The complexity of photocathode designs and detector materials, and the need to model their performance for short pulse durations, the response to high-frequency photons, the presence of coatings and/or thinness of the absorptive layer, necessitates modifications to three-step and moments models of photoemission that are used in simulation codes. In this study, methods to include input from computationally intensive approaches, such as density functional theory to model optical properties and transfer matrix approaches to treat emission from the surface or transport past coatings, by means of parametric models are demonstrated. First, a technique to accurately represent optical behavior so as to model reflectivity and penetration depth is given. Second, modifications to bulk models arising from the usage of thin film architectures, and a means to rapidly calculate them, are provided. Third, a parameterization to model the impact of wells associated with coatings and surface layers on the transmission probably is given. In all cases, the methods are computationally efficient and designed to allow for including input from numerically intensive approaches that would otherwise be unavailable for simulations.

Models of photo-excitation and electron transport are needed to predictively estimate the quantum efficiency QE and emittance εn,rms of photoemitters with coatings or heterostructures that meet the needs of future x-ray free-electron lasers such as the Linac Coherent Light Source II (LCLS-II) and Dynamic Mesoscale Material Science Capability (DMMSC) xFELs.1–3 The same methods are argued to be useful to respond to the demands of other technologically important applications and developments, such as (i) beam optics codes, such as MICHELLE4–8 for the treatment of pulsed or density modulated beams, (ii) simulations treating emission through and from negative electron affinity (NEA) and/or roughened surfaces and semiconductors,9–18 (iii) simulations in which processes generate a range of photon energies in exotic bulk materials that leads to internal photo-excitation effects as in detectors, photovoltaics, and optoelectronic devices,19,20 (iv) photo-enhanced emission mechanisms for materials with coatings or heterostructures used for energy conversion,21,22 and (v) photoemission processes modified by multi-photon emission,23 short pulse effects,24,25 or rapid heating from high intensity lasers.26–28 Such examples share one or more features of light absorption over a range of energies, transport within bulk materials, and/or electron emission through or past surfaces with coatings or heterostructures in a manner that complicates phenomenological models often used to treat photoemission and generally referred to as Three-step Models (TSMs)29–34 or Simple Moments models (SMMs).35,36

The present work reformulates a program began in a prior work37 to upgrade the SMM to account for increasingly complex physics associated with modern photocathode candidates; to enable a better account of the physics of absorption, transport, and emission; and to implement the theory in simulation methods compatible with beam optics code requirements and the utilization of the theory for characterization and analysis of sources. We first briefly recount the application of SMM to bulk materials such as metals (e.g., copper34) and semiconductors (e.g., multialkali antimonoides with or without cesiated surfaces38 or graphene layers2,39 or perovskites20,40,41), in Sec. II. It then undertakes specific modifications to extend the applicably of the underlying physics models and thereby address opportunities associated with recent developments. In particular, improvements to the SMM will account for materials physics, optical parameters, and scattering; extend the optical models to frequencies for which measured data are not available; evaluate the transmission probability D(E) using exact methods for surface barriers more complex than the simple triangular and metallic (or “Schottky–Nordheim”42) barriers from which analytical models43 can be developed; and accounts for modifications associated with thin photocathodes.44 Such modifications are needed for modeling photoemission when conditions are rapidly changing because small duration charge bunches37 are drawn from metals and semiconductors with coatings and layers.2,45–47 The changes fall into five categories:

  1. parameterizations of the optical constants and laser penetration depth extended to high energy regimes discussed in Sec. III,

  2. material physics based parameters affecting, e.g., density of states factors, effective masses, and scattering terms based on density functional theory (DFT) and other methods outlined in Sec. III C,

  3. replacement of the (infinite extent) bulk semiconductor material by a layer (or a thin film) of finite thickness presented in Sec. IV,

  4. more complex transmission probabilities, accounting for applied fields, surface coatings, resonances, and reflectionless well/barrier combinations that require exact evaluation described in Sec. V, and

  5. meso-scopic factors caused by surface contamination by high work function materials such as carbon, different crystal faces, effects of temperature and thermal-field emission, and field enhancement due to surface roughness or geometric emitters28,48 contribute and require additional considerations.

The first four are treated in the present study; the last is treated separately17,49 or part of on-going studies (e.g., temperature-dependence of scattering factors for novel materials and their impact on the transport models).

The extensions to SMM are constrained by two requirements: first, their implementation is guided by the needs of simulation and design, in particular, by the needs of beam optics codes and device modeling requirements and second, they remain open to incorporating input from experiments, measurements, and details provided by computationally intensive approaches such as density functional theory50 and Monte Carlo simulations11 using parametric methods.

The need for computational expediency in contrast to comprehensive theoretical models51–54 is made apparent by efforts to model the complications of surface roughness, work function and other emission site statistical variation, delayed emission, and space charge forces17,55 that are implicated in emittance growth, halo formation,56,57 and non-optimal launch times associated with short pulse length demands. The relentless demand for higher beam brightness necessitates simulations of beam dynamics where fluctuations and initial particle distribution affect the prediction of photoinjector performance.28 How the beam optics codes treat the emission process, therefore, has bearing. A 3D beam optics simulation using a modern particle-in-cell (PIC) code for photoemission in an RF photocathode can have upward of 330×106 macro-particles or more per beam bunch. Similar large scale macro-particle counts are required when modeling micro- to meso-scale rough surfaces features and/or surface material property variations in order to predict transverse energy spread. For very short pulses, where the transport time of a photoexcited electron to the surface is comparable to the duration of the laser pulse or when scattered electrons contribute significantly to an emission tail existing after the laser pulse duration is over,7,9 a history of intensity and field strength conditions (to say nothing of temperature) at earlier times and the repetitive calculation of emission at each time step during the beam pulse further complicate the nature of the emission models that can be employed to model the origin of fluctuations. A computationally expedient and simpler methodology that permits generalizations enabling minimizing computation time per emitted particle is needed. The reformulations developed in the present work are restricted by computational expediency in how they fulfill that need yet bring in the desired physics. The additional demands associated with internal photoemission effects at all frequencies associated with x-ray and γ-ray detectors add to the complexity in a different but no less complicated manner.58 

A Simple Moments Model (SMM) derives from the same framework responsible for the Fowler Nordheim and Richardson equations59,60 and is structurally analogous to Three-Step Models (TSMs) of photoemission,30,31,33,61,62 a phenomenological approach in which separate models for absorption of photons, transport of electrons through bulk material, and emission of electrons past a surface barrier are combined. The nomenclature “Moments” is used because both quantum efficiency and emittance are proportional to the moments kz and k2 for QE and εn,rms, respectively.36,63 In either case, it is the distribution of emitted electrons over which the moments are taken that require the extensions to the SMM developed herein, with which the momentum component (and to what power) is considered being of secondary importance. For that reason, even though the SMM methodology applies equally to QE and εn,rms, for narrative simplicity, it is both more convenient and likely more familiar to focus on the former. In its zero temperature limit, SMM models quantum efficiency (QE) by

QEmetal=(1R(ω))ω(2μω)μ+ϕωμEdEFλ(xm,E),
(1)
QEsemi=(1R(ω))(ωEg)2EaωEgEdEFλ(xm,E),
(2)

where the scattering factor Fλ(x,E) is given by64 

Fλ(xm,E)xm1fλ(x,E)xdx,
(3)

where ω is the photon energy, R(ω) is the reflectivity, R(ω) ¼ reflectivity, D(E) is the density of states (assumed parabolic), μ is the chemical potential (or Fermi energy at T=0), Φ is the work function, E~=E+ω, and Eg,Ea are the bandgap and electron affinity. The height above the Fermi level for metals ϕ=Φ4QF is the work function reduced by the Schottky barrier lowering factor 4QF, where Qq2/16πε0=αfsc/4 and αfs1/137.036 is the fine structure constant.65 The factor xm(E)=cosθm defines the “escape cone” or the minimum angle for which Ez=Ecos2θ exceeds the barrier height and is

xm={[(μ+ϕ)/(E+ω)]1/2(metal),[Ea/E]1/2(semi)
(4)

for metals characterized by a work function Φ or semiconductors characterized by an electron affinity Ea. The scattering factor fλ depends on the ratio between the laser penetration depth δ(ω) [see Eq. (32)] and the mean free path between scattering events =(k/m)τ, where k/m and τ are the velocity and the associated relaxation time of the photoexcited electron, resulting in

fλ(cosθ,E)1δ0exp[xδ(1+pcosθ)]dx=cosθcosθ+p(ω,E),
(5)

where p(ω,E)=δ(ω)/(E~) (the tilde indicating that it is the energy of the photoexcited electron that is used). It assumes that electrons that have undergone a scattering event no longer have sufficient energy to surmount the surface barrier as governed by the transmission probability D(E~z), resulting in limits on the angular and energy integrations on which QE depends. The integral in Eq. (3) is analytically given by

Fλ(x,p)=12(1x)(1+x2p)p2ln(x+p1+p).
(6)

The SMM often makes further approximations to obtain analytic forms of Eqs. (1) and (2). For metals, near threshold, ω is close to ϕ, and therefore, fλ(cosθ,E)cosθ/[1+p(ω,μ)]. Doing so results in

QE(1R(ω))3ω(2μω)(1+p)μ+ϕωμE(1xm3)dE,

where p=p(ω,μ). Replacing the integral with its trapezoidal approximation gives to the leading order,59,66

QEmetal(1R(ω)1+p(ω,μ))μ(ωϕ)24ω(μ+ω)(2μω),
(7)

which recovers the Fowler–DuBridge approximation QE(ωϕ)2 to leading order. Equation (7) enjoys good correspondence with data: a comparison with the measured data of Dowell et al.34 using representative values of the various frequency-dependent factors for copper (Cu) gives the correspondence shown in Fig. 1, for which the value of p=mδ/kτ used entails that τ1.3 fs assuming δ=12.9 nm and k=2m(μ+ω) (evaluated for λ=266 nm or ω=4.661 eV): this value is lower than, but nevertheless comparable to, the relaxation times in copper for similar conditions.34,66 It does not, however, account for variations in δ(ω), R(ω), or τ(μ+ω) that will cause changes as a function of frequency.

FIG. 1.

Quantum efficiency for copper34 (data courtesy of D. Dowell) compared to the simple moments model of Eq. (7) for μ=7 eV, Φ=4.6 eV, F=50 eV/μm (corresponding to a field of 50 MV/m and implying ϕ=Φ4QF=4.3317 eV), R(ω)=0.336, and p=5. Also shown are Fowler–DuBridge (FD) relations pinned to QE values at the squares at ω=4.66 eV and 6.2 eV.

FIG. 1.

Quantum efficiency for copper34 (data courtesy of D. Dowell) compared to the simple moments model of Eq. (7) for μ=7 eV, Φ=4.6 eV, F=50 eV/μm (corresponding to a field of 50 MV/m and implying ϕ=Φ4QF=4.3317 eV), R(ω)=0.336, and p=5. Also shown are Fowler–DuBridge (FD) relations pinned to QE values at the squares at ω=4.66 eV and 6.2 eV.

Close modal

A similar analysis for semiconductor parameters can be performed using a triangular barrier model for the transmission probability,2 giving

QEsemi(1R1+p)2s5s2(1+s)(s+s),
(8)

where s=[(ωEg)/Ea]1/2, s=[(ωEgEa)/Ea]1/2, and Ea and Eg are the electron affinity and bandgap, respectively, which recovers the parametric form suggested by Spicer29 and which compares favorably to measured values (Figs. 8 and 9 of Ref. 2).

For bulk photocathodes under long illumination conditions, low intensities that do not heat the photocathode, and photon energies comparable to the surface barrier, the simple forms of Eqs. (7) and (8) are well-suited for simulation and characterization. Complications such as heating due to high intensity lasers leading to additional emission mechanisms,26,27,35,67 ultrafast pulses23,24,68 and delayed emission,7,9,37 evolving surface conditions during operation or designed structures,2,69 large area emission from thin materials44 with irregular surface conditions for which coatings70 or layers are present,17,39,45,46 or illumination at short wavelengths40 for which optical and material parameters are unavailable or inadequate require that the SMM be reformulated and methods developed for it to utilize more varied sources of parameters and models. An auxiliary goal of the present work is to ensure that the reformulated methods do not demand a computationally prohibitive numerical overhead or introduce black-box components that would undermine their utility in beam optics codes, device simulations, and characterization efforts (e.g., optimization routines available in symbolic and computational packages for the determination of parameters).

A simple model of the dielectric constant ε(ω)=K(ω)ε0 is derivable from a damped harmonic oscillator model applied to a free-electron gas and can account for features of the optical properties of metals and semiconductors with large carrier concentrations,71,72 but the same underlying Lorentzian forms naturally emerge from a quantum theory of free-carrier absorption73,74 applied to semiconductors for a wide range of photon energies. Rakić et al.75–77 have developed algorithms for modeling peaks in the imaginary part of the dielectric function using a small number of Lorentzians for metals and semiconductors parameterized by optimization methods. Here, a means to incorporate many Lorentzians is developed, allowing excellent matching with measured data over a large energy range and, importantly, providing a means to parameterize Density Functional Theory (DFT) results for which data do not exist or have not been extended to frequency regimes of interest. For comparisons to experimentally measured dielectric constants in Figs. 2–8, 11, and 12, data have been aggregated (from tabulated data or digitally extracted) from commonly used sources for metals and semiconductors77–83 and compared to online databases.84 

FIG. 2.

Ki(ω) as a function of photon energy for copper. Dots: measured data.77,79,80 Line: sum of the simple Drude and Lorentzian components evaluated via Eqs. (10) and (11). Yellow dots are locations of ωj for the Lorentz resonances.

FIG. 2.

Ki(ω) as a function of photon energy for copper. Dots: measured data.77,79,80 Line: sum of the simple Drude and Lorentzian components evaluated via Eqs. (10) and (11). Yellow dots are locations of ωj for the Lorentz resonances.

Close modal
FIG. 3.

The data associated with Y0(xi) for copper, obtained by subtracting the red line values from the black dot values in Fig. 2. Green vertical lines show the locations of the xj resonances.

FIG. 3.

The data associated with Y0(xi) for copper, obtained by subtracting the red line values from the black dot values in Fig. 2. Green vertical lines show the locations of the xj resonances.

Close modal
FIG. 4.

The data of Fig. 2 for the middle-range energies and showing the contributions of the resonant Lorentzians of Eq. (11) (R) and the asymptotic (LD) lines for copper.

FIG. 4.

The data of Fig. 2 for the middle-range energies and showing the contributions of the resonant Lorentzians of Eq. (11) (R) and the asymptotic (LD) lines for copper.

Close modal
FIG. 5.

Comparison of measured K(ω) to results of the light-dependent resistor (LDR) analysis (Steps 1–4) for both real and imaginary components for copper data.77,79,80 The absolute value of the real parts is shown.

FIG. 5.

Comparison of measured K(ω) to results of the light-dependent resistor (LDR) analysis (Steps 1–4) for both real and imaginary components for copper data.77,79,80 The absolute value of the real parts is shown.

Close modal

A dielectric function K(ω) can be written as78 

K(ω)=1+χf(ω)+χb(ω),
(9)

where χf(ω) and χb(ω) are intraband and interband electric susceptibilities. The intraband susceptibility is described parametrically by the free-electron Drude–Zener model with the oscillator strength f0 and the damping rate Γ0,76 

χf(ω)=f0ωp2ω(ω+iΓ0),
(10)

whereas the interband susceptibility is described parametrically by the simple semi-quantum model resembling the Lorentz result for insulators77 given by

χb(ω)=j=1Nbfjωp2ω2ωj2+iωΓj,
(11)

where

ωp2=q2ε0mρo16πQmρo
(12)

is the plasma frequency for ρo electrons per unit volume and Nb is the number of interband transitions with the frequency ωj, oscillator strength fj, and damping rate Γj.

The total dielectric constant is then the sum of the free and bound components, or K(ω)=Kf(ω)+Kb(ω), both of which are complex. Having obtained the dielectric function, the optical parameters n and k are evaluated from its real and imaginary parts K(ω)=Kr(ω)+iKi(ω) by

2n2=Kr2+Ki2+Kr,2k2=Kr2+Ki2Kr.
(13)

In turn, the reflectivity R(ω) and penetration depth δ(ω) are obtained from

R(ω)=(n1)2k2(n+1)2k2,δ(ω)=c2kω
(14)

[compare Eq. (32)]. For example, by adding two phenomenological Lorenztian terms, the prediction of reflectivity for copper (see Fig. 60 of Ref. 35) is improved. Systematic methods to evaluate and optimize the weighting factors fj, damping factors Γj, and resonant frequencies ωj have been treated by Rakić et al.77 and Adachi et al.85 and can be applied to measured79–82,86 or calculated (e.g., by DFT) values for both metals and semiconductors. In contrast, the rapid and repetitive determination of R(ω) and δ(ω), as demanded by predictive models of photoemission and simulation models used in beam optics codes, requires a reconsideration of how the factors (fj,Γj,ωj) may be determined, with methods allowing for flexibility and speed taking precedence and (depending on application) extended to frequencies for which measured data may not be available. A computationally expedient and simpler methodology that permits generalizations is needed.

The method is first demonstrated for metals, beginning with copper as a representative and well understood test case. ε(ω) is taken to be the sum of three classes of terms: a low frequency Drude component based on Eq. (10) and characterized by fd and Γd, a high-frequency Lorentz component based on Eq. (11) characterized by fo, Γo, and ωo such that the latter two are in some sense “large”, and resonant components based on the j(1,N) frequencies ωj and associated with fj and Γj. They are found for metals by matching the imaginary part Ki(ω) to Eqs. (10) and (11) because the Kramers–Kronig relations71 ensure that Kr(ω) can be determined from the same parameters. By introducing x=ω and function y(x), the forms of which are fashioned after the behavior of Ki(ω), and by defining γ=Γ/,

y(x;a)γx(x2a2)2+(γx)2.
(15)

Furthermore, let the measured (or calculated) data Ki(ω) be processed such that it is mapped onto a regularly spaced set of values ωixi (where Δxi+1xi) using, e.g., cubic spline fitting: the “i” subscript will refer to that set of data, whereas the “j” subscript is reserved to correspond to the resonant levels determined by ωj=xj. Such a mapping enables methods drawn from finite difference techniques87 to be used for approximating derivatives and finding local maxima. Let yiy(xi;a). The factors (fj,Γj,ωj) are then determined as follows. Observe in advance that the fj so chosen will be replaced by a fourth step and also that the plasma energy ωp evident in Eq. (11) has been folded into its definition. The method of extracting Lorentzian parameters is adapted from techniques familiar to spectral analysis88 and is undertaken in four steps:

  1. evaluation of Drude parameters,

  2. evaluation of Lorentz (ωo,γo,fo) parameters,

  3. evaluation of Lorentz (ωj,γj,fj) parameters, and

  4. correction to the resonant fj parameters.

These steps are described next.

1. Drude

Set a0 and γγd in Eq. (15) with

γd[xn3ynx13y1x1y1xnyn]1/2,fdx1y1γd(x12+γd2)
(16)

for a suitably chosen small xn (n2), where lny is approximately linear in lnx. Although fd so evaluated gives a good fit to Ki(ω) for a small ω, the non-optimization of the algorithm and its reliance on potentially noisy data result in estimates of Kr(ω) being slightly off there. That small offset [which affects R(ω)] can be mitigated by fdhfd with h close to unity after Step 4 is performed: for copper, h0.9 is used in Figs. 5 and 6, whereas h=1.28 is used for Figs. 7 and 8.

