Calculations of secondary electron yield of graphene coated copper for vacuum electronic applications Open Access

Secondary electron emission (SEE) is known to be a major cause of performance degradation in various electronic devices. An example of the negative effects of SEE is the deleterious electron cloud effect in the context of accelerators.^1,2 This effect is a consequence of the strong coupling between a charged-particle beam and the enclosing vacuum chamber that is detrimental to the performance of storage rings. The multipactor effect is another discharge phenomenon that depends on secondary electron emissions and occurs in radiofrequency (RF) components operating under vacuum conditions.³ In these situations, the free electrons can synchronize with the RF field oscillations in the device, leading to time-dependent increases in the electron population. Consequences include greater losses, enhanced temperatures, potential outgassing at the walls, and even ionization breakdown which may physically damage the structure.⁴ Multipactor can also be an important issue in satellite microwave components and lead to electrostatic discharges on the outer surface of satellites.⁵ The electrostatic discharges initiated by such electron emission events have been proven to cause power losses, and even lead to failure of satellite missions.⁵ Multipactor is also known to initiate window breakdown on the vacuum side of microwave systems,^6,7 and is observed in the context of accelerator structures,⁸ microwave tubes,^9,10 and antennae or transmission lines.¹¹ 

Multipactor was first recognized by Farnsworth¹² decades ago in 1934. Much of the work on this subject has been reviewed by Vaughan¹³ and Kishek et al.¹⁴ More recent work includes the development of an analytical theory to study this phenomenon in terms of interaction of electron sheets with an external circuit.^15,16 The electron impact energy necessary for multipactor was shown to equal the lowest value at which the secondary electron yield was unity. Other numerical physics-based studies lead to similar qualitative results.¹⁷ More recently, it was shown that multipactor is very sensitive to the low energy secondary electron yield.¹⁷ Thus, it becomes important not only to mitigate the secondary electron yield (SEY) in a general sense, but also suppress this value at the lower energies.

The deleterious nature of SEE on system reliability has led to numerous investigations into devising techniques for SEY suppression. Most have focused on surface treatment techniques or have used geometric modifications. For example, Pivi et al.¹⁸ obtained low SEY values on the basis of a grooved emitting structure with the suppression depending on the angle and aspect ratio of the grooves. Ye et al.¹⁹ used a roughened surface with a micro-porous array. A more recent report was that by Aguilera et al. who used copper oxide nanowires grown on copper to strongly reduce secondary electron emission from the sample surface. An array of free-standing velvet whiskers also resulted in a reduced secondary yield.²⁰ Finally, ion-textured graphite coatings have been shown to significantly lower the SEY down to values of around 0.2, though this requires ion-sputtering to produce the texture on the graphite.²¹ Even more dramatic improvements in SEY values to about 0.1 were obtained by depositing a few monolayers of graphene on the electrode.²² However, these experiments were conducted at a fixed incident electron energy of 1 keV, and the energy-dependent behavior was not probed.²³ The advantage of using graphene is that neither any post-treatment, nor ion-sputtering, is required for such an approach. This appears to be very promising given the exceptional material strength, high electrical and thermal conductivities, and remarkable electronic properties (such as significant mobilities and extremely low resistivities) of graphene.^24,25 More recent studies of energy-dependent SEY for graphene coated copper systems have been reported, and they all point to the benefits of this approach.^26–28 However, numerical studies of SEY on graphene coated copper have not been performed, nor have careful comparisons with experimental data been made. The present study represents an effort in this direction.

The exciting reduction in SEY observed experimentally in graphene coated copper typically use graphene layer thicknesses in the range of 0.3-0.5 nm. Hence, based on simplistic reasoning, the attenuation of incident electrons through such an overlayer can be expected to be negligible. The strong SEY reductions observed then suggest that the main effect of the graphene layer very likely originates from substantial reductions in the escape probability of the secondary (and tertiary) electrons created within the copper.

