^{1,a)}, Eric J. Montgomery

^{2}, Donald W. Feldman

^{2}, Patrick G. O’Shea

^{2}, John R. Harris

^{3}, John W. Lewellen

^{3}and Nathan Moody

^{4}

### Abstract

An oft used approximation to predict quantum efficiency (QE) from bare metals or those with a low work function coating such as cesium is to assume that photo-excited electrons have not scattered prior to their emission. Monte Carlo simulations are used to assess that approximation, and show that, while good for bare metals, for cesiated metals a photoexcitedelectron may undergo several scattering events and yet be emitted. Neglecting scatteredelectrons therefore underestimates QE. Emitted electrons that have undergone scattering before emission elongate the response time by giving rise to a long time tail, low energy contribution to the faster non-scattered emission, for which a model is developed. The theory is applied to study variations in QE as a function of wavelength measured from cesiated metal surfaces. The extension of the findings to semiconductorphotocathodes is briefly discussed.

We thank the *Joint Technology Office* (JTO) and the *Office of Naval Research* (ONR) for their support of this work. We thank D. Dowell for access to the experimental data shown in Fig. 18, and both him and J. Smedley (Brookhaven National Laboratory) for discussions.

I. INTRODUCTION

II. METHODS

A. Scattering factor and quantum efficiency

B. Monte Carlo Implementation

1. Initial distribution

2. Relaxation times

3. Collisions and scattering

III. RESULTS

A. Fatal approximation and its limitations

B. Current and time response

C. Energy distribution

D. Including scatteredelectrons in the moments model

1. Bare metals

2. Cesiated metals

E. Implications for semiconductors

IV. CONCLUSION

### Key Topics

- Electron scattering
- 149.0
- Photoemission
- 19.0
- Copper
- 18.0
- Photoexcitations
- 18.0
- Work functions
- 18.0

## Figures

(Color online) Scattering times as a function of energy for acoustic (green square), electron-electron (blue diamond), and total (red circle) for 77 K (dash) and 300 K (solid line). At 1.6 eV above the Fermi level, the *e-e* and acoustic relaxation times are 20.4 and 107 fs, respectively, at 300 K.

(Color online) Scattering times as a function of energy for acoustic (green square), electron-electron (blue diamond), and total (red circle) for 77 K (dash) and 300 K (solid line). At 1.6 eV above the Fermi level, the *e-e* and acoustic relaxation times are 20.4 and 107 fs, respectively, at 300 K.

(Color online) Time slices at every 6 fs (starting at 2 fs) of 1000 photoexcited electrons in bare copper for the Fatal approximation (blue top), allowing one scattering event (green middle) and allowing all scatterings (red bottom). All electrons whose energy falls below the surface barrier are removed from visualization. Red electrons move toward the surface (to the right) and blue electrons move away (to the left). The images are similar, implying the Fatal approximation is reasonable. Length of simulation region is approximately 12 nm.

(Color online) Time slices at every 6 fs (starting at 2 fs) of 1000 photoexcited electrons in bare copper for the Fatal approximation (blue top), allowing one scattering event (green middle) and allowing all scatterings (red bottom). All electrons whose energy falls below the surface barrier are removed from visualization. Red electrons move toward the surface (to the right) and blue electrons move away (to the left). The images are similar, implying the Fatal approximation is reasonable. Length of simulation region is approximately 12 nm.

(Color online) Same as Fig. 2, but for Cs–Cu. The time slices are every 8 fs starting at 4 fs. The figures are different, implying that scattered electrons may nevertheless still be emitted if conditions are favorable. Length of simulation region is approximately 16 nm.

(Color online) Same as Fig. 2, but for Cs–Cu. The time slices are every 8 fs starting at 4 fs. The figures are different, implying that scattered electrons may nevertheless still be emitted if conditions are favorable. Length of simulation region is approximately 16 nm.

(Color online) Evolution in mean energy of electrons for which and a standard deviation *σ* for Cu parameters. After two scatterings, less than 1% of the initial 10^{6} electrons remain.

(Color online) Evolution in mean energy of electrons for which and a standard deviation *σ* for Cu parameters. After two scatterings, less than 1% of the initial 10^{6} electrons remain.

(Color online) Same as Fig. 4 but for Cs–Cu parameters. After eight scattering events, less than 1% of the initial 10^{6} electrons in the simulation remain, but assigning a % loss factor after each scattering is impractical because the mean energy changes and therefore also changes.

(Color online) Same as Fig. 4 but for Cs–Cu parameters. After eight scattering events, less than 1% of the initial 10^{6} electrons in the simulation remain, but assigning a % loss factor after each scattering is impractical because the mean energy changes and therefore also changes.

(Color online) The emitted electrons as a percentage of the initial number for Cu parameters (blue square) or Cs–Cu (red circle). Primary (solid symbols and lines) indicates photoexcited electrons; secondary (open symbols and dashed lines) indicates electrons excited by collisions with the photoexcited electrons.

(Color online) The emitted electrons as a percentage of the initial number for Cu parameters (blue square) or Cs–Cu (red circle). Primary (solid symbols and lines) indicates photoexcited electrons; secondary (open symbols and dashed lines) indicates electrons excited by collisions with the photoexcited electrons.

(Color online) The ratio of the number of emitted secondaries to number of emitted primaries for the Cs–Cu data of Fig. 6 as a function of photon energy, and compared to Eq. (5).

(Color online) The ratio of the number of emitted secondaries to number of emitted primaries for the Cs–Cu data of Fig. 6 as a function of photon energy, and compared to Eq. (5).

(Color online) Emitted electrons are from two population types: the left are modeled as an expanding spherical *shell* of charge; the right are modeled as a diffusively expanding *sphere* with a Gaussian distribution of charge. The origin of each (circle) is within the copper (grey) surface.

(Color online) Emitted electrons are from two population types: the left are modeled as an expanding spherical *shell* of charge; the right are modeled as a diffusively expanding *sphere* with a Gaussian distribution of charge. The origin of each (circle) is within the copper (grey) surface.

(Color online) Emitted charge as a function of time for bare copper parameters (266 nm photons, 4.5 eV work function, 10 MV/m field). The inset equation is Eq. (6). The initial number of electrons is 10^{6}.

(Color online) Emitted charge as a function of time for bare copper parameters (266 nm photons, 4.5 eV work function, 10 MV/m field). The inset equation is Eq. (6). The initial number of electrons is 10^{6}.

(Color online) Same as Fig. 9, but for cesiated copper parameters (266 nm photons, 1.6 eV work function, 10 MV/m field). Using Eqs. (6) and (7) with the parameters *Qs* = 14 828, *Qd* = 69 127, and *τ** = 2.53 fs gives the best least squares correspondence (*t _{0} * is fixed at 3.42 fs).

(Color online) Same as Fig. 9, but for cesiated copper parameters (266 nm photons, 1.6 eV work function, 10 MV/m field). Using Eqs. (6) and (7) with the parameters *Qs* = 14 828, *Qd* = 69 127, and *τ** = 2.53 fs gives the best least squares correspondence (*t _{0} * is fixed at 3.42 fs).

(Color online) Same as Fig. 10, but on linear axes to show long time behavior.

(Color online) Same as Fig. 10, but on linear axes to show long time behavior.

(Color online) Current as obtained from the time derivative of Eqs. (6) and (7) using parameters extrapolated from Fig. 11.

(Color online) Current as obtained from the time derivative of Eqs. (6) and (7) using parameters extrapolated from Fig. 11.

(Color online) Histograms of energy distribution of emitted electrons from copper: (left) total energy; (middle) transverse energy; (right) normal energy. Legend is labeled by number of scattering events plus one: *n* = 1 (no scattering) is the fatal approximation; *n* = 2 is fatal + one scattering event, as considered by Berglund and Spicer in Ref. 8 For copper further changes beyond *n* = 3 are not perceptible.

(Color online) Histograms of energy distribution of emitted electrons from copper: (left) total energy; (middle) transverse energy; (right) normal energy. Legend is labeled by number of scattering events plus one: *n* = 1 (no scattering) is the fatal approximation; *n* = 2 is fatal + one scattering event, as considered by Berglund and Spicer in Ref. 8 For copper further changes beyond *n* = 3 are not perceptible.

(Color online) Same as Fig. 13 but for cesium on copper. The energy distribution is much more broad due to the small work function, and scattering up to *n* = 10 give visible changes.

(Color online) Same as Fig. 13 but for cesium on copper. The energy distribution is much more broad due to the small work function, and scattering up to *n* = 10 give visible changes.

(Color online) Percentage of emitted electrons as a function of number of tangible scattering events they had. For copper (blue square), less than 1% of the total number of electrons have had more than one scattering; for cesium on copper (red dot), roughly half of the emitted electrons have experienced one or more scatterings before emission.

(Color online) Percentage of emitted electrons as a function of number of tangible scattering events they had. For copper (blue square), less than 1% of the total number of electrons have had more than one scattering; for cesium on copper (red dot), roughly half of the emitted electrons have experienced one or more scatterings before emission.

(Color online) Emitted charge (black squares) is the sum of two populations: those electrons that do not scatter before emission (red circles) and those that do (blue diamonds). When , the contribution of scattered electrons is small, i.e., the Fatal approximation is good. When , the two populations are comparable. Each simulation began with 10^{5} electrons.

(Color online) Emitted charge (black squares) is the sum of two populations: those electrons that do not scatter before emission (red circles) and those that do (blue diamonds). When , the contribution of scattered electrons is small, i.e., the Fatal approximation is good. When , the two populations are comparable. Each simulation began with 10^{5} electrons.

(Color online) Using Monte Carlo, the ratio of the total emitted charge with the unscattered charge, anticipated to be proportional to the ratio of the full QE with the QE calculated using the Fatal approximation. Variations in wavelength for a given work function are black solid and open circles. Variations in work function for a given wavelength for cesium on copper (Cs–Cu) parameters are colored diamond, square, and triangle symbols. An analytical fit to the shorter wavelength Cs–Cu conditions is also shown.

(Color online) Using Monte Carlo, the ratio of the total emitted charge with the unscattered charge, anticipated to be proportional to the ratio of the full QE with the QE calculated using the Fatal approximation. Variations in wavelength for a given work function are black solid and open circles. Variations in work function for a given wavelength for cesium on copper (Cs–Cu) parameters are colored diamond, square, and triangle symbols. An analytical fit to the shorter wavelength Cs–Cu conditions is also shown.

(Color online) Comparison of the moments-based predictions of QE from bare copper using the nonfatal approximation (red solid line) and the fatal approximation (blue dashed line) with data from Fig. 8 of Ref. 21 (data courtesy of D. Dowell).

(Color online) Comparison of the moments-based predictions of QE from bare copper using the nonfatal approximation (red solid line) and the fatal approximation (blue dashed line) with data from Fig. 8 of Ref. 21 (data courtesy of D. Dowell).

(Color online) Work function of submonolayer coatings of cesium on copper (dashed lines) and tungsten (solid lines) for various *f* (labeled by crystal face) using Gyftopoulos–Levine theory. Experimental measurements herein correspond to the [B] line.

(Color online) Work function of submonolayer coatings of cesium on copper (dashed lines) and tungsten (solid lines) for various *f* (labeled by crystal face) using Gyftopoulos–Levine theory. Experimental measurements herein correspond to the [B] line.

(Color online) Images of the surface of a sintered tungsten dispenser cathode surface. Different crystal faces have different gray scales associated with them.

(Color online) Images of the surface of a sintered tungsten dispenser cathode surface. Different crystal faces have different gray scales associated with them.

(Color online) Same as Fig. 17, but *R*(*x*) for cesium on tungsten (Cs–W) parameters. The coefficients and power differ, and there is some variation with wavelength, but a useful analytical fit is shown.

(Color online) Same as Fig. 17, but *R*(*x*) for cesium on tungsten (Cs–W) parameters. The coefficients and power differ, and there is some variation with wavelength, but a useful analytical fit is shown.

(Color online) Comparison of the QE obtained by experiment at UMD for various wavelengths (shown on left-hand-side in the same color and relative height as the peak QE). Theory was calibrated by relative peak location and height of the green data line. It is seen that the *R*(*x*) prefactor better accounts for both shape and comparative magnitudes of the different QE relations.

(Color online) Comparison of the QE obtained by experiment at UMD for various wavelengths (shown on left-hand-side in the same color and relative height as the peak QE). Theory was calibrated by relative peak location and height of the green data line. It is seen that the *R*(*x*) prefactor better accounts for both shape and comparative magnitudes of the different QE relations.

(Color online) The normalized current functions *B*(*x*,*n*) and *D*(*x*) for unscattered and diffusive emission. “Shell” refers to the *B* function with the number following “shell” indicating the value of *n*. The dashed line corresponds to the diffusive emission term of Eq. (11).

(Color online) The normalized current functions *B*(*x*,*n*) and *D*(*x*) for unscattered and diffusive emission. “Shell” refers to the *B* function with the number following “shell” indicating the value of *n*. The dashed line corresponds to the diffusive emission term of Eq. (11).

## Tables

Parameter values and units and definitions and representative values of parameters used in calculations.

Parameter values and units and definitions and representative values of parameters used in calculations.

Copper, tungsten, and cesiated values. Terms used in the evaluation the relaxation times for copper, cesiated copper, tungsten, and cesiated tungsten in both Monte Carlo and moments QE numerical evaluations (adapted from Refs. 29, 38 except for μ(W) which is taken from Ref. 76).

Copper, tungsten, and cesiated values. Terms used in the evaluation the relaxation times for copper, cesiated copper, tungsten, and cesiated tungsten in both Monte Carlo and moments QE numerical evaluations (adapted from Refs. 29, 38 except for μ(W) which is taken from Ref. 76).

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