Theory of photoemission from cathodes with disordered surfaces

The development of advanced accelerator applications such as x-ray free electron lasers (XFEL),¹ energy recovery linacs (ERL),² and ultrafast electron diffraction (UED)³ depends to a large extent on the brightness of electron sources used to generate electron bunches. High density bunched electron beams for such applications are typically produced using photoinjectors, which consist of a photocathode placed in an accelerating electric field $E$ along with electron optics to mitigate the brightness-degrading effects of space charge.

Electron beam brightness is a key figure of merit for these applications. The maximum charge density that can be extracted from a photoinjector is proportional to the $n$th power of the accelerating electric field $E$⁠. Hence, the maximum brightness achievable from a photoinjector is proportional to the nth exponent of the electric field $E$ and is inversely proportional to the Mean Transverse Energy (MTE) of the photoemitted electrons,⁴ 

where $n$ is a real number between 1 and 2, depending on the design of a photoinjector. The MTE is related to the intrinsic emittance $εi$⁠, which is a measure of the volume occupied by the beam in phase space, via the following expression:

where $σx$ is the rms transverse beam size, $m$ is the rest mass of a free electron, and $c$ is the speed of light.

The maximum possible accelerating field depends on the design of the photoinjector, but the MTE is a property of the cathode material, its surface, as well as laser driving conditions. It is imperative to reduce the MTE in order to maximize the beam brightness. The MTE is often lowered by operating the cathode near its photoemission threshold, at the cost of its Quantum Efficiency (QE).⁵ MTE and QE, which are two intrinsic properties of the photocathode, can potentially limit the beam brightness and are studied in order to understand the photoemission process.

The Dowell–Schmerge (D–S) scheme,^6,7 which is based on Sommerfeld’s free electron gas theory in solids, coupled with Spicer’s three-step model⁸ is often used to explain photoemission from most cathodes in use today. Under the assumptions of a nearly free electron gas model, parabolic dispersion relations, conservation of transverse momentum, constant density of states (DOS), and zero lattice temperature,^6,7 the expressions for QE and MTE were analytically calculated by D–S.

In the D–S scheme, the QE was shown to vary quadratically with the excess energy $Eexcess$ as

where the excess energy is defined as the difference between the incident photon energy $ℏω$ and the effective material work function $ϕ$⁠, after accounting for the Schottky reduction.⁹ Well above the photoemission threshold, i.e., at large excess energies, the MTE was found to vary linearly with the excess energy as

A wide range of polycrystalline photocathodes like Sb,^5,7 Cu,^7,10 Cs$2$Te,¹¹ and alkali antimonides like Cs$3$Sb¹² and Na$2$KSb¹³ have demonstrated MTE proportional to one-third of the excess energies at large excess energies, as predicted by the D–S scheme.

A more recent work by Vecchione et al.,¹⁴ often referred to as the extended D–S scheme, has included the effect of non-zero temperature on the Fermi–Dirac distribution of electrons, which leads to the polylogarithmic expressions for QE and MTE,

where $kB$ is the Boltzmann constant, Li$2$ and Li$3$ are the polylogarithm functions of order 2 and 3, respectively, $S$ is a constant, and $T$ is the temperature of the electrons in the crystal. For smaller laser fluences, $T$ is the lattice temperature as the electrons can be considered to be in equilibrium with the lattice. Near the photoemission threshold, where the excess energies are practically zero or negative, emission essentially occurs from the Fermi tail, which limits the value of MTE to the thermal limit of $kBT$⁠. The thermal limit for MTE has been experimentally demonstrated from thin polycrystalline Sb films at room temperature⁵ and Cu (100) cathodes at cryogenic temperatures.¹⁰ 

A large number of assumptions⁶ have been considered in the D–S scheme, which do not hold true for the aforementioned cathodes with disordered surfaces. For example, the transverse momentum is not a conserved quantity for polycrystalline cathodes with disordered surfaces.¹⁵ For semiconductor cathodes, the valence density of states close to the Fermi level, from which photoexcitation takes place near the threshold, cannot be assumed to be a constant. Also, parabolic dispersion relations are not obeyed by the band structures of most cathodes, especially near the vacuum level. Thus, the assumptions mentioned above, which are intrinsic to the D–S scheme, cannot be applied to many of the cathodes that have been experimentally studied. Nonetheless, all of the cathodes mentioned earlier have experimentally demonstrated QE and MTE near and above the threshold, as predicted by the D–S scheme.

The D–S scheme, through the unphysical assumptions of the single-band-parabolic dispersion relation and the conservation of transverse momentum, happens to arrive at a distribution of emitted electrons in which the parabolic dispersion relationship of the emitted (free) electrons is uniformly filled, giving the QE and MTE relationships. We realized that such a uniformly filled dispersion relation is also possible due to a disordered surface—which is the case for all photocathodes used in guns thus far. An atomically disordered surface can cause electrons to be scattered uniformly in all energetically allowed momenta.

In this paper, within the framework of Spicer’s three-step formulation,⁸ we develop a photoemission model, which reproduces expressions for QE and MTE that were obtained by the D–S scheme, without resorting to the unrealistic assumptions. The first two steps of optical excitation and electron transport have been treated in the traditional way as in the D–S scheme. The third step of emission has been treated in an unconventional manner by incorporating the quantum mechanical effect of scattering from a disordered surface. Section II describes the three steps of our photoemission model in detail. We then apply this model to Cs$3$Sb cathodes in Sec. III and show how QE and MTE near and below the threshold can be dominated by defect/impurity density of states lying within the bandgap of Cs$3$Sb cathodes.

In this section, we discuss the three steps of photoemission: optical excitation, transport, and emission in detail and show how we can derive the expressions for QE and MTE from cathodes with disordered surfaces.

The first step is the optical excitation of electrons from the occupied valence band density of states (DOS) into the vacant conduction band DOS due to the absorption of photons with energy $ℏω$ as indicated by Step 1 in Fig. 1. The probability for this transition to occur from the state with energy $E$⁠, measured with respect to the Fermi level energy, to $E+ℏω$ follows from Bergund and Spicer’s formalism.¹⁶ For polycrystalline cathodes in which the band structure is not well-defined, the probability is proportional to the number of electrons in the occupied states at energy $E$⁠, which is given by the product term $D(E)f(E)$⁠, and the number of unoccupied states at energy $E+ℏω$⁠, which is similarly given by $D$(⁠$E+ℏω$⁠)[$1−f(E+ℏω)$]. Here, $D$(⁠$E)$ denotes the density of states at initial energy of the electrons E; similarly, $D$(⁠$E+ℏω$⁠) represents the density of states at energy $E+ℏω$⁠. Here, $f(E)$ is the Fermi–Dirac distribution which is used to define the occupation probability of the electrons,

where $Ef$ is the position of Fermi level. The probability of excitation to occur can be written as

For single crystalline cathodes, excitation can be modeled using vertical transitions in the band structure, obeying the law of conservation of energy and momentum.¹⁵ However, we do not consider the single crystalline case in this paper.

The second step is the transport of these photoexcited electrons to the cathode–vacuum interface. In the case of metallic cathodes, electron–electron (e–e) scattering is the most dominant scattering mechanism during transport. When such scattering events occur, the electrons excited within a few 100 meV above the threshold lose enough energy that they fall below the vacuum level and do not get emitted at all. For photon energies within a few 100 meV of the threshold, the fraction of electrons that are lost due to such an e–e scattering process can be treated as a constant invariant of the photon energy and electron energy.¹⁷ 

In the case of semiconductor cathodes with a low electron affinity, the e–e scattering is prohibited due to the lack of final states into which scattered electrons can relax, owing to the presence of the forbidden bandgap energy.¹⁷ This makes electron–phonon/electron–plasmon/electron–impurity (e–p/e–i) scattering processes dominant in semiconductor materials.^18–20 The energy distribution of the electrons, which reach the interface, gets modified due to these scattering processes. However, given the small thickness of these cathodes and low energy transfer rates, a relatively large fraction of the scattered electrons retain enough energy to still get emitted, resulting in a higher overall QE compared to metallic cathodes. The losses suffered due to these scatterings and the resulting change in the electron energy distribution during transport can be studied in greater detail by employing the Monte Carlo techniques to solve the Boltzmann transport equation.^18–20 If the cathode thickness is smaller than the mean free path of excited electrons (typically in the range of several 10 s to 100 s of nm for semiconductors), the number of scattering events before emission can be assumed to be very small such that it will not affect the energy distribution dramatically. In such situations, we can assume that the probability of an excited electron reaching the surface is invariant with the electron and photon energy.