Changes in workfunction, in the case of graphene deposited on SiC, were invoked as a possible mechanism for decreases in SEY.²⁹ A change of about 7% in the electron affinity with an increasing number of graphene layers on SiC material was shown to reduce the secondary electron spectral distribution. However, actual SEY values were not obtained. Also, peculiarities of the graphene bandstructure or changes in its workfunction upon deposition, arising from charge transfers associated with physisorption, were ignored. Due to its two-dimensional nature, graphene is highly sensitive to the environment, and its properties are strongly influenced when creating contact with a metal. First-principle calculations at the level of density functional theory have shown that when contacting with copper,^30,31 the difference in work functions causes charge transfer through the interface. Due to the small density of states in graphene near the Dirac energy, almost all the charge (and potential shift) resides in the graphene, and is weakly screened.³² Despite charge transfer and the development of a built-in potential, its magnitude (and that of the work-function change) can be expected to be small for the scenario of a one to two monolayers of graphene deposition. Studies, for example by Song et al.,³³ point to relatively small work-function changes. Thus it remains unclear why the modest shifts might cause large changes in SEY values from over 1.0 for pure copper, as reported.^22,26–28

Here we focus on the two-dimensional nature of the graphene layers. Transmission of secondary electrons incident on the metal-graphene interface from the metal would conserve momentum along the parallel direction during their transit. Electron energy in the two-dimensional graphene (E_gr) can be approximated to: E_gr = ℏ v_g k_||, where ℏ is the reduced Planck’s constant, v_g is the electron group velocity assumed to be about 8.6x10⁵ m/s,³⁴ and k_|| the wavevector in the graphene plane. Furthermore, we denote the graphene workfunction as W_G(d), with d being the graphene thickness. For a thin layer of graphene, typically around 5 Å as noted by Luo et al.,²² the value of W_G(d) would be close to the workfunction of copper (W_Cu). In any case, from energy considerations, one gets the condition:

where ΔV is the interfacial potential step due to charge transfers between copper and graphene, and ΔE_F = W_Cu–W(d)-ΔV. The value of ΔE_F depends on the separation distance,³¹ and would roughly be around 0.4 V in the present case. The parallel wavevector k_|| for electrons incident at the interface surface from the copper side is related to the incident angle θ as: k_||/(k_||²+k_z²)^½= sin(θ), leading to: E_||/E = sin²(θ). Hence, eqn. (1) can be expressed as: E sin²(θ) - ΔE_F> 0. There is thus an energy threshold for emission based on the angle of electrons incident from the copper side, at the interface.

Numerical computations to probe electron escape then require knowledge of the momentum distribution of all electrons incident on the copper-graphene interface. These would include any possible incident primary, and all exiting secondary and tertiary electrons. Here, Monte Carlo (MC) simulations as outlined by numerous authors^35–37 were used to track incident particles. In this approach, elastic collisions of electrons were treated in a detailed manner, whereas the energy loss due to inelastic collisions were assumed to be continuous along a free path of each electron between elastic collisions. Unlike the case of high-energy electrons where the number of elastic collisions can be very high and requires special schemes to reduce the computational costs;³⁸ such methods were not necessary in our case. For elastic scattering the screened Rutherford formula³⁹ was used, while inelastic collisions were treated based on energy transfer given by the Bethe relation^40,41) in copper. Formation of secondary and tertiary electrons in the continuous-slowing-down approximation also invoked energy-dependent stopping powers. All radiation processes were neglected since these only become important for high energies exceeding hundreds of keV.⁴² It must be mentioned for completeness, that the continuous-slowing-down approximation overlooks the detail that energy is actually lost in discrete collisions. The latter implies that the number of such collisions can fluctuate, and so some electrons can occasionally suffer very large energy losses in close collisions with the atomic electrons. Though improved Monte Carlo formulations exist,^43,44 our primary goal is to qualitatively probe the physical mechanism behind SEY suppression by graphene monolayers. Refinements of our Monte Carlo including can easily be implement subsequently. For example, inelastic scattering could be treated based on the dielectric function modelling theory by Penn,⁴⁵ and the imaginary part of the requisite dielectric function obtained from Density Functional Theory rather than using approximations.^43,44 Here, a more simplistic treatment is provided merely to obtain the qualitative trend of the SEY response in ultrathin graphene coated copper.