FIG. 6.

Reflectivity R(ω) and penetration δ(ω)/δo as a function of wavelength for copper, with δo=20 nm. Points marked “exp” are evaluated using the data of Fig. 5.

FIG. 6.

Reflectivity R(ω) and penetration δ(ω)/δo as a function of wavelength for copper, with δo=20 nm. Points marked “exp” are evaluated using the data of Fig. 5.

Close modal
FIG. 7.

Comparison of measured K(ω) to results of the LDR analysis (Steps 1–4) for both real and imaginary components for gold data.77,79,80

FIG. 7.

Comparison of measured K(ω) to results of the LDR analysis (Steps 1–4) for both real and imaginary components for gold data.77,79,80

Close modal
FIG. 8.

Reflectivity R(ω) and penetration δ(ω)/δo as a function of wavelength for gold,77,79,80 with δo=25 nm.

FIG. 8.

Reflectivity R(ω) and penetration δ(ω)/δo as a function of wavelength for gold,77,79,80 with δo=25 nm.

Close modal

2. Lorentz

Set aωoxo and Γγo/. Approximate x/y(x) by a quadratic in x2, or x/yA+Bx2+Cx4 for a range of xaxixb with xa and xb chosen to span a region for which the polynomial approximation to x/y(x) in x2 is reasonable. Letting g(x)=1/(xy(x)), then

ωoxaxb,ga=gbgaxbxa,γo[gagaxa+2(ωo)2]1/2,fo1γoga.
(17)

In Fig. 2, xa=30 eV and xb=70 eV were chosen for the evaluation. While the resonant γj will be small (O(1)), the γo will be large (O(10)).

3. Resonant

Finding the xj first requires removing the Drude and Lorentz components: this is accomplished by subtracting the theory (red line) from the measured data line in Fig. 2. Let Y0 be the resulting set of data points, with the “0” indicating that only the Drude and Lorentz components have been subtracted, but no resonant term has as yet been removed. Y0 is shown in Fig. 4. The curve Y0(x) shows a series of peaks taken to be near the xj, the selection of which is an esthetic choice (e.g., the highest of the peaks and/or a shape that appears to match a Lorentzian and/or progress toward a desired behavior): “esthetic” means that no algorithm is involved, but rather, the xj is selected on the basis of how they affect the shape of y(x) when the xj resonant contribution is removed. This entails that the xj is not selected in the order of magnitude, but rather in the order that they best remove peaks and features. This is acceptable for present purposes because although such a process may be automated (in principle), (i) it meets the immediate needs of developing a library of optical parameters that can be called as required by simulation codes, (ii) it is accurate to the extent needed for simulations of photocathode yield at a given wavelength, and (iii) simulations of detector performance sum over a (large) range of frequencies responsible for photoexcited electrons so that small variations as a function of frequency are smoothed. The by-hand approach is, therefore, preferable for present purposes as it focuses on important features and does not preclude future automation of the process. The locations of the xj for copper are indicated by the yellow-filled circles in Fig. 2. Let xn be the local extremum (peak or valley) of a set of five points, and let y~n be the corresponding values of Y0(x), shown in Fig. 3, such that there are five pairs of coordinates {(xn2,y~n2)(xn+2,y~n+2)}. Using the five-point finite difference approximation to the derivatives [Appendix (A1.3.2) of Ref. 59],

yn=y~n+2+8y~n+18y~n1+y~n212Δ,yn=y~n+2+16y~n+130y~n+16y~n1y~n212Δ2,

then the location of the extremum xm [where y(xm)=0] is approximated by

xm=xnynyn.
(18)

The use of a 5-point scheme is advantageous when the shapes of the extrema are more hyperbolic than parabolic, but the scheme can occasionally introduce undesired effects for closely spaced extrema due to the overlapping of the resonant terms (as occurs when ω is small). Introduce the coefficients A and B defined by

A=yn+12(xm2xn)yn+12xn(xm+xn)yn,B=xnynyn2xm.

Introduce an initial value of sA/2Bxm2, which is then iterated according to

sA(s+2)(s+1)(s2+2s+4)2Bxm2(3s5+15s4+33s3+44s2+28s+8),

with four iterations generally being sufficient. Then, it is found that

xj(1+s)xm.
(19)

This approach is an iterative method to find a=xj in Eq. (15) when the location of its maximum is known. The last step is to remove the resonant term from the Y0(xi) data points, or

Yj+1(xi)=Yj(xi)fjy(xi;xj),
(20)

after which the process is repeated to remove the next desired local extremum at xj+1 and y~ drawn from Yj+1 until the resulting YNb is of a desired flatness, where Nb is the number of resonant terms required.

4. Weighting correction

The values of fj returned in Step 3 were temporary, only serving to remove the influence of a local maximum or minimum: each subsequent one corrupts the earlier fj because of the finite width of y(x;xj) as governed by γj. The final step is, therefore, to renormalize the weighting factors fj. With γj and xj determined, the fj are found by inverting the matrix equation

Y=Mf,
(21)

where Y and f are vectors of the length Nb (number of resonant terms) such that [Y]i=Y0(xi), [f]i=fi, and [M]ij=y(xi;xj)/y(xm;xj) is an Nb×Nb matrix, the form of which is determined from Eq. (15). Note that xm is evaluated for each associated xj. The values of (ωj,δj,fj) as determined by these steps, along with the Drude and Lorentz parameters, are shown in Table I for copper.

TABLE I.

Optical parameters for copper using the Lorentz–Drude-resonant model of Eqs. (10) and (11) as evaluated in Steps 1–4. Units are in eV. The R-terms are evaluated in the order they appear in the j column. The D- and L- rows are Drude and Lorentz factors, respectively. The fD value is shown with the factor of h implicitly included.

jωjγjfjjωjγjfj
D 0.0888 79.710 L 45.826 210.34 2611.5 
11 1.435 0.8263 0.5554 8.789 5.070 0.771 8 
1.901 0.9819 −2.370 11.96 1.602 0.112 4 
2.695 1.223 4.158 14.41 6.247 0.659 3 
3.616 1.178 1.486 17.44 4.418 0.095 79 
10 4.768 2.017 −0.4978 25.43 3.842 0.239 7 
5.06 2.363 4.396     
jωjγjfjjωjγjfj
D 0.0888 79.710 L 45.826 210.34 2611.5 
11 1.435 0.8263 0.5554 8.789 5.070 0.771 8 
1.901 0.9819 −2.370 11.96 1.602 0.112 4 
2.695 1.223 4.158 14.41 6.247 0.659 3 
3.616 1.178 1.486 17.44 4.418 0.095 79 
10 4.768 2.017 −0.4978 25.43 3.842 0.239 7 
5.06 2.363 4.396     

The same exercise can be performed for other metals such as gold, for which the parameters are shown in Table II, the behavior of K(ω) being depicted in Fig. 7, and the R(ω) and δ(ω) plotted in Fig. 8. The range in ω over which K(ω) values are measured is smaller than that for copper (90 eV for Cu compared to 30 eV for Au), making the determination of the L-parameters affected. No effort was made to mitigate the defect in order to show the impact of non-optimized parameters: it is seen that the evaluation remains reasonably robust for the estimation of R(ω) and δ(ω). Optimization beyond the straightforward implementation of Steps 1–4 is deferred to a separate study. The absence of an optimization step results in some metals (such as lead) being more challenging to parameterize well but enables the steps to be implemented far more easily. Specifically, metals such as lead can be characterized as having higher parameter sensitivity, which is a numerical, rather than physical, characteristic of their dielectric response and thus require more refined parameter optimization.

TABLE II.

Optical parameters for gold using the Lorentz–Drude-resonant model of Eqs. (10) and (11) as evaluated in Steps 1–4. Units are in eV. The R-terms are evaluated in the order they appear in the j column. The D- and L- rows are Drude and Lorentz factors, respectively. The fD value is shown with the factor of h implicitly included.

jωjγjfjjωjγjfj
D 0.0262 70.564 L 25.691 42.307 801.98 
17 1.909 1.044 −0.6409 9.32 0.5973 −0.154 6 
16 2.392 0.6644 −0.6887 10.28 6.402 0.874 5 
12 2.783 0.5378 2.258 12.3 2.601 −0.515 2 
13 3.108 0.681 1.451 13.22 2.988 0.857 8 
11 3.887 1.737 4.258 17.43 2.403 −0.247 1 
14 4.705 1.310 0.5908 18.45 1.449 −0.215 1 
15 5.856 2.995 0.894 20.37 13.42 1.084 
7.329 4.783 0.2931 10 20.85 1.485 0.035 35 
18 7.685 1.152 0.5327 24.77 2.820 −0.515 3 
jωjγjfjjωjγjfj
D 0.0262 70.564 L 25.691 42.307 801.98 
17 1.909 1.044 −0.6409 9.32 0.5973 −0.154 6 
16 2.392 0.6644 −0.6887 10.28 6.402 0.874 5 
12 2.783 0.5378 2.258 12.3 2.601 −0.515 2 
13 3.108 0.681 1.451 13.22 2.988 0.857 8 
11 3.887 1.737 4.258 17.43 2.403 −0.247 1 
14 4.705 1.310 0.5908 18.45 1.449 −0.215 1 
15 5.856 2.995 0.894 20.37 13.42 1.084 
7.329 4.783 0.2931 10 20.85 1.485 0.035 35 
18 7.685 1.152 0.5327 24.77 2.820 −0.515 3 

