Investigation into the mechanism of surface atom emission from an individual cathode spot using molecular dynamics simulation

Contact erosion on the cathode of a vacuum arc is determined by the behaviours of the cathode spots, where the plasma–surface interactions take place. A comprehensive model of a single cathode spot is developed in this work based on the molecular dynamics method, where an atomic copper substrate in the size of nanometres is built and the contributions to the development of cathode spot from leftover plasma ions, surface electron emission, surface atom emission, back ions, Nottingham heating and Joule heating are integrated. Defined based on the surface temperature distribution, a cathode spot is observed in the simulation results. The surface atom emission, which is the origin of mass loss, can be directly detected by the atoms being isolated from the surface. Two routes of surface atom emission are observed as the sources of mass loss, including evaporation, and atom sputtering or splashing. It is found that in the high-temperature region, atom sputtering or splashing dominates the surface atom emission, which leads to considerable mass losses. The simulation results are consistent with previous experimental and other simulation findings, providing fundamental insights into the cathode spot formation mechanism from a microscopic perspective.


Introduction
Contact erosion is one of the main causes that reduce the dielectric strength of vacuum circuit breakers.Therefore, alleviating the contact erosion is essential to extend the electrical lifetime of the vacuum circuit breaker and increase its Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.application voltage level [1,2].From experiments, it was found that the vacuum arc attachment on the cathode is in the form of multiple cathode spots, which were observed as luminous spots during the arc and small surface craters after the arc [3,4].Cathode spots which are in the size of hundreds of nanometres to tens of micrometres, conduct a limited amount of current depending on the cathode material and exist for tens of microseconds [5].The contact erosion is determined by the behaviours of the cathode spots, including the development of an individual cathode spot, and the cathode spot dynamics such as the formation of a new cathode spot from the previous one.Therefore, the characteristics of cathode spots have been of great interest in recent years, such as the motion of cathode spots [6,7] and the plasma expansion based on the cathode spot [8].
It is commonly perceived that the second-type cathode spot is ignited by the ions from a leftover plasma produced by the previous cathode spots [9].Under the thermal and pressure effects of ion bombardment, the cathode spot gradually expands, with sufficient electron emission current to maintain the current continuity on the cathode surface.The surface atom emission during the lifetime of the cathode spot leads to the mass loss and produces the leftover plasma ions to induce the next cathode spot [10,11].
As the connection between the plasma and the cathode spot, the surface layer decides the intensity of the input of leftover plasma ions on the cathode surface.Due to the kinetic and collisional behaviours of the various species in the surface layer, such as plasma ions, surface-emitted atoms, and electrons, the surface layer is dynamic.For example, the potential fall and electric field across the sheath are fluctuating, and the possibility of fast ions exists since the ion velocities are uncertain due to the collisions.Moreover, along with the development of the surface crater, the surface deformation could lead to a deformed surface layer.In the cathode spot simulation, the characteristics of the surface layer need to be considered to define the intensity of the plasma input.Detailed surface layer characteristics could be calculated by solving the Poisson equation.However, in the cathode spot simulations, a simpler static surface layer model was usually adopted to simplify the calculation.
The simulations of an individual cathode spot have been achieved by hydrodynamics (HD) method [12][13][14], with a simplified static surface layer consideration.These simulations proposed that the droplet detachment from the periphery of the cathode crater is the main contribution to the mass loss.However, the mechanisms of this droplet detachment were not agreed.In [12] it appeared along the crater expansion, while in [13] it only occurred after the crater expansion stopped.Moreover, limited by the method, the previous HD models of cathode spot were incapable of observing the other route of surface atom emission, which was commonly assumed by the theoretical evaporation expressed by the Langmuir equation [15].Therefore, the mechanism of surface atom emission is still unclear and requires further investigation.
A new simulation method, molecular dynamics (MD), was first applied to investigate cathode spot formation in [16] and was recently developed by [17].In MD models, the effects of plasma ions and surface atom emission can be realised.Therefore, the mass input from leftover plasma ions and the mass output by surface atom emission can be considered which were ignored in all previous cathode spot simulations.In this work, a further developed cathode spot model is established by MD.To reach a comprehensive model of cathode spot, the contributions of leftover plasma ions, surface electron emission, surface atom emission, back ions, Nottingham heating, Joule heating, and heat conduction are all coupled.From this model, the formation of the surface crater is observed, and the distribution of surface temperature as well as the electron current density are obtained as the basic features to assess the cathode spot.To investigate the contact erosion of a cathode spot, the phenomenon of surface atom emission is focused in the results as it is the origin of the mass loss.As discussed in [17], although the spatial size of the MD model is limited to nanometres, the characteristics of cathode spot formation are the same, hence the analysis helps understand the phenomena of the real cathode spot.

