Simulations of Charge Gain and Collection Efficiency from Diamond Amplifiers Dimitre A. Dimitrov1, Richard Busby1, John R. Cary1, Ilan Ben-Zvi2,3, John Smedley2, Xiangyun Chang2, Triveni Rao2, Jeffrey Keister2, Erik Muller2,3, and Andrew Burrill2 1Tech-X Corporation, Boulder, CO, U.S.A. 2Brookhaven National Laboratory, Upton, NY, U.S.A. 3Stony Brook University, Stony Brook, NY, U.S.A. ABSTRACT A promising new concept of a diamond amplified photocathode for generation of high-current, high-brightness, and low thermal emittance electron beams was recently proposed and is currently under active development. To better understand the different effects involved, we have been developing models, within the VORPAL computational framework, to simulate secondary electron generation and charge transport in diamond. The implemented models include inelastic scattering of electrons and holes for generation of electron-hole pairs, elastic, phonon, and charge impurity scattering. We will discuss these models and present results from 3D VORPAL simulations on charge gain and collection efficiency as a function of primary electron energy and applied electric field. The implemented modeling capabilities already allow us to investigate specific effects and compare simulation results with experimental data. INTRODUCTION

To address the need for high average-current, high brightness electron beams in current and future accelerator-based systems (e.g. electron cooling of hadron accelerators, energy-recovery linac light sources, and ultra-high power free electron lasers), a new design for a photoinjector with a diamond amplifier was proposed [1] and is under intensive development [2,3]. The technologically important material properties of diamond have been actively researched for emission [4] and detector applications. Moreover, diamond is currently being investigated to develop high-flux x-ray monitors for synchrotron beam lines [5-7].

The new photoinjector concept has important advantages [1] compared to existing metallic and semi-conductor photocathodes. The idea of its operation is to first generate a primary electron beam using a conventional photocathode and inject it into diamond. The primary electrons scatter in diamond, generating a cascade of secondary electrons. More recently, photons are also being considered in experiments as a source for generation of secondary electrons instead of using primary electrons.

The secondary electrons drift through the diamond under the acceleration of an applied electric field. The transported secondary electrons are emitted from a diamond surface with a negative electron affinity into the accelerating cavity of an electron gun. Over two orders of magnitude charge amplification (number of generated secondary electrons emitted relative to the number of primary electrons used as input) could potentially be achieved using this approach.

In order to better understand the phenomena involved in electron amplification and emission from diamond, we have been implementing models [8,9], within the VORPAL [10] computational framework, for simulation of secondary electron generation and charge transport. Here, we report results on gain obtained by considering only the loss of electrons due to electron cloud expansion near the diamond surface where primary electrons enter.

We compare simulation results with results from experiments [2,3] on electron gain when using primary electrons to generate the secondaries. The implemented modeling capabilities

Mater. Res. Soc. Symp. Proc. Vol. 1203 © 2010 Materials Research Society 1203-J17-43

allow us to consider, at least qualitatively, physical properties (collection efficiency, in particular) related to how diamond responds to x-ray photons. We will also discuss initial results from our simulations on studying collection efficiency (CE).

THEORY When a primary electron with sufficiently high energy enters into diamond, it undergoes

inelastic scattering leading creation of electron-hole (e-h) pairs. Moreover, some of the secondary electrons and the holes are created with sufficiently large energies to produce additional e-h pairs.

These inelastic processes continue until the free charge carriers (electrons in the conduction band and holes in valance band) no longer have enough energy to cause the transition of electrons from the valance to the conduction band. The minimum energy for such transitions is equal to the energy gap in diamond, EG = 5.47 eV, at room temperature. However, the mean free time between such inelastic scattering events (it can be obtained from the total cross section for this process [11,8]) increases towards infinity when the energy of the charge carrier decreases towards EG. Effectively, when the energy of the free carriers is close to 10 eV, the further relaxation of the electron energies towards the bottom of the conduction band is due to low energy inelastic scattering with phonons (and similarly for holes).

For the results presented here, we used our implementation of Ashley's model [11] for the inelastic scattering leading to e-h generation at 300 K. For the low-energy inelastic scattering with optical and acoustic phonons, we implemented Monte Carlo algorithms based on the models given by Jacoboni and Reggiani [12]. Our electron-phonon scattering implementation leads to electron drift velocities that are in agreement with band structure calculations and with experimental data [13]. We have coded a general Monte Carlo algorithm to handle all scattering processes. In between scattering events, the particle-in-cell algorithms in VORPAL [10] provide the capabilities to self-consistently move particles interacting with electromagnetic fields.

From the time a primary electron starts moving in diamond, the secondary electron generation (initiated by the primary electron) is close to complete after 200 fs. The code switches electrons and holes to phonon scattering only, once their energy falls below 11 eV during the first 400 fs. The simulations for studying electron gain starts with a primary entering the diamond surface set at the x=0 side of the simulation domain.

The simulations for evaluating CE start with a distribution of electrons (inside diamond) located at different depths from a surface considered as sink for particles. The initial part of the simulations is done with a time step Δt small enough (of the order of 0.01 fs) to resolve the scattering rate of the high-energy charge carriers.

At 400 fs, we dump the simulation state and restart with all particles switched to inelastic scattering with phonons only. These scattering processes have several orders of magnitude larger mean free time. This allows selecting a larger time step after the restart. We have used up to Δt = 2 fs. The cell size was varied in the range from 0.1 to 2 µm. In all simulations, the cell sizes were equal along the x, y and z directions. DISCUSSION Charge Gain

In the simulations, we estimate gain by counting the number of free electrons that successfully separate from the domain of the diamond surface where the secondary electrons are

generated. These results can be compared, at least qualitatively, with the gain measured from transmission mode [2] experiments (metal contacts on opposite diamond surfaces are used to create an electric field of up to several MV/m). Primary electrons are injected through one of the metal contact surfaces (at x = 0 in the simulations with the applied field in the x direction). The generated secondary electrons and holes move in opposite directions due to the applied field but also expand diffusively due to scattering. In the initial stage of the charged cloud evolution, some of the electrons will reach into the metal contact at the x = 0 surface and will be lost from diamond. The free electrons that are successfully separated from the domain of generation drift towards the opposite metal contact. We count these electrons as the realized gain. The hole cloud also expands diffusively but it drifts towards the initial metal contact surface and is extracted there. We model this behavior by using a sink boundary condition at the x = 0 side of the simulation box. The primary electrons are started at this side with an initial velocity along the positive x axis.

In Fig. 1, we show the time evolution of the number of electrons and holes for 3 and 0.1 MV/m applied electric fields in diamond. As expected, the rate at which electrons drift away from the diamond surface increases when increasing the field. The less amount of time the electrons are close to the metal contact surface, the less number of electrons are lost from diamond, thus, leading to higher gain.

Figure 1: The evolution of the electron and hole numbers vs time is as expected – these numbers reach saturation faster when increasing the applied electric field.

We compare gain results from the simulations to experimental data [2] for primary electron energies of 0.7, 1.7, 2.7, 3.7, and 4.7 keV on the left plot in Fig. 2. The simulations show the same overall dependence on the applied field as the experiments. The results from the simulations and the experiments are in good agreement for the 2.7 keV case. They agree qualitatively at the lower and higher energies investigated. For the 3.7 and 4.7 keV cases, the simulations predict 16 and 25 % higher gain than observed in the experiments.

The higher gain seen in the simulations for these two cases could be due to two reasons. In the simulations, the gain was determined by the number of electrons that are successfully separated from the metal contact at the diamond surface near which the generation happens. Any additional electron loss during the propagation to the other particlesmetal contact is not taken into account (e.g., due to trapping or recombination with holes). The gain in the experiments is determined by the total transmitted charge collected at the opposite metal contact. Secondly, the gain in the experiments was determined [2] assuming that the primary electrons lose a constant

amount of energy when going through the metal contacts before entering in diamond. However, this loss is generally dependent on the energy of the particles.