The Drude–Lorentz (DL) susceptibility model provides a suitable phenomenological parametric representation of experimental values for semiconductor dielectric functions.89 The application of Steps 1–4 can be extended to the treatment of semiconductors but subject to modifications. In insulators and semiconductors, where Van Hove singularities [non-smooth points in the density of states (DOS) where |k(Ek)| vanishes, and thus, Dj(ω) is not a smooth function of its argument] are of interest, this model can achieve the more detailed representation of band edges in the optical spectrum by including critical points associated with these singularities in the joint density of states.90 For the interband transitions between two bands, c and v, with energies Ec(k) and Ev(k), the susceptibility in Eq. (11) can be written as

χb(ω)=kfcv(k)ωp2ω2ωcv2(k)+iωΓk,
(22)

where ωcv(k)=Ecv(k)/, Ecv(k)Ec(k)Ev(k), fcv(k)=2|Pcv|2/(mEcv(k)), and Pcv is the matrix element of the momentum operator. Accordingly, close to the band edge, the imaginary and real parts of the interband dielectric function ΔK(ω)=4πχb(ω) with susceptibility from Eq.(22) are given, respectively, by85,91,92

ΔKi(ω)=πq2m2ω2Vk|Pcv|2δ[Ecv(k)ω],ΔKr(ω)=q22mVk|Pcv|2Ecv2(k)(ω)22mEcv(k),
(23)

where V is the volume and the δ-function ensures energy conservation. The sum over the δ-function in Eq. (23) can be turned to the joint density of states D by

Dj(ω)=1Vkδ[Ecv(k)ω]=14π3Ecv=ωdSk|kEcv|,
(24)

where Sk is the surface defined by constant Ecv(k) and the index j labels critical points known as Van Hove singularities93 for which kEcv=0. In three dimensions, the energy near critical points can be expanded as

Ecv(k)=Ecv(0)+22mi=13αiki2.

Four possible Van Hove singularities exist, classified according to the number of negative coefficients Mj:j{0,1,2,3}, where M0 and M3 represent a maximum and a minimum in the spectrum where all αi are positive (maximum) or negative (minimum), and M1 and M2 are saddle points where one or two of the αi are negative. Various combinations of critical points can be used to fit the dielectric function for a large number of direct bandgap semiconductors.85,94–98 Near the M0 critical point, in the range of energy where the expansion above is valid, the imaginary part of K(ω) can be written as

Ki(ω)=A(ω)pωEgΘ(ωEg),
(25)

where p=2, Eg=Ecv(0) is the bandgap, and Θ(x) is the Heaviside step function. For all 3D critical points Mj, a general form of the dielectric constant is99 

K(ω)=1+Aij(ω+iη)p{2EgEgωiηω+Eg+iη},
(26)

with η vanishing. For the model leading to Eq. (26), the energy is assumed to be relatively close to the critical point such that typically |Egω|Eg. In the case of a resonance structure possessing features that are of interest, accurate parametric representations exist.91,92 The present intent, however, is to construct a dielectric response function over a wide range of energies by means of interpolation between the M0 band edge, where KωEg, and regions of a high frequency, where ωEg and Kωp. In particular, capturing general trends of K(ω) is desirable, consistent with the inherent sensitivity of integral formulations of QE implicit in, e.g., Eqs. (1) and (2).

The extension of the Drude–Lorentz model to semiconductors has additional complications due to the lower free-carrier density of electrons in comparison with metals. The validity of the (quasiclassical) Boltzmann transport equation (and hence the Drude–Zener theory100) used in the modeling of solids in the determination of the dielectric function is altered by quantum mechanical effects, as can be shown using either the density matrix or the second order perturbation theory (both giving the same result): the relaxation time at high frequencies becomes frequency dependent, and this in turn alters the absorption coefficient. Briefly, the absorption coefficient’s relation to the optical conductivity, which is proportional to the relaxation time τ(ω) [Eqs. (114) and (126) of Ref. 101], entail that the frequency dependence of τ(ω) transfers to the absorption coefficient. A 1/ω3 dependence describes the absorption coefficient instead of the 1/ω2 predicted by the quasi-classical Boltzmann equation and is revealed by a quantum mechanical calculation of the absorption coefficient for the polar optical scattering mechanisms for, e.g., GaAs, InAs, InP, and CdTe.101 At higher carrier concentrations, impurity scattering increases in importance, and the characteristic dependence of the product of the absorption coefficient with the index of refraction varies from ω4 to ω2 and is dependent on the ratio of the photon energy to the initial electron energy.73,74,100,102 As a result, a higher power of ω can appear in the denominator of Eq. (25). Practically, it means that the value of p in Eq. (25) takes into account the quantum effects discussed in Sec. III C, for which p is expected to range from 3 to 4 instead of 2. Because measured data at large ω can be lacking, a demonstration of the dependence will include a description of the DFT methods that are used to estimate the behavior of the optical constants at higher energies than are available from experimental data so far considered.

DFT calculations were performed considering an ideal system for calculating optical properties of the alkali antimonides Cs3Sb and CsK2Sb, as shown in Fig. 9. For present purposes, the parameterization analysis is only conducted for Cs3Sb. Electronic structure calculations were done using the Vienna Ab initio Simulation Package (VASP),103–105 including core state effects via the VASP implementation106 of Projector Augmented-Wave (PAW) methods.107 We used the Local Density Approximation (LDA)108,109 to DFT110,111 for basic relaxation of structures, choosing a plane wave cutoff energy of 520 eV. Lattice constants of 0.91 nm were chosen for Cs3Sb and 0.86 nm for CsK2Sb, in the Fm3_m space group, to approximate experimental values. For all structures, a k-mesh of 8×8×8 was used for non-local exchange, resulting in actual k-spacings of 0.3×0.3×0.3 per 0.1 nm. The k-mesh was forced to be centered on the Γ point.

FIG. 9.

Measured Ki(ω) data compared to DFT Cs3Sb ( and red line) and CsK2Sb (° and blue line). The DFT ω axis has been shifted by ωω0.7 eV (see the text for discussion). Gray lines show energies characteristic of the harmonic frequencies of an Nd:YAG laser for comparison.

FIG. 9.

Measured Ki(ω) data compared to DFT Cs3Sb ( and red line) and CsK2Sb (° and blue line). The DFT ω axis has been shifted by ωω0.7 eV (see the text for discussion). Gray lines show energies characteristic of the harmonic frequencies of an Nd:YAG laser for comparison.

Close modal

The properties calculated determine the proper choice of density functional, as well as the particular system of interest. Calculations for graphene, as well as other metallic and dielectric systems, have been treated previously.112–115 For the present optical calculations, we chose a hybrid functional (HSE06)116 that combines a screened Hartree–Fock approach within DFT. The expression for the complex imaginary dielectric function was obtained by summing over conduction bands:117 in a formulation reflecting conventions of VASP calculations and at the risk of some confusion, it is commonly represented as

εαβ(2)(ω)=4πe2Ωlimq01q2v,c,k2ωkδ(ϵckϵvkω)×uck+qeα|uvkuvk|uck+qeβ,
(27)

where for this equation only, e is the electron charge, uck and similar are periodic parts of Bloch wave functions, Ω is the cell volume, ωk is a multiplicity factor for each k-point, and transitions are made from occupied to unoccupied states within the first Brillouin zone.

Real and imaginary parts of dielectric functions are connected by Kramers–Kronig relations118 from which the optical constants R(ω) and δ(ω) are calculated as per Eqs. (13) and (14). A comparison of measured vs calculated values of Ki(ω) for both Cs3Sb and CsK2Sb are shown in Fig. 9. The DFT data for ω are shifted, given the inability of semi-local density functional models to reproduce the absolute value of the bandgap: such shifts in the x-coordinate are not uncommon. It is assumed, however, that for the present study, DFT is sufficiently accurate for qualitatively estimating the functional character of Ki(ω), which can then be adjusted parametrically to fit a target functionally.

Although experimentally measured Ki(ω) data exist for the multi-alkali antimonides, the energy range is generally not as extensive as simulation desires. Extending the energy range of Ki(ω) using DFT, for the purposes of estimating its functional character at higher energies, and performing a parameter fit results in the family of curves shown in Fig. 10, where a conventional Eg=1.6 eV value is chosen, and where the p-curves are normalized so that they pass through an anchor point chosen to be 90% of Ki(ω) at ω=1.7478 eV, although the actual power that is selected depends on how the Lorentz–Drude factors are chosen and used in Fig. 11. The closest fits are for 3<p<4, compatible with the discussion regarding the Drude–Zener theory. The p=3.5 curve passes through the high Ki(ω) values well so that p is chosen. Performing the ωj analysis analogous to Steps 3 and 4 using the parameters shown in Table III, but using Eq. (25) instead of the Drude model of Eq. (10), gives rise to the red “Theory” line and seen to match the data well. From the theory line, the R(ω) and δ(ω) relations can be obtained, as in Fig. 12.

FIG. 10.

Measured Ki(ω) ()79,81 compared to Eq. (25) for different values of p. The value of A for all p-lines was determined by demanding that the curve pass through 90% of the Ki(ω) value for ω=1.7478 eV, shown as a yellow dot.

FIG. 10.

Measured Ki(ω) ()79,81 compared to Eq. (25) for different values of p. The value of A for all p-lines was determined by demanding that the curve pass through 90% of the Ki(ω) value for ω=1.7478 eV, shown as a yellow dot.

Close modal
FIG. 11.

Measured Ki(ω) ()79,81 compared to the resonant terms of Eq. (11), but instead of Eq. (10), Eq. (25) is used for the red line, for which p=3.5 (as in Fig. 10).

FIG. 11.

Measured Ki(ω) ()79,81 compared to the resonant terms of Eq. (11), but instead of Eq. (10), Eq. (25) is used for the red line, for which p=3.5 (as in Fig. 10).

Close modal
FIG. 12.

Reflectivity R(ω) and penetration δ(ω)/δo as a function of wavelength for Cs3Sb found from standard optical data,79,81 with δo=75 nm.

FIG. 12.