Model
The MD model to study the surface atom emission of an individual cathode spot is built in LAMMPS and is viewed by OVITO [18].The initial model with the x-y cross-sectional view is presented in figure 1(a).The central red region is the cathode substrate with a size of 22.4 × 22.4 × 11.2 nm.There is a surrounding thermostat layer marked in blue with a thickness of 3.615 nm in the three directions.In the ±x and ±y directions the periodic boundary condition is adopted, while in the ±z directions the open boundaries are set.To avoid the atoms leaving the simulation domain from the bottom plane, a fixed layer is attached at the bottom.The timestep of the simulation is 1 fs. Figure 1(b) presents an example of the model during simulation, with the leftover plasma ions, emitted atoms, and back ions distinguished by colours.
Initially, this model is thermally relaxed at 300 K. Subsequently, the other effects come into play, with the thermostat layer maintained at 300 K and the whole system set under micro-canonical ensemble.As the atoms and ions involved in this model are both copper, a potential that combines the embedded atom method potential and the Ziegler-Biersacke-Littmard potential is adopted to calculate the interatomic forces [19].The former is effective for the atoms under stable metallic structure with normal interatomic distances, while the latter is effective for the atoms with short interatomic distances, which happens during the bombardment process.The effective interatomic distance range of this potential is 0.02 − 6 Å.
Among all the involved effects, heat conduction is inherently realised by the interatomic thermal collisions, and the surface atom emission is observed directly as the isolated atoms, hence they do not require special treatments.The other effects considered are the leftover plasma ions, the surface electron emission, the Nottingham heating, the back ions, and the Joule heating.The calculation procedures of these effects are shown in figure 2. The leftover plasma ions with assumed spatial distribution are inserted into the headroom above the copper substrate.Then during the ion bombardment, the temperature distribution of the deformed surface is obtained, based on which the surface electron emission is calculated, including the electron current density, and the Nottingham heat.In the meantime, the emitted atoms are detected above the cathode surface, and then they are converted into back ions, where back ions current density is calculated.The total ion current density contributing to the Joule heating includes the leftover ions current density, back ions current density, and electron current density.In each timestep, the Joule heating is added to all the substrate atoms, while the Nottingham heat is subtracted from the surface atoms.In this work, the contribution of the leftover plasma ions is included using the same methodology as [17], with ion energy and ion insertion frequency adopted as the input parameters.To simplify the calculation of the ion energy, the static surface layer model without deformation is assumed.The characteristics of the surface layer are represented by several parameters including sheath voltage fall U, electron temperature T e , ion density n i , and the mean charge number of the ions z i , which were measured from the experiments to be varied over certain ranges [3].According to the experiment measurements, the values of these parameters adopted in this simulation are shown in table 1.Moreover, in this work, the electric field is assumed to be solely dependent on the leftover plasma ions.Since the input ion parameters are fixed, the spatial distribution of the electric field is also static.Therefore, in this work, the simulation results based on certain ion energy reflect the cathode spot formation under a static surface layer.On the other hand, the influence of the dynamic surface layer on the cathode spot formation could be reflected by the comparisons of the simulation sets with different ion energies, which will be discussed in section 4.1.
The details of other effects are introduced as follows.

Surface electron emission
According to the T-F emission theory [4], the surface electron emission current density J em is calculated as: where λ R and A G are both constants decided by the cathode material.W is the work function of the cathode material, ϵ 0 is the vacuum permittivity.Surface electron emission takes place on the surface atom layer.Therefore, the determination and tracking of the surface atom layer after surface deformation is critical.To achieve this, the substrate atoms are meshed into chunks on the x-y plane, with the x-y size of each chunk 3.615 × 3.615 Å.The x-y size of the chunk is set the same as the lattice constant of the cathode material, so that in each atom layer there is no more than one atom filled in each chunk.Referenced to the maximum z-position in each chunk, the uppermost atoms in the chunks are determined as the surface atoms.An example of the determined surface atoms layer after the surface is deformed is shown in figures 3(a) and (b).It shows the applicability of this method to track surface atoms even though the surface is greatly deformed.
The surface temperature is calculated by the thermal velocity of the selected surface atoms.When the incident ions arrive on the surface, intense transient interatomic collisions will happen locally, leading to an enormous kinetic velocity of the bombarded atoms.The surface temperature should be the stable thermal temperature of a group of substrate atoms.
Therefore, the atoms with an enormous kinetic velocity are regarded as under the transient collisions and are temporarily ignored when selecting the surface atