ( E a ). The denominator corrects an erroneously neglected 2 on ( s + u ) 2 in the presentation of Q E s e m i in earlier works.2,18,31 Equation (12) is notable because it separates the quantum efficiency into a product of two terms: h ( s ) governs the impact of the surface barrier E a and the bandgap E g compared to the photon energy ω. The coefficient in square brackets, on the other hand, is dependent on the dielectric properties of the material, particularly its reflectivity and laser penetration depth. As a result, the optical constants primarily impact the coefficient. By contrast, the behavior of h ( s ) is parametrically simple, governed as it is by quantities that do not depend on the laser frequency apart from the photon energy.
The function h ( s ) has asymptotic limits given by
(13)
These limits are seen to be in contrast to a relation suggested by Spicer57 that can be recast as53 
(14)
where the factor of 5 in the denominator gives a reasonable overlap with Eq. (13) and is obtained by requiring h s p ( 7 / 3 ) h ( 7 / 3 ) for a representative point ( s = 7 / 3 ) in the knee of the curve. The curve labeled “mid” corresponds to the mid-range fit h ( s 1 ) and is based on h ( 1.6 ). In both Eqs. (13) and (14), values of h ( 1 ) = 0.035 533 9, h ( 1.6 ) = 0.083 922 9, and h ( 4 ) = 0.178 186 are from Eq. (12). A comparison of h ( s ) with its small, mid-range, and large s asymptotic expansions, as well as with Spicer’s relation, are shown in Fig. 2.
FIG. 2.

The function h ( s ) [Eq. (12)] for the calculation of QE. (a) h ( s ) compared to its small (blue), large (green), and mid-range expansions (gray dashed) of Eq. (13). Also shown is the representation due to Spicer [Eq. (14), red thick]; (b) Same as (a) but on a log–log scale. Green and yellow dots are the locations used to generate mid-range approximation at s = 8 / 5 and Spicer approximation h s p ( s ) at s = 7 / 3, respectively.

FIG. 2.

The function h ( s ) [Eq. (12)] for the calculation of QE. (a) h ( s ) compared to its small (blue), large (green), and mid-range expansions (gray dashed) of Eq. (13). Also shown is the representation due to Spicer [Eq. (14), red thick]; (b) Same as (a) but on a log–log scale. Green and yellow dots are the locations used to generate mid-range approximation at s = 8 / 5 and Spicer approximation h s p ( s ) at s = 7 / 3, respectively.

Close modal

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 3Sb 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 Q E means that finding the impact on the optical constants on Q E 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 Q E, 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 ( n ^ ) = ( n + i k ) = k below.

The complex index of refraction n ^ determines two crucial optical parameters for cubic materials considered herein and the multialkali antimonides59 to be considered separately (which do not exhibit anisotropy in the complex dielectric function): the reflectivity R ( ω ) and the penetration depth parameter δ ( ω ) (also known as the optical attenuation length17 or characteristic penetration depth/inverse absorbance27) governing how light is attenuated in a material. The latter is found from how the electric field E ( x , t ) component of the photon declines with distance and is31,
(15)
The former leads to Beer’s equation,26,27,60
(16)
The factors n and k are both deduced from the complex dielectric function ε ^ K ^ ε 0 by solving the equations
(17)
where K ^ = K r + i K i. It appears that K r and K i must be measured separately, but in fact, they are related by the Kramer–Kronig relations,61,
(18)
where P ( ) is the principal part of the integral, and the same equation holds after replacing K r K i. As a result, only measurements of K i ( ω ) are required to determine both. Those measurements, combined with treating electrons in materials as free and bound particles undergoing damped motion in response to a force F ( t ) generated by an oscillating electric field, provide a parameterized model that allows for good representations of K i ( ω ). The damped harmonic oscillator equation is for motion along the z ^ axis,
(19)
where τ is a relaxation time constant in the Drude–Zener theory,62, ω o is the optical frequency, and the three terms on the right hand side correspond to acceleration, dampening, and restoring forces, respectively. A Fourier transform renders z ( t ) as
(20)
If the complex index of refraction is rendered as n ^ ( ω ) 2 1 + χ ^ ( ω ), where χ ^ is the susceptibility arising from the relationship between the macroscopic polarization and the electric field63 given by P = ε 0 χ ^ E. Let ρ o be the (number) density of electrons in the material. It then follows18,
(21)
where Γ 1 / τ, and the plasma frequency ω p is given by
(22)
e.g., for copper ( μ = 7 eV), then k F = 2 m μ / = 13.5546 [1/nm], ρ o = k F 3 / 3 π 2 = 84.1089 [1/nm 3 ], and ω p is as given in Table I. The form of Eq. (21) is the basis of introducing K ^ = 1 + K ^ f + K ^ b to account for both free and bound electrons.26,31,64–66 The free or Drude part for metal-like behavior takes ω o ω d = 0. The bound or Lorentz part for insulator-like behavior considers a large ω o ω l. In the visible part of the spectrum, numerous peaks in K i ( ω ) are due to terms resembling Eq. (21). And so, the Lorentz–Drude–Resonant model represents the dielectric function as K ^ ( ω ) = 1 + χ ^ f ( ω ) + χ ^ b ( ω ) where
(23)
(24)
(25)
where ( D , L , R ) subscripts are for (Drude, Lorentz, and Resonant) respectively, f j is the weighting factor, and ω j and Γ j for j ( 1 , , N b ) are interband transition frequencies and damping rates, respectively. In a small departure from prior methods,31 the Drude parameters are subscripted by “ D” and the Lorentz parameters by “ L.” Subtracting the LD model [constructing K i ( ω ) from Eqs. (23) and (24) only] from measured data gives an indication of the contribution of Eq. (25) to the optical parameters.
TABLE I.