Figure 2: Comparison of electron gain from transmission experiments [2] with

simulation results (left) show qualitative agreement overall and quantitative for the middle of the energy range. The maximum number of electrons separated from the generation domain (right) shows two order of magnitude charge gain for primary electron energy larger than 2 keV.

The dependence of the maximum gain on the primary electron energy from the

simulations is shown in the right plot of Fig. 2. The data indicate that at higher energies, the diffusive motion does not affect the gain as much as at lower energies. For a 4.7 keV primary electrons, the average energy to generate a secondary electron is about 15 eV. However, for a 700 eV primary electron, it is 38 eV. Note that simulations started with a primary electron in bulk diamond (using the Ashley model for the inelastic scattering) give about 14 eV average e-h pair generation energy. This value is within the range from 10 to 17 eV reported [11] previously by different experimental, theoretical, and computational studies. Collection Efficiency

The currently implemented modeling capabilities also allow us to directly estimate the CE as a function of the depth at which photons are absorbed since we can start the simulations with an initial distribution of e-h pairs at any given depth inside diamond. The collection efficiency of a detector is defined as the ratio η = Qc/Q of collected charge Qc to the total charge Q generated in response to the absorbed photons. The obtained CE vs depth (measured from the surface photons enter into diamond) for 600 eV photons and several values of the applied electric field are shown in the left plot of Fig. 3. In the simulations here, we determine the CE by measuring the number of electrons successfully separate from the domain of generation (after the loss of electrons to the metal contact has completed) and separately the total number of electrons generated by the input energy of the absorbed photons. As expected, the deeper the absorption occurs in diamond and the higher the applied field, higher CE is obtained.

Figure 3: Simulations allow direct investigation of CE vs absorption depth – results for different depths and fields for 600 eV photons (left plot). Experimental data on CE vs field for 600 eV photons are in overall qualitative agreement with the simulation results (right plot).

In Fig. 3 (right plot), we compare simulation and experimental data on the dependence of the CE vs electric field for 600 eV photons. Overall, they are in qualitative agreement (and in quantitative at lower fields). However, the most important insight from these simulations is new understanding about the experimental data on responsivity vs photon energy. The results from the VORPAL simulations show that the observed responsivity (S), measured in A/W, can be described by the expression

€

S ν( ) =1WC

exp − tMλM ν( )

⎛

⎝ ⎜

⎞

⎠ ⎟ 1− exp −

tCλC ν( )

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟ CE ν,E( ) , (1)

where WC is the mean energy to create an e-h pair, tM and tC are respectively the thickness lengths of the metal contact layer and the diamond sample; λM and λC are the corresponding photon energy-dependent absorption lengths. The energy and field dependent collection efficiency function,

€

CE ν,E( ), is obtained by an empirical fit to the VORPAL simulations data. The VORPAL results show that the loss of charge carriers to the metal contact (due to the diffusive expansion of the charge cloud) provide an explanation for the observed data. This charge loss effect is most important for low photon energies, less than 1 keV. For these energies, the e-h pairs are generated close to the metal contact and the diffusive expansion allows some charge carriers to reach the metal contact. The details of this approach to describe the observed data are given by Keister et al. [14].

CONCLUSIONS We reported here initial simulation results on charge gain and CE from diamond. Higher

gain can be achieved by increasing the primary electron energy and the magnitude of the applied electric field. However, increasing the field will lead to increasing the effective temperature of the drifting electrons [12,13,9], thus, increasing the thermal emittance of electron beams. At the intermediate primary electron energies considered, the gain from the simulations agrees well with experimental data. Overall, the agreement is qualitative. However, in the simulations we have considered only loss due to the diffusive motion near the diamond surface where secondary

electrons are generated. The simulations on CE provided the important understanding that the experimental data on responsivity can be understood by taking into account the loss of charge to the metal contacts due to diffusive expansion of the created e-h clouds.

ACKNOWLEDGMENTS We are grateful to the DOE for supporting this work under grant DE-FG02-06ER84509.

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