Reflectivity R(ω) and penetration δ(ω)/δo as a function of wavelength for Cs3Sb found from standard optical data,79,81 with δo=75 nm.

Close modal
TABLE III.

Optical parameters for Cs3Sb. DL parameters are absent; instead, Eq. (25) is used with A = 152.1982, p = 3.5, and Eg = 1.6 eV. j is the order in which the resonant terms are subtracted in removing the peaks.

jωjγjfjjωjγjfj
35 1.336 0.322 3 2.981 32 3.306 0.096 01 0.656 5 
34 1.538 0.248 2 5.702 3.394 0.133 4 1.791 
14 1.595 0.043 19 4.071 17 3.496 0.089 38 0.832 6 
33 1.639 0.082 28 −0.2193 3.825 0.221 1.488 
36 1.655 0.758 4 −1.765 25 3.975 0.088 26 0.725 5 
15 1.797 0.003 73 0.1212 4.087 0.114 2 2.868 
16 2.099 0.111 9 1.405 18 4.24 0.056 75 0.270 
2.224 0.165 4 2.939 4.418 0.291 5 2.625 
2.337 0.170 6 12.19 10 4.583 0.116 3 1.174 
29 2.394 0.116 1 −1.865 19 4.703 0.102 6 0.179 6 
2.551 0.158 3 6.065 11 4.837 0.146 9 0.421 4 
28 2.693 0.093 57 −0.8654 21 4.953 0.113 4 0.099 34 
2.792 0.128 4 7.723 20 5.097 0.123 5 0.158 5 
27 2.839 0.171 3 −1.645 12 5.441 0.091 88 0.582 8 
30 2.894 0.077 55 0.2366 23 5.599 0.098 76 −0.113 5 
31 3.041 0.131 0 0.1525 24 5.709 0.122 5 −0.077 27 
3.112 0.336 4 6.963 22 5.883 0.123 3 −0.217 9 
26 3.232 0.123 1 −1.847 13 6.01 0.185 9 0.775 4 
jωjγjfjjωjγjfj
35 1.336 0.322 3 2.981 32 3.306 0.096 01 0.656 5 
34 1.538 0.248 2 5.702 3.394 0.133 4 1.791 
14 1.595 0.043 19 4.071 17 3.496 0.089 38 0.832 6 
33 1.639 0.082 28 −0.2193 3.825 0.221 1.488 
36 1.655 0.758 4 −1.765 25 3.975 0.088 26 0.725 5 
15 1.797 0.003 73 0.1212 4.087 0.114 2 2.868 
16 2.099 0.111 9 1.405 18 4.24 0.056 75 0.270 
2.224 0.165 4 2.939 4.418 0.291 5 2.625 
2.337 0.170 6 12.19 10 4.583 0.116 3 1.174 
29 2.394 0.116 1 −1.865 19 4.703 0.102 6 0.179 6 
2.551 0.158 3 6.065 11 4.837 0.146 9 0.421 4 
28 2.693 0.093 57 −0.8654 21 4.953 0.113 4 0.099 34 
2.792 0.128 4 7.723 20 5.097 0.123 5 0.158 5 
27 2.839 0.171 3 −1.645 12 5.441 0.091 88 0.582 8 
30 2.894 0.077 55 0.2366 23 5.599 0.098 76 −0.113 5 
31 3.041 0.131 0 0.1525 24 5.709 0.122 5 −0.077 27 
3.112 0.336 4 6.963 22 5.883 0.123 3 −0.217 9 
26 3.232 0.123 1 −1.847 13 6.01 0.185 9 0.775 4 

Experimental data for the dielectric function of cesium lead halide perovskites is available, as is computational data for the cubic phase.41,119,120 Cesium lead halide perovskites arrange in lower symmetry phases at room temperature, with the experimental data most likely measured for these phases. In the absence of more conclusive experimental and computational studies, approximate fits analogous to the methods above can be constructed using the available information with regard to the bandgap Eg and the high-frequency dielectric function K=ε/ε0.

The bandgap in cesium lead halide perovskites can be tuned by replacing one halogen with another120 from the largest in CsPbCl3, Eg3.1 eV, to a smaller in CsPbBr3, Eg2.4 eV, to the smallest in CsPbI3, Eg1.7 eV. These approximate values can be estimated using combinations of experimental121,122 and computational41,120 data.

TABLE IV.

Optical parameters for lead halide perovskite PbCI3N2H5. DL parameters are absent; instead, Eq. (25) is used with A = 90, p = 2.6, and Eg = 2 eV.

jωjγjfjjωjγjfj
38 −0.9919 1.434 0.071 98 12.94 2.627 0.497 6 
56 1.940 0.5407 −0.102 2 19 13.7 0.9547 −0.050 37 
31 2.434 0.6001 0.509 3 18 14.54 0.7026 0.098 8 
55 2.495 0.4193 −0.240 5 50 14.97 0.4653 0.134 8 
54 2.649 0.1511 −0.209 7 52 15.25 0.2310 0.045 29 
11 2.792 0.6273 2.601 20 15.68 1.3720 0.236 2 
2.959 0.1687 2.922 53 16.15 0.6062 0.014 08 
32 3.049 0.2493 −0.821 8 51 16.54 0.4444 0.081 02 
3.255 0.5134 −1.508 17.02 0.7430 0.333 2 
33 3.769 0.4166 0.350 2 17.69 0.4592 0.591 6 
12 4.002 0.539 0.535 2 21 18.74 1.518 0.238 6 
34 4.168 0.4703 0.425 9 22 19.14 0.6398 0.061 48 
41 4.516 0.4247 −0.220 3 28 19.94 0.5764 0.110 7 
15 5.127 1.0100 −1.117 10 20.98 1.570 0.279 4 
35 5.300 0.5067 0.467 4 23 22.09 1.656 0.114 4 
42 5.669 0.4418 −0.271 8 25 22.90 1.394 0.127 1 
36 6.449 0.3569 0.013 81 24 23.92 0.8345 0.146 1 
37 6.568 0.4259 0.019 61 26 25.52 1.294 0.084 56 
16 6.679 2.5940 −1.198 27 26.64 0.7474 0.061 6 
13 7.429 0.2069 1.064 44 27.82 0.5566 0.024 73 
8.043 2.0560 3.225 47 28.28 0.8834 0.022 8 
14 8.912 0.3308 0.392 5 48 31.20 0.9443 0.019 05 
9.424 1.1630 1.277 46 32.12 0.8113 0.020 17 
17 10.18 0.8626 0.610 3 29 33.17 0.6884 0.010 66 
39 10.36 0.1339 0.059 74 43 34.37 0.8436 0.024 05 
10.93 0.6245 1.087 45 35.43 0.5430 0.022 2 
11.37 0.5368 0.325 1 49 39.89 1.589 0.014 61 
40 11.75 0.1583 0.068 8 30 40.91 0.7115 0.023 62 
jωjγjfjjωjγjfj
38 −0.9919 1.434 0.071 98 12.94 2.627 0.497 6 
56 1.940 0.5407 −0.102 2 19 13.7 0.9547 −0.050 37 
31 2.434 0.6001 0.509 3 18 14.54 0.7026 0.098 8 
55 2.495 0.4193 −0.240 5 50 14.97 0.4653 0.134 8 
54 2.649 0.1511 −0.209 7 52 15.25 0.2310 0.045 29 
11 2.792 0.6273 2.601 20 15.68 1.3720 0.236 2 
2.959 0.1687 2.922 53 16.15 0.6062 0.014 08 
32 3.049 0.2493 −0.821 8 51 16.54 0.4444 0.081 02 
3.255 0.5134 −1.508 17.02 0.7430 0.333 2 
33 3.769 0.4166 0.350 2 17.69 0.4592 0.591 6 
12 4.002 0.539 0.535 2 21 18.74 1.518 0.238 6 
34 4.168 0.4703 0.425 9 22 19.14 0.6398 0.061 48 
41 4.516 0.4247 −0.220 3 28 19.94 0.5764 0.110 7 
15 5.127 1.0100 −1.117 10 20.98 1.570 0.279 4 
35 5.300 0.5067 0.467 4 23 22.09 1.656 0.114 4 
42 5.669 0.4418 −0.271 8 25 22.90 1.394 0.127 1 
36 6.449 0.3569 0.013 81 24 23.92 0.8345 0.146 1 
37 6.568 0.4259 0.019 61 26 25.52 1.294 0.084 56 
16 6.679 2.5940 −1.198 27 26.64 0.7474 0.061 6 
13 7.429 0.2069 1.064 44 27.82 0.5566 0.024 73 
8.043 2.0560 3.225 47 28.28 0.8834 0.022 8 
14 8.912 0.3308 0.392 5 48 31.20 0.9443 0.019 05 
9.424 1.1630 1.277 46 32.12 0.8113 0.020 17 
17 10.18 0.8626 0.610 3 29 33.17 0.6884 0.010 66 
39 10.36 0.1339 0.059 74 43 34.37 0.8436 0.024 05 
10.93 0.6245 1.087 45 35.43 0.5430 0.022 2 
11.37 0.5368 0.325 1 49 39.89 1.589 0.014 61 
40 11.75 0.1583 0.068 8 30 40.91 0.7115 0.023 62 

Similarly, the high-frequency dielectric constant can be deduced from experimental measurements of the exciton binding energy, provided the effective masses are known,41 or directly calculated120 using the density functional perturbation theory.123 Again, the exciton binding energy is most likely measured in low symmetry phases: 20 meV (I), 40 meV (Br), and 70 meV (Cl),124 which is consistent with binding energies reported earlier.122 Estimates based on experiments are given in Ref. 41, where K=4.5 (Cl), 4.8 (Br), and 5.0 (I). For CsPbI3, however, the dielectric constant of 5.0 is taken from measurements on hybrid organic–inorganic perovskite MAPbI3, which forms a cubic structure similar to CsPbI3, with Cs replaced by the organic cation of methylammonium (CH3NH3+ or MA). MA is one of two widely used organic cations (the other is formamidinium NH2CH=NH2+ or FA) to stabilize the cubic phase. In general, the dielectric constant in the range of 4.55 is consistent with the results of first-principles calculations for several halides being 4.1 (Cl), 5.0 (Br), and 6.3 (I).120 

A simple fit is based on approximation of the dielectric function near the M0 van Hove critical point at the edge of the bandgap, believed to be formed by the valence and conduction band at the high symmetry point R of the Brillouin zone. At the M0 point, and introducing sω/Eg, the real and imaginary parts can be approximated as

Kr(ω)=a+bsp[21+s1sΘ(1s)],Ki(ω)=bsps1Θ(s1),
(28)

where Θ(x) is the Heaviside step function and a and b are numerically determined by matching the LDR model with DFT simulations (or measured data), but which are understood as arising from the coefficients of Eq. (26). For the three materials shown in Fig. 13, a=1 for all, and b= 12.4, 16.0, and 21.2 for CsPbCl3, CsPbBr3, and CsPbI3, respectively. For small ω, p=2, but as with the Drude–Zener discussion (Section III C), its value shall be set by fits to the data.