Copper parameters used for the creation of Fig. 4 lines based on aggregated data for Copper. “D” refers to Drude, L to Lorentz, and number indices to Resonant parameters; ω p is the plasma frequency. R-parameters are ad hoc, not using a numerical procedure but rather fitted by eye. Units of ω j and Γj are [eV]; fj is dimensionless.

Lorentz, Drude, and plasma
Termj = Dj = Lj = p
ω j … 54.1756 10.7691 … … 
Γj 0.091 241 87.8489 … … … 
fj 0.735 981 8.972 11 … … … 
Lorentz, Drude, and plasma
Termj = Dj = Lj = p
ω j … 54.1756 10.7691 … … 
Γj 0.091 241 87.8489 … … … 
fj 0.735 981 8.972 11 … … … 
Resonant
Termj = 1j = 2j = 3j = 4j = 5
ω j 2.60 5.10 8.90 14.5 26.0 
Γj 2.00 3.10 5.00 10.0 6.00 
fj 0.18 0.55 0.30 1.30 0.50 
Resonant
Termj = 1j = 2j = 3j = 4j = 5
ω j 2.60 5.10 8.90 14.5 26.0 
Γj 2.00 3.10 5.00 10.0 6.00 
fj 0.18 0.55 0.30 1.30 0.50 

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 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.

FIG. 3.

Copper: (a) Imaginary part of the dielectric function ( K i ): aggregated measured data42,43,64 from the literature is black , the blue line is the Drude (D) model based on Eq. (23), the green line is the Lorentz (L) model based on Eq. (24), and the red line is the sum (D+L). (b) K i K d K l Δ K is the aggregated measured data minus the LD fit and shown as black . Lines are ad hoc peaks adjusted by hand for five different resonant peaks listed in Table I.

FIG. 3.

Copper: (a) Imaginary part of the dielectric function ( K i ): aggregated measured data42,43,64 from the literature is black , the blue line is the Drude (D) model based on Eq. (23), the green line is the Lorentz (L) model based on Eq. (24), and the red line is the sum (D+L). (b) K i K d K l Δ K is the aggregated measured data minus the LD fit and shown as black . Lines are ad hoc peaks adjusted by hand for five different resonant peaks listed in Table I.

Close modal
FIG. 4.

Ad hoc theoretical model to create Actual data for N b = 6 resonances after Eq. (25) and their sum to create Δ K. Parameters are in Table II under Actual heading. Data shown are reproduced in Fig. 5 for comparison purposes.

FIG. 4.

Ad hoc theoretical model to create Actual data for N b = 6 resonances after Eq. (25) and their sum to create Δ K. Parameters are in Table II under Actual heading. Data shown are reproduced in Fig. 5 for comparison purposes.

Close modal
FIG. 5.

Successive removal of individual Lorentzians after the finding of the y n maximums, where y j is Δ K y j before the removal of the ( ω j , Γ j , f j ) Lorentzian. Large filled symbols (e.g., ) atop gray dashed lines show the location of the maximum peak.