In this paper, we assume that the cathodes are either metallic or semiconducting thin films with a thickness smaller than the mean free path of the electrons. Thus, the energy distribution of the excited electrons does not undergo any significant changes during transport. Hence, the probability of an excited electron reaching the surface with enough energy for emission can be accounted for by a constant, invariant of the photon energy, and the electron energy. This probability has been denoted by a constant $Ct$ at all photon energies

The third step is the emission of electrons at the cathode–vacuum interface into vacuum by overcoming the work function/photoemission threshold of the material. Here, we make the assumption that an electron, incoming toward the surface with any wave vector, scatters uniformly into all energetically possible wave vectors with equal probability. This assumption can be justified due to a quantum mechanical scattering effect of the electrons from a disordered surface.

Consider a surface with a disorder in the form of a very small sinusoidal variation in height or work function with a period comparable to that of the de Broglie wavelength of the emitted electrons. For emitted electrons with kinetic energies in the 1 meV–1 eV range, which is typical for near-threshold emission from photocathodes, the de Broglie wavelength is around 1–40 nm. Electrons, scattering quantum mechanically from such a surface, can scatter into a transverse wave vector equal to the transverse (parallel to the surface) part of the incoming electron wave vector plus harmonics of the wave vector of the periodic disorder on the surface, so long as it is energetically allowed.²¹ Extending this approach to a randomly disordered surface that is composed of all wave vectors, one can envision that any incoming electron wave vector will be scattered into all energetically allowed transverse wave vectors with equal probability, justifying our assumption.

The conservation of energy imposes an upper limit on the values of transverse momentum $k⊥$⁠, with which the electrons can be emitted. The maximum transverse momentum $k⊥max$ is given by the kinetic energy $Ekinetic$ as

where $ℏ$ is the reduced Planck’s constant. The emission probability of electrons with a particular energy $E$ is then proportional to the number of vacuum states it can scatter into and can be expressed as

where $Ekinetic=E+ℏω−Evacuum$ is the kinetic energy of the electrons and $Evacuum$ is the vacuum level position of the cathode material measured with respect to the valence band maximum (VBM) and $Ce$ is a constant that involves the 2D density of states (for the two transverse directions) and will have units of 1/energy.

The variation of the work function being considered here is on a spatial scale smaller than or comparable to the de Broglie wavelength of the emitted electrons. Hence, we do not expect this variation to cause significant spatial non-uniformity in the emitted current, unlike in the case where the work function variation is on the larger micron spatial scales.²² In our case, we can assume the work function to have a constant average value and account for the variation only in the surface scattering of the electrons during emission.

The emission probability of the electrons, which reach the cathode–vacuum interface, is, thus, proportional to their kinetic energy. This is indicated by Step 3 in Fig. 1. This also implies that only those electrons, which have kinetic energies greater than the vacuum level energy, will get emitted.

Combining the probabilities for excitation, transport, and emission to occur as discussed above, the expression for the number of electrons emitted can be written as

Similarly, the number of electrons excited is given by

Quantum efficiency can be defined as the ratio of the number of electrons emitted to the number of electrons excited,

where $α$ accounts for optical absorption by the cathode.