FIG. 13.

Real (thin dashed lines) and imaginary (thick solid lines) K(ω) for three perovskite materials CsPbCl3, CsPbBr3, and CsPbI3, based on Eq. (28).

FIG. 13.

Real (thin dashed lines) and imaginary (thick solid lines) K(ω) for three perovskite materials CsPbCl3, CsPbBr3, and CsPbI3, based on Eq. (28).

Close modal

For the purposes of demonstrating the methodology on perovskites, therefore, first consider the more tractable lead halide perovskite PbCI3N2H5, as shown in Fig. 14, to demonstrate performance, with the results shown in Figs. 15 and 16 using Table IV parameters. The method is then applied to cesium lead halide cases CsPbX3 for X= (Br, Cs, I), in Fig. 17. In these applications, the method is clearly not optimized but is rapid and flexible and provides a reasonable behavior of the critical parameters R(ω) and δ(ω), particularly for high photon energies ω. Moreover, it allows for a substantial adjustment of parameters in an algorithmic manner and, therefore, may lead to an automated determination of the resonant and DL parameters, closer to the spirit of Rakić et al.,75,77 but for many more ωj components. For example, many of the more rapid variations in Ki(ω) can be mapped by artificially restricting the size of γj and using more resonant terms (ωj) where a detailed structure is present. It is cautioned that attention to fine structure, however desirable as a matter of accuracy, is not necessary in simulations that span a broad range in the spectrum: variation within that range is of greater interest and is more consequential when a range of frequencies are present (as for detectors) or can be investigated for particular frequencies (as for photocathdoes). Therefore, the LDR method of extracting K(ω), and thereby R(ω) and δ(ω), is significantly advantageous for the simulation of internal photoemission processes, characterization of photocathode materials, and utilization by beam codes. Further optimizations are not precluded and will be taken up elsewhere. Finally, DFT simulations of perovskites with a top cesium layer cause changes in the work function58 and in turn introduce changes in the LDR parameters that can presumably be accommodated: such a study will be reported separately, but an indication of how surface layers affect emission are considered in Sec. V.

FIG. 14.

Lead halide perovskite PbCI3N2H5.

FIG. 14.

Lead halide perovskite PbCI3N2H5.

Close modal
FIG. 15.

Ki(ω) determined using DFT-calculated optical values for lead halide perovskite PbCI3N2H5. Equation (25) is used for the blue line, for which p=2.6.

FIG. 15.

Ki(ω) determined using DFT-calculated optical values for lead halide perovskite PbCI3N2H5. Equation (25) is used for the blue line, for which p=2.6.

Close modal
FIG. 16.

δ(ω) determined using DFT-calculated optical values for lead halide perovskite PbCI3N2H5. Copper shown for comparison in R(ω) but is not visible for δ(ω) for the limits shown.

FIG. 16.

δ(ω) determined using DFT-calculated optical values for lead halide perovskite PbCI3N2H5. Copper shown for comparison in R(ω) but is not visible for δ(ω) for the limits shown.

Close modal
FIG. 17.

R(ω) and δ(ω) evaluated using DFT-calculated optical values for cesium lead halide perovskites CsPbX3 with X= (Br, Cl, I).

FIG. 17.

R(ω) and δ(ω) evaluated using DFT-calculated optical values for cesium lead halide perovskites CsPbX3 with X= (Br, Cl, I).

Close modal

A thin film model differs from the bulk calculation: in the former, reflections occur at the substrate (back contact), and such reflections from a metallic substrate have been long suspected to be potentially important.125 Specifically, with proper engineering, an increase in the quantum efficiency is expected.44 Such “etalon” cathodes70 further decrease the response time of a photocathode by curtailing how far into a bulk material electrons are photoexcited and thereby limiting the transport time to the surface.7 The methodology to analyze them is first applied to the well-known bulk-vacuum problem and then extended to treat a thin film on a metallic substrate.

From Maxwell’s equations for the monochromatic electromagnetic field of the frequency ω, c×H+iωK(ω)E=0 and c×Eiωμ(ω)H=0,118 where in non-magnetic materials’ permeability is μ0=1/ε0c2, it follows that

[c22+ω2K(ω)]E=0.
(29)

In general, the dielectric function, K(ω)=1+χf+χb, includes both intra- and interband susceptibilities. In conductors, however, the imaginary part of the intraband component dominates according to Eq. (10) as χf=iσ/ω, where σ=f0ωp2/(4πΓ0) is the Drude conductivity and Eq. (29) then turns into [c22+ω2]E=iσωE. Assuming the incident laser is along the x^-direction, the electric field y^E=Ey is given by Ey(x,t)=Eoeiκxiωt, with the associated magnetic field Hz=Hoexp(iκxiωt) with Ho=(cκ/ω)Eo, where κ is the wave number, and then Eq. (29) for the homogeneous medium entails,

c2κ2+ω2K(ω)=0.
(30)

The complex index of refraction n^n(ω)+ik(ω) is then

n^cκω=K(ω).
(31)

The imaginary part (n^)=k(ω) leads to dampening as light penetrates the bulk material. The penetration depth introduced in Eq. (14) is found from how intensity decays, which is proportional to |E|2, and therefore,

xln(|E(x)|2)1δ(ω)=2ωkc.
(32)

Using the convention that terms in vacuum are designated by a “0” subscript and those in bulk by a “1” subscript, then when the wave is incident on the surface, part of the wave is reflected or Er=rE0exp(iκ0xiωt) and part is transmitted or Et=tE0exp(iκ1xiωt). Because E(x)=t1Eoeiκx in bulk, δ(ω) is independent of x. Demanding continuity of E and xE across the surface results in equations readily handled by introducing a matrix M and a coefficients vector C defined by

Mn(x)[eiκnxeiκnxiκneiκnxiκneiκnx],Cn=[tnrn]
(33)

for which the relations of continuity at the surface are compactly expressed as

M1(0)C1=M0(0)C0,
(34)

with the boundary conditions t01 and r10. The solution of the matrix equation results in the commonly known relations

t1=2n^0n^1+n^0,r0=n^0n^1n^0+n^1,
(35)

in terms of which the reflection coefficient R0=|r0/t0|2 is given by the well-known result,

R0(ω)=|n^0n^1n^0+n^1|2=(n11)2+k12(n1+1)2+k12,
(36)

where n^=nik and n0=1 have been used.

Now, let the photocathode material be deposited to a thickness L, which can be nanometers to micrometers in thickness, on the surface of a metal substrate. This is schematically shown in Fig. 18. The coefficients Cn are altered by the introduction of another relation that must be satisfied, in addition to Eq. (34), of the form

M2(L)C2=M1(L)C1
(37)

but now with the boundary conditions t01 and r20. The resulting matrix equation,

C2=M2(L)1M1(L)M1(0)1M0(0)C0,
(38)

with t0=1 and r2=0, may be solved for r0 and t2. Introducing the compact forms [recall Eq. (31)]

A=(n^1n^2)(n^0+n^1)exp(iκ1L),B=(n^1+n^2)(n^0n^1)exp(iκ1L),
(39)

where n^0=1, then it is found

r0=B+A(n^0+n^1n^0n^1)B+(n^0n^1n^0+n^1)A,
(40)
t2=(4n^1n^0A+B)r0exp(iκ2L).
(41)

Alexander et al.44 observe that the simple vacuum-bulk solution of Eq. (35) is recouped in the limit that k1L, but an equivalent method is to take n^2n^1 for which A0, and Eq. (35) is directly recovered. Continuing, the coefficients for the photocathode film are given by

C1=[t1r1]=12n^1[(n^1+n^0)+(n^1n^0)r0(n^1n^0)+(n^1+n^0)r0].
(42)

The reflection coefficient at the film-substrate boundary is then

R1(ω)=|r1eiκ1Lt1eiκ1L|2=|n^1n^2n^1+n^2|2,
(43)

as may have been anticipated by Eq. (36). The electric field E in the film is the top element of M1(x)C1 or

E1(x)=Eo(t1eiκ1x+r1eiκ1x)
(44)

for which the intensity Iω(x)|E(x)|2 may be evaluated given the material parameters n^2, n^1, and n^0=1. The interference between the transmitted and reflected waves, however, means that a constant dampening factor similar to Eq. (32) may not always be possible: depending on n1 and k1, absorption may peak within the film rather than at the surface, with consequences for quantum efficiency QE and emittance εn,rms due to losses associated with photoexcited electrons transporting back to the surface.

FIG. 18.

Schematic of incident light on a photocathode thin film of the thickness L deposited on a (metal) substrate. Coefficients of the transmitted and reflected waves are labeled by the region (0= vacuum, 1= photocathode, 2= substrate). No wave is incident from the right (r2=0), and normalization is such that t0=1.