Figure 1 shows our Monte Carlo simulation results, obtained for the secondary electron yield from pure copper without any graphene, as a function of the incident energy. This represents a well-known situation, and serves as a validity check of our model implementation. Normal incidence was assumed and parameters for copper were taken from Joy.³⁶ The simulation predictions are in fairly close agreement with the published report by Ding et al.⁴⁶ Furthermore, as evident from Fig. 1, the SEY values for copper are generally high. Simulation results for different incident angles were also obtained and are shown in Fig. 1(b). As might qualitatively be expected, the SEY increases with enhanced incident angle, since a higher angle leads to shallower penetration and makes it easier for the generated carriers to exit the copper and re-emerge. Furthermore, the difference due to angular change at lower energies is not appreciable since the penetration is not very large anyway, and not many secondaries are created.

Next, MC simulations with a graphene monolayer on copper were carried out. In the simulations, the value of W_G(d)–W_Cu was taken to be 0.4 eV. The results obtained are shown in Fig. 2, which also includes experimental data reported by Wang²⁷ and Xie.²⁸ The following aspects are evident from the plots: (a) Predictions of the SEY values are generally in good agreement with the measurements for a graphene coated copper substrate. However, it must be mentioned for completeness that the secondary emission is somewhat sensitive to surface treatments,⁴⁷ and so it is likely that the exact values could vary for diverse samples under different conditions. Hence, the present results should be viewed more as a qualitative predictor of the trend rather than a precise evaluation that comprehensively factors in all the material-dependent physics. (b) The SEY magnitudes with the graphene are much lower than those predicted for copper, especially at the lower energies. For example, while values around 110 are predicted for copper, the corresponding SEY with the graphene coating is as small as 55 for the low energy regime. This is especially beneficial from the standpoint of mitigating multipactor, since it has been shown that this phenomenon is sensitive to the low energy secondary electron yield.¹⁷ Qualitatively this outcome is entirely expected since the low energy electrons emerging from the copper may not have the requisite transverse energies to transcend the differential workfunction [= W_G(d)–W_Cu] barrier. The low energy range can be quite relevant to RF excitations in the context of satellites in space.

So in summary, mitigation of secondary electron yield was demonstrated for a thin 0.5 nm graphene coating on copper. The highest degree of SEY suppression was predicted at the lower energies, though the simulation results consistently yielded lower SEY for the entire 0-500 eV energy range. Furthermore, values smaller than unity were predicted up to 125 eV. This represents a lower limit for potential multipactor. Specifically, in comparison to copper, the graphene coating is shown to result in SEY reductions of almost 50 percent. Though this result is promising, an important practical consideration would be the degree of ruggedness and stability that the graphene nano-membranes could offer. Such reliability and resilience against aging and long term fatigue or degradation are important if graphene nano-coatings are to be an effective route to SEY and multipactor suppression. For a more rigorous and in-depth analysis though, it would be necessary and meaningful to conduct thorough analyses of potential impact damage due to charged particles incident on such graphene monolayers. This issue is especially germane for space applications. However this aspect, including the possibility of local defect formation due to external charge bombardment, will be studied and reported elsewhere based on Molecular Dynamics simulations. It is very possible that an energy threshold for damage exists, or that the potential for defects and damage depends on the thickness of the graphene coating. These are related aspects that merit further detailed analyses.

This research was supported in part by a Department of Defense MURI Grant No. FA9550-18-1-0062 on ″Multipactor and Breakdown Susceptibility and Mitigation in Space-Based RF Systems.″