Using Eqs. (8) through (14), expression for the QE as a function of the photon energy can be written as

Similarly, the expression for MTE can be formulated by calculating the average transverse momenta squared,

which, by substituting the value of $k⊥max$ from Eq. (10), reduces to

We first consider a constant density of states and finite lattice temperature in order to compare our simulations for QE and MTE to the extended D–S scheme. To begin with, we solve the expressions for QE and MTE in Eqs. (15) and (17) analytically in the following manner. Under the assumption of a constant DOS, Eq. (15) reduces to

At zero temperature, Eq. (18) gives

Equation (19) can be integrated to obtain the well-known dependence of QE on excess energy as predicted by the D–S scheme

The excess energy is defined as the difference between the incident photon energy and the work function of the material. Typically, work function is defined as the vacuum level energy with respect to the Fermi level. In the case of semiconductor photocathodes, the Fermi level is not too well-defined as it often lies within the forbidden bandgap. Hence, it is reasonable to modify the expression of work function to be the difference in energy between the vacuum level and the highest occupied electron level, which in this case is the VBM instead of the Fermi level. Therefore, for semiconductor cathodes, the excess energy can be defined as the difference in energy between the incident photon energy and the vacuum level energy. The VBM is considered to be the zero level for reference.

Under the same assumption of a constant DOS, the expression for MTE in Eq. (17) reduces to

Assuming a zero temperature dependence, Eq. (21) reduces to

Integrating Eq. (22) results in MTE being proportional to one-third of the excess energy obtained by the D–S scheme

Equations (18) and (21) can be numerically integrated to obtain the spectral response of QE and MTE under the assumption of a constant DOS for non-zero temperatures. This has been plotted in Figs. 2 and 3, respectively. We see in Fig. 2 that QE varies as square of the excess energy as was predicted by the D–S scheme. MTE varies linearly with the excess energy as E$excess$/3 beyond threshold and goes down to $kBT$ at zero/ negative excess energies as shown in Fig. 3.

Thus, our photoemission model reproduces the same experimentally measured QE and MTE expressions as the D–S scheme without the use of unphysical assumptions like parabolic dispersion relations and conservation of transverse momentum for disordered surfaces. Furthermore, our model can easily be extended to explain near the threshold photoemission from in-gap defect states in semiconductor cathodes. In Sec. III, we use this model to show that the experimentally measured MTE from alkali-antimonide semiconductor cathodes can be limited by such defect states.

High QE photocathodes like Cs$3$Sb are an excellent choice for electron sources in different high beam brightness applications. Due to their high QE, large laser power does not have to be used to extract the required charge densities, thereby significantly mitigating the possibility of MTE growth via different non-linear effects.^23,24

In an attempt toward maximizing brightness by reducing MTE, the MTE from Cs$3$Sb photocathodes close to the threshold has been measured at both room and cryogenic temperatures. At both temperatures, MTE did not go all the way down to the thermal limit $kBT$ as was predicted by the extended D–S scheme. It was measured to be 40 meV at room temperature and 22 meV at 90 K—significantly larger than $kBT$⁠, which is 25 and 8 meV at room temperature and 90 K, respectively.²⁵ This discrepancy was attributed to the surface roughness and work function variations,²⁵ and defect states within the bandgap.⁸ While the effects of the surface roughness and work function variations on MTE have been quantitatively investigated in great detail,^21,26,27 the effect of in-gap defect states on QE and MTE has never been investigated. In this section, we apply our photoemission model to investigate the effect of the defect states on QE and MTE and show how larger MTE could arise from the in-gap defect states.

In our model, we use the valence and conduction band DOS of Cs$3$Sb calculated from the density functional theory (DFT).²⁸ The DFT calculations have been performed, assuming that these cathodes are single crystalline. In order to account for the polycrystalline and disordered nature of these cathodes, an exponential tail of width 0.04 eV has been added to the valence band DOS. This tail width has been used to obtain the best possible fit with experimental data. Such a value is also typical for the exponential tail width of disordered semiconductors.^19,29

Figure 4 shows the density of states that has been used to model photoemission from Cs$3$Sb cathodes. The approximate position of the Fermi level $Ef$ and the vacuum level $Evacuum$⁠, which have been used in the simulation, is indicated by dotted lines in Fig. 4. Table I gives the values of these two material-dependent parameters, which have been used in the simulation of QE and MTE. These values have been used as they provide the best fit to the QE experimental data. A variation of 0.1–0.2 eV in the position of the Fermi level is common among semiconductors over a temperature range between 20 and 300 K.³⁰ All energies are measured with respect to the VBM.