FIG. 18.

Schematic of incident light on a photocathode thin film of the thickness L deposited on a (metal) substrate. Coefficients of the transmitted and reflected waves are labeled by the region (0= vacuum, 1= photocathode, 2= substrate). No wave is incident from the right (r2=0), and normalization is such that t0=1.

Close modal

Multi-alkali antimonide photocathode materials (e.g., Cs3Sb and CsK2Sb) are under active consideration to meet the needs of future x-ray Free-Electron Lasers (xFELs) because they (along with III–V semiconductors such as GaAs) are well-established high quantum efficiency photocathodes once the appropriate surface treatments are implemented (although at the cost of limited lifetime).2 Two examples are indicative for parameters drawn from the parameters for Cs3Sb-on-Cu for λ=532 nm and CsK2Sb-on-Ag for λ=650 nm, referred to as cases A and B, respectively (compare to, for example, Cs2Te, with 0.8n1.8 and 0.3k0.7 for λ=254 nm126), compare to values shown in Table V for common photocathode materials. In both cases, let L=λ/3. The intensity is proportional to |E1(x)|2 with E1(x) given by Eq. (44). Also shown is the L (bulk) calculation, where the decay is governed by Eq. (32). Although the values of n(ω) in both cases are approximately λ/L, the higher value of k(ω) in Case A leads to a rapid decline as x increases so that reflection off the substrate and the interference effects it entails in E1(x) are mitigated. In Case B, constructive interference effects can occur so that the intensity has peaks further from the surface. As a result, the escape cone is narrowed, and therefore, the QE and εn,rms will be affected.

TABLE V.

Approximate values of real and imaginary parts of n^ for typical wavelengths of technological interest. Data for GaAs, approximated from Table V of Ref. 86, shown for comparison.

λ (nm)1064800650532405355266206
ω (eV) 1.17 1.55 1.91 2.33 3.06 3.49 4.66 6.02 
Ag n 0.405 0.469 0.571 0.74 1.10 1.28 1.51 1.56 
Ag k 7.36 5.28 4.06 3.08 2.11 1.81 1.44 1.23 
Au n 0.118 0.144 0.216 0.406 1.34 1.51 1.37 1.16 
Au k 6.73 4.75 3.53 2.43 1.89 1.49 1.64 1.06 
Cu n 0.414 0.328 0.365 0.582 1.26 1.25 1.54 1.13 
Cu k 7.44 5.31 4.01 2.87 2.20 1.93 1.77 1.66 
Pb n 2.97 2.19 2.76 2.79 1.56 1.23 1.05 0.708 
Pb k 6.50 4.48 3.72 4.05 3.76 3.25 2.50 2.16 
Cs3Sb n 3.61 3.73 3.81 2.96 1.98 1.37 1.01 0.378 
Cs3Sb k 0.0434 0.512 1.26 2.12 2.24 1.98 1.70 1.16 
CsK2Sb n 3.19 3.17 3.11 3.21 2.34 1.68 1.23 0.717 
CsK2Sb k 0.123 0.313 0.564 0.992 2.06 1.84 1.55 1.47 
GaAs n … 3.68 3.83 4.13 4.42 3.54 3.66 1.26 
GaAs k … 0.09 0.18 0.34 2.07 2.02 3.34 2.47 
λ (nm)1064800650532405355266206
ω (eV) 1.17 1.55 1.91 2.33 3.06 3.49 4.66 6.02 
Ag n 0.405 0.469 0.571 0.74 1.10 1.28 1.51 1.56 
Ag k 7.36 5.28 4.06 3.08 2.11 1.81 1.44 1.23 
Au n 0.118 0.144 0.216 0.406 1.34 1.51 1.37 1.16 
Au k 6.73 4.75 3.53 2.43 1.89 1.49 1.64 1.06 
Cu n 0.414 0.328 0.365 0.582 1.26 1.25 1.54 1.13 
Cu k 7.44 5.31 4.01 2.87 2.20 1.93 1.77 1.66 
Pb n 2.97 2.19 2.76 2.79 1.56 1.23 1.05 0.708 
Pb k 6.50 4.48 3.72 4.05 3.76 3.25 2.50 2.16 
Cs3Sb n 3.61 3.73 3.81 2.96 1.98 1.37 1.01 0.378 
Cs3Sb k 0.0434 0.512 1.26 2.12 2.24 1.98 1.70 1.16 
CsK2Sb n 3.19 3.17 3.11 3.21 2.34 1.68 1.23 0.717 
CsK2Sb k 0.123 0.313 0.564 0.992 2.06 1.84 1.55 1.47 
GaAs n … 3.68 3.83 4.13 4.42 3.54 3.66 1.26 
GaAs k … 0.09 0.18 0.34 2.07 2.02 3.34 2.47 

The emergence of the constructive interference effects as n^1 changes can be investigated. Figure 19 considers a hypothetical example for which (n^p)=k stays fixed at k=0.3, and the index of refraction (n^p)=n changes from 1.0 to 4.5 for the case of a bulk material (L) and a thin film (L=λ/3) with λ=600 nm and n^s=0.5+5.0i. When k is increased, the oscillations are damped and the bulk behavior recovered; conversely, when it is smaller, interference effects are stronger and the oscillations, similar to those in Fig. 20, are correspondingly larger, particularly when L/λ is close to an integer.

FIG. 19.

Intensity Iω(x)|E(x)|2 in arbitrary units as a function of the distance x (nm) from the surface and λ (nm) for a Cu substrate with a Cs3Sb thin film of thickness L=160 nm. (top) Iω(x) (a.u.) (bottom) ln(Iω(x)).

FIG. 19.

Intensity Iω(x)|E(x)|2 in arbitrary units as a function of the distance x (nm) from the surface and λ (nm) for a Cu substrate with a Cs3Sb thin film of thickness L=160 nm. (top) Iω(x) (a.u.) (bottom) ln(Iω(x)).

Close modal
FIG. 20.

Intensity Iω(x)|E(x)|2 for (A) Cs3Sb-on-Cu for λ=532 and (B) CsK2Sb-on-Ag for λ=650 nm. In both cases, the film is of thickness L=λ/3. Values of n^(ω) for both the substrate and the film are drawn from Table V.

FIG. 20.

Intensity Iω(x)|E(x)|2 for (A) Cs3Sb-on-Cu for λ=532 and (B) CsK2Sb-on-Ag for λ=650 nm. In both cases, the film is of thickness L=λ/3. Values of n^(ω) for both the substrate and the film are drawn from Table V.

Close modal

For actual materials, the dependence of both n and k on ω results in maps of Iω(x) of greater complexity. The example of a layer of Cs3Sb, of thickness L=160 nm, on a substrate of copper in Fig. 21 shows both Iω(x), which better shows the interference effects giving rise to peaks in the intensity, and ln(Iω(x)) in Fig. 19, which better gives a sense of the decay of the intensity into the film. Practically, the actual behavior of n^p(ω) affects choices regarding the thickness of the film effect and the optimal choice of the substrate. Theoretically, the departure of ln(Iω(x)) from linearity affects both TSM and moments models2,35,37 of the quantum efficiency QE, and the presence of peaks in Iω(x) within the film has implications for emittance εnorms.44 Both the exploration of configurations and material choices and the simulation of photoemitter performance are, therefore, enhanced by having a computationally efficient model of the optical properties and an ability to predictively estimate them by DFT as in Sec. III E.

FIG. 21.

Hypothetical intensity Iω(x)|E(x)|2 in arbitrary units for λ=600 nm and k=0.3 as a function of the distance x from the surface and n=(n^). (top) L or a bulk material and (bottom) l=λ/3 or a film.

FIG. 21.

Hypothetical intensity Iω(x)|E(x)|2 in arbitrary units for λ=600 nm and k=0.3 as a function of the distance x from the surface and n=(n^). (top) L or a bulk material and (bottom) l=λ/3 or a film.

Close modal

Insofar as a high quantum efficiency is a primary metric of photocathode utility for photoinjectors,69 a notable feature of the highest QE photocathodes is the incorporation of cesium in the stoichiometry and/or its presence as a surface layer127–130 to reduce the work function or eliminate it as for the III–V negative electron affinity, or NEA, photocathodes.131,132 For the latter cathodes, Spicer131 and Fisher et al.133 proposed a narrow, or “interfacial,” barrier at the surface, often represented as a tall triangular barrier but of sufficient thinness that tunneling electrons mostly pass through. For other metals and semiconductors for which a submonolayer covering of cesium acts to reduce the barrier height to emission, the manner by which the barrier is reduced, is a consequence of dipole effects described successfully by phenomenological theories.66,134,135 Our investigations of coatings on surfaces using DFT have suggested that the hypothetical triangular barrier may, in fact, be better treated as a potential well37,43 for which the qualitative features of a triangular barrier are mimicked. The barrier-well structure resulting from those simulations qualitatively resembled a model of photoemission from a metal with a partial alkali or alkali earth coating on the surface proposed by Fowler in his treatment of “selective” photoemission (the transmission of electrons from the conduction band of a metal first through a barrier then past a well with the barriers and wells being square, or constant potential, regions), which, although challenged, matched data.136 Here, a quantification of the behavior resulting from a coating layer is undertaken. The evaluation of the transmission probability D[E(k)] demonstrates that the electrons that most contribute to QE are those for which the normal energy Ez=2kz2/2mn [in contrast to E that appears in the mean transverse energy (MTE) associated with emittance]. These electrons are sufficiently above the conduction band minimum that effective mass modifications do not arise; that is, the “effective” mass of the electron can be taken as the free-electron mass (nearly free-electron approximation). This approximation is made throughout. In addition, for the one-dimensional (1D) transmission problem, it is convenient to use kz rather than Ez and to suppress the z-subscript, as shall be done below.