FIG. 5.

Successive removal of individual Lorentzians after the finding of the y n maximums, where y j is Δ K y j before the removal of the ( ω j , Γ j , f j ) Lorentzian. Large filled symbols (e.g., ) atop gray dashed lines show the location of the maximum peak.

Close modal

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 , 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 , ω j , Γ j } to create a Δ K obtained from a sum of Lorentzians over a range of ω.

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 Δ K = K i K d K l. Particular features are purposely introduced, such as narrow and broad peaks or “problematic” closely spaced resonant terms ω 2 and ω 3.

The model parameters are found as follows. The pseudo-experimental data, called Actual, is mapped over to uniformly spaced values [ ω l , Δ K l ], e.g., ω l = l ω max / N, where N is chosen to be suitably large and l ( 0 , 1 , N ). To indicate “data,” the ω 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 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 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 Δ K 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.

1. Prediction stage

Create x l = ω l and y ( x l ) = Δ K ( ω l ). Let δ ω m a x / N [recall that there are ( N + 1 ) points if l = 0 is included in the allowed indices]. The largest value of Δ K l is found, and its index is designated n, or Δ K n = max ( Δ K l ). Derivatives of y ( x ) at x = x n are given by
(26)
Observe that 2 δ = x n + 1 x n 1. Designate the local maximum near y ( x n ) to be at x m, where d y / d x | x = x m = 0. It follows from a Taylor expansion of y ( x ) that
(27)
which allows the parameters of the j t h Lorentzian to be determined from
(28)
In practice, this procedure produces Γ j that are slightly large because of neglecting consideration of the other Lorentzians that can infect the estimate. A method to offset that tendency is to replace Γ j λ Γ j, where λ is of order unity, and a value of λ 0.90 is the value of use here. The j t h peak is then removed by
(29)
where the fraction being removed is the imaginary part of a Lorentzian in the sum of Eq. (25). The process is repeated until all the positive peaks have been eliminated to a level deemed sufficient (negative peaks are dealt with in during the Correction phase), generally signaled by when the predicted weighting factor f j is small. The number of times the process is iterated is taken as a measure of N r. The resonances are sorted to ensure that ω j < ω j + 1. Doing so on the Actual example of Fig. 4 produces a sequence of curves shown in Fig. 5.

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.

2. Correction stage

The Prediction stage generally encounters some Lorentzian components that are broad ( Γ j is in some sense large), and their broadness will contribute to peaks that the Prediction stage for finding f j does not account for. Therefore, erroneous peaks (even negative ones) can be introduced. The Correction stage is an effort to eliminate them and provide better correspondence with the Actual data. Multiplicative weighting factors are appended to each one of the ( ω j , Γ j ) pairs, or ( ω j , Γ j ) ( ϵ j ω j , g j Γ j ), where the correcting scale factors ( ϵ j , g j ) = ( 1 , 1 ); that is, they are initialized to unity. The peaks are then examined sequentially: if the n t h resonant term is examined, then ( ϵ n , g n ) is adjusted until the shape of the prediction better matches the pseudo-data in the vicinity of ω n. After each modification of a pair, all the f j terms are recalculated according to the solution of a matrix equation given by
(30)
The judgment of a “better match” is necessarily a visual one, and the adjustment of the ( ϵ n , g n ) is by hand. Even so, the process is relatively rapid, although its potential for automation has not been explored. Adjusting the ( ϵ j , g j ) pairs in pursuit of visual agreement, the choices,
(31)
provide reasonable agreement, resulting in the Correction parameters of Table II and the Correction subplot of Fig. 6(b). The Correction values are not identical to the Actual values, nor are they expected to be given that changes are made sequentially instead of en masse, but their reasonably close correspondence gives an indication of the utility of the procedure.
FIG. 6.

(a) Prediction algorithm results for the determination of the Lorentzian parameters of Fig. 4. A log plot brings out differences between the Lorentzian components with the Actual model. Thin dashed lines bounding filled color areas are the Actual curves with their sum shown by open ( °), corresponding to the curves shown in Fig. 4. Thick solid lines (no fill) and black ( ): results of the Prediction algorithm. Prediction parameters used to create solid lines given in Table II. (b) Same, but now using the Correction algorithm results.

FIG. 6.