The spectral response of QE from Cs$3$Sb cathodes is calculated using Eq. (15). Figure 5 shows a close resemblance of the calculated values of QE with experimental values at two sets of temperature: room temperature 300 K³¹ and cryogenic temperature 77 K.⁸ The photoemission threshold is marked by the knee around $∼2$ eV. Below the threshold, the QE drops sharply. Without the presence of defect states, the QE should drop to zero right at the threshold. However, the exponentially decaying tail in QE below the threshold is due to the defect density of states, which decreases exponentially with increasing energy. It is also affected by the location of the Fermi level to some extent.

A reduction in the QE is observed when the cathode is cooled down to 77 K.⁸ This is explained by the shorter Fermi tail leading to the reduction in the number of defect states filled with electrons, with energies enough to be excited above the threshold. Figure 6 shows the behavior of calculated MTE values at large excess energies [using Eq. (17)] compared with experimentally obtained values from Cs$3$Sb cathodes at room temperatures.¹² Figure 7 shows the comparison of MTE at incident photon energies very close to and below the threshold at both room temperature and cryogenic temperature of 90 K.²⁵ It can be seen that the calculated values of MTE from our model are in close agreement with those observed experimentally at low/negative excess energies.

Thus, we see that our theoretical model predicts that cathodes with disordered surfaces and defect states in the bulk, when operated near the threshold may yield MTE values, which are above $kBT$ even below the threshold. Furthermore, these higher MTE values are related to the exponential decay of the defect density of states, which also determines the exponential decay of the QE below the threshold. Thus, cathodes with a longer QE decay below the threshold can be expected to have a higher MTE at or below the threshold. This increase in MTE will be beyond the increase caused due to the surface roughness of such cathodes.

We have developed a new photoemission model based on Spicer’s three-step formalism to explain photoemission from photocathodes with disordered surfaces. Our model obtains the same QE and MTE relationships as the D–S scheme without the need to use unphysical assumptions like parabolic dispersion relations or conservation of transverse momentum and can be applied to metals as well as semiconductor cathodes.

We have used this model to obtain a theoretical understanding of photoemission from polycrystalline alkali-antimonide photocathodes at photon energies close to and below the threshold. Below the threshold, photoemission is due to the excitation of electrons from states populated with electrons from the Fermi tail as well as an exponentially decaying density of states above the VBM caused due to defects and impurities. Our model predicts values of QE and MTE from such polycrystalline Cs$3$Sb cathodes, which are in excellent agreement with those observed experimentally at both room and cryogenic temperatures. This also suggests that the MTE from such cathodes may not reach the thermal limit, when operated near the threshold due to such defect states and a more detailed study of the effects of such defect states on MTE is necessary to maximize the brightness of electron beams.

This work was supported by the U.S. National Science Foundation under Award No. PHY-1549132 and the Center for Bright Beams, the DOE under Grant Nos. DE-SC0021092 and DE-SC0020575.

The authors have no conflicts to disclose.

Pallavi Saha: Conceptualization (supporting); Data curation (lead); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Oksana Chubenko: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (equal); Validation (supporting); Visualization (supporting); Writing – review & editing (equal). J. Kevin Nangoi: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (equal); Validation (supporting); Visualization (supporting); Writing – review & editing (equal). Tomas Arias: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (equal); Validation (supporting); Visualization (supporting); Writing – review & editing (equal). Eric Montgomery: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (equal); Supervision (supporting); Validation (supporting); Visualization (supporting); Writing – review & editing (equal). Shashi Poddar: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (equal); Validation (supporting); Visualization (supporting); Writing – review & editing (equal). Howard A. Padmore: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (equal); Supervision (supporting); Validation (supporting); Visualization (supporting); Writing – review & editing (equal). Siddharth Karkare: Conceptualization (lead); Data curation (supporting); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Software (equal); Supervision (lead); Validation (supporting); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.