The Pöschl–Teller (PT) potential well,137 aka sech2 potential,138 is modified (mPT) to be

Vpt(x)=2ν(ν+1)2ma2sech2(x/a).
(45)

For integer ν in Eq. (45), D(k)1 such that all incident electrons for a particular k are transmitted. We have shown previously that analogous behavior occurs for rectangular, triangular, and parabolic wells,37,43 wherein for an arc of k values, reflectionless transmission occurs. For the purposes of constructing an analytic representation of D(k) useful for the simulation of QE, however, the behavior of D(k) with respect to a particular k for the sech2 potential is advantageous, and therefore, it shall be used exclusively here. The justification is because macro-averaging of the potentials associated with coatings, as obtained from DFT, closely conforms to the sech2 potential.37 

For the non-integer ν in Eq. (45), then reflection can occur and can be determined using the transfer matrix approach (TMA) following methods described previously.43,59,139 For a potential mimicking a coating, V(x) is given by

Vpt(x)ΔVsech2(xxoan),
(46)

where xo specifies the location of the monolayer coating and an is ostensibly given by the Bohr radius for the nth shell of the coating atom or an=n/αfsmc. For cesium, n=6; therefore, a6=0.3175 nm. Also, a representative potential is shown in Fig. 22.

FIG. 22.

Vpt(x) for ΔV=Ry as per Eq. (22), composed of Np=12 discrete line segments for TMA analysis. The spacing of xj is chosen so that the points are more densely spaced near xo.

FIG. 22.

Vpt(x) for ΔV=Ry as per Eq. (22), composed of Np=12 discrete line segments for TMA analysis. The spacing of xj is chosen so that the points are more densely spaced near xo.

Close modal

As in Ref. 43, D(k) can be plotted with increasing ΔV to reveal when reflectionless conditions occur: an example for cesium-like parameters is shown in Fig. 23. Clearly, specific values of ΔV are associated with D(k)=1 for all k. In between, reflection for small k occurs. ΔV is correlated to the strength of the dipole associated with sub-monolayer coatings of cesium on surfaces in Gyftopoulous–Levine theory,59,66,135 and is, therefore, related to the degree of submonolayer coverage of cesium. Therefore, a representative selection of ΔVRy/4 needs to be parameterized. A contour representation of that region in shown in Fig. 24.

FIG. 23.

Lines of D(k) as ΔVj increases from 0 to Ry=13.6057 eV (fj increases from 0 to 1), clearly showing values of ΔV for which reflectionless transmission [D(k)=1 for all k] occurs.

FIG. 23.

Lines of D(k) as ΔVj increases from 0 to Ry=13.6057 eV (fj increases from 0 to 1), clearly showing values of ΔV for which reflectionless transmission [D(k)=1 for all k] occurs.

Close modal
FIG. 24.

The region of Fig. 23 for which ΔV0.25Ry rendered as a surface plot and emphasizing the region between the ν=1 (ΔV/Ry0.055) and ν=2 (ΔV/Ry0.161) reflectionless ridges.

FIG. 24.

The region of Fig. 23 for which ΔV0.25Ry rendered as a surface plot and emphasizing the region between the ν=1 (ΔV/Ry0.055) and ν=2 (ΔV/Ry0.161) reflectionless ridges.

Close modal

The region between the maximum reflection near ΔV=0.1Ry and the reflectionless condition near ΔV=0.164Ry shows conditions ranging from significant reflection for a small k to near total transmission for all k. For a step-function potential barrier, D(k) is known analytically59 to be given by D(k)=4kk/(k+k)2 with k=k2ko2 and 2ko2/2m=Vo equal to the height of a step-function barrier, corresponding to the asymptotic limit (as the field vanishes) of Fowler and Nordheim’s triangular barrier DFN(k) [compare Eqs. (15) and (23) of Ref. 37]. Such limits have but one independent variable (the barrier height), whereas the ad hoc triangular barrier of Spicer and Fisher et al. are further characterized by a field. As previously shown,37 the triangular barrier model is a fair representation of the sech2 well in that the behavior of D(k) in between the reflectionless (integer ν) regions qualitatively resembles the step potential D(k). Therefore, a rapid two-parameter model of D(k) for regions where reflection is significant for the sech2 potential wells is sought so as to quantitatively account for the presence of coatings.

Let each line shown in the waterfall plot of D(k) of Fig. 24 be indexed by j with the depth of the well given by ΔVjfjRy and fj=jfmax/Nv, for which Fig. 24 sets fmax=0.25 and Nv=96. The reflectionless conditions then occur for j=21 and 62 (corresponding to f21=0.0547,ν=1 and f62=0.1615,ν=2). The Dj(k) for 39j62 are shown in Fig. 25, approximately corresponding to the region where 1.5ν2. In examining the reflections that occur in that region, a sampling of which are shown in Fig. 25, it is evident that a better approximation to D(k) for the values of well depth that show reflection is reminiscent of the step potential. A two-parameter candidate is

D(k)kr(k2r+ka2r)1/2,
(47)

where ka and r are to be determined (and not to be confused with the notation for the optical constants in Secs. III and IV). Define a new kj by the relation Dj(kj)=1/2 whereby ka2r=3kj2r and resulting in

Dja(k)krj(k2rj+3kj2rj)1/2.
(48)

The best least-squares values of the remaining parameter rj are straightforward, resulting in the approximations shown by lines in Fig. 25.

FIG. 25.

Dots: values of Dj(k) determined by using TMA on the potential of Fig. 23 using ΔVj with (39j62). Lines: best least-squares fit using Eq. (48). The legend denotes a value of j from Fig. 24.

FIG. 25.

Dots: values of Dj(k) determined by using TMA on the potential of Fig. 23 using ΔVj with (39j62). Lines: best least-squares fit using Eq. (48). The legend denotes a value of j from Fig. 24.

Close modal

Beam simulations4,7 demanding evaluations of QE over sub-μm-scale areas and sub-ps time steps entail a large number of calls to the algorithms for generating QE when modeling an electron bunch: the number scales as Nx2Nt, with Nx=L/Δx and Nt=T/Δt, with L and T being the diameter of the emission area and the duration of the pulse, respectively. Device simulations for detectors, for which pair production and Compton scattering can result in energetic electrons that pass a bulk material/a contact barrier,40 have analogous demands. Consequently, even though an approximate form of D(k) given by Eq. (48) has become available, the requirements to perform a least-squares fit to the critical r-parameter and extract kj from a TMA evaluation, are substantial obstacles to its usage. It is necessary to either pre-load values for representative conditions and interpolate between them or reduce the time required for an evaluation. The latter approach motivates additionally approximating r and kj by simple functions. As shown in Fig. 26, the approximations,

sj=(j39)/23,rj=0.8891sj7/50.8176,kj=1.7219sj20.6935,
(49)

perform adequately. Therefore, a means to rapidly modify D(k) for varying submonolayer coatings is available, or, conversely, to rapidly determine parameters for a given configuration. More importantly here, however, is that an approximation to D(k) that replaces the ad hoc triangular barrier model is now made available and accounts for the effect of a coating or a surface layer (like graphene37,45,46) so as to predictively estimate QE from surfaces that have been modified. Rapid methods to treat interface barriers are described separately.140 

FIG. 26.

Dots: values of rj and kj for sj(j39)/23. Lines: approximations given by Eq. (49). “j” refers to Dj(k) from Fig. 24.

FIG. 26.

Dots: values of rj and kj for sj(j39)/23. Lines: approximations given by Eq. (49). “j” refers to Dj(k) from Fig. 24.

Close modal

The calculation of quantum efficiency and photoexcitation in complex multicomponent materials and structures requires numerically efficient algorithms for beam optics and device simulation codes to model the generation of electron beams from a photocathode, predict detector behavior, or characterize performance. Three-step models (TSMs) and simple moments models (SMMs) used to treat bulk materials provide an effective and widely used basis for simulation. However, these approaches require extensions to their underlying physics models to treat novel and coated materials with surface modifications, particularly if exploiting wave interference (an “etalon” thin film cathode) to enable fast responding structures with improved quantum efficiency is desired or if extending predictive capabilities to frequency regimes where optical or material properties are lacking is needed. In this contribution, modifications to the SMM model to allow for the effects of thin films were developed and demonstrated across a number of technologically relevant materials.

The predictive capabilities of DFT to treat the optical properties of materials, and of TMA to model emission past barrier structures, have been shown here to satisfy the physics modeling needs. Their computational cost (ill-afforded in device simulation and beam optics codes) has been mitigated by developing parametric models. For DFT, optical parameters such as the reflectivity R(ω) and penetration depth δ(ω) of Eq. (14) are well-described by parameterizing a Drude–Lorentz fit with additional resonant terms for both metals and semiconductors. A method to identify and evaluate the required frequency, damping term, and weight, or (ωj,γj,fj) parameters was presented, vetted on metals and semiconductors used in photocathodes, and projected to treat perovskites where the optical constants are extended to large frequencies using DFT methods. For TMA, the types of barrier modifications associated with coatings were parameterized by a Poschl–Teller well representation and their relation to the transmission probability associated with triangular barriers for non-reflectionless conditions. These techniques (in conjunction with methods treating interface transport and temperature effects) are being adapted for inclusion into characterization and simulation codes, the performance of which shall be reported in a separate contribution.

This work was supported in part by the Los Alamos National Laboratory (LANL) Directed Research and Development Funds (LDRD) and partly conducted at the Center for Nonlinear Studies (CNLS) and the Center for Integrated Nanotechnologies (CINT), U.S. Department of Energy (DOE). This research used resources provided by the LANL Institutional Computing (IC) Program. LANL is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy (Contract No. 89233218NCA000001). This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. We thank Dr. Oksana Chubenko (Arizona State University) for discussions regarding the optical constants and their modeling.

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

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