(a) Prediction algorithm results for the determination of the Lorentzian parameters of Fig. 4. A log plot brings out differences between the Lorentzian components with the Actual model. Thin dashed lines bounding filled color areas are the Actual curves with their sum shown by open ( °), corresponding to the curves shown in Fig. 4. Thick solid lines (no fill) and black ( ): results of the Prediction algorithm. Prediction parameters used to create solid lines given in Table II. (b) Same, but now using the Correction algorithm results.

Close modal
TABLE II.

Actual (ωj, Γj, fj) parameters (ad hoc) used to create ΔK in Fig. 4. Prediction parameters are returned from the peak-search algorithm. Correction parameters are obtained by using ϵ j , g j modifications to the Prediction (ωj, Γj) parameters as per Eq. (31).

Actual
Termj = 1j = 2j = 3j = 4j = 5j = 6
ω j 3.1 4.9 6.3 9.4 15.1 18.8 
Γj 1.5 0.8 3.1 4.5 5.3 6.1 
fj 0.2 0.3 1.3 0.7 1.1 0.3 
Actual
Termj = 1j = 2j = 3j = 4j = 5j = 6
ω j 3.1 4.9 6.3 9.4 15.1 18.8 
Γj 1.5 0.8 3.1 4.5 5.3 6.1 
fj 0.2 0.3 1.3 0.7 1.1 0.3 
Prediction
Termj = 1j = 2j = 3j = 4j = 5j = 6
ω j 2.985 5.029 6.528 9.239 14.94 18.49 
Γj 1.315 1.641 2.575 4.161 4.959 5.522 
fj 0.1438 0.7381 0.803 0.6628 1.080 0.3875 
Prediction
Termj = 1j = 2j = 3j = 4j = 5j = 6
ω j 2.985 5.029 6.528 9.239 14.94 18.49 
Γj 1.315 1.641 2.575 4.161 4.959 5.522 
fj 0.1438 0.7381 0.803 0.6628 1.080 0.3875 
Correction
Termj = 1j = 2j = 3j = 4j = 5j = 6
ω j 3.045 5.029 6.463 9.239 14.94 18.49 
Γj 1.289 1.149 3.606 4.161 5.207 6.626 
fj 0.1541 0.4780 1.282 0.5304 1.122 0.3916 
Correction
Termj = 1j = 2j = 3j = 4j = 5j = 6
ω j 3.045 5.029 6.463 9.239 14.94 18.49 
Γj 1.289 1.149 3.606 4.161 5.207 6.626 
fj 0.1541 0.4780 1.282 0.5304 1.122 0.3916 

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 + i k 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 2KSb to create (Cs)Na 2KSb changes the lattice constant from 0.7727 to 0.7745 nm68 with similar changes in related materials, such as the cubic K 2CsSb, which is stable under alkali excess;69 changes in stoichometry result in bandgap changes, such as E g = 0.57 eV for Cs 2KSb but 1.62 eV for Cs 2KSb;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.

FIG. 7.

Reflectivity R ( ω ) ( °) and penetration depth δ ( ω ) ( ) as a function of photon energy ω using the Actual factors of Table II: so as to have both on the same plot, δ has been scaled by a factor of 1 / 20. The values of ( ω 2 , Γ 2 , f 2 ) were sequentially varied. Holding all other factors constant: R ω and δ ω used ω 2 = 5.3 eV; R g and δ g used Γ 2 = 1.1 eV; and R f and δ f used f 2 = 0.5.

FIG. 7.

Reflectivity R ( ω ) ( °) and penetration depth δ ( ω ) ( ) as a function of photon energy ω using the Actual factors of Table II: so as to have both on the same plot, δ has been scaled by a factor of 1 / 20. The values of ( ω 2 , Γ 2 , f 2 ) were sequentially varied. Holding all other factors constant: R ω and δ ω used ω 2 = 5.3 eV; R g and δ g used Γ 2 = 1.1 eV; and R f and δ f used f 2 = 0.5.

Close modal

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 Q E ( 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 antimonides5–9 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 3Sb 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 3Sb 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.

FIG. 8.

K i ( ω ) for Cs 3Sb evaluated using density functional theory (DFT): data courtesy of E. R. Batista and S. Aryal (LANL). Copper data are shown for comparison.

FIG. 8.

K i ( ω ) for Cs 3Sb evaluated using density functional theory (DFT): data courtesy of E. R. Batista and S. Aryal (LANL). Copper data are shown for comparison.

Close modal

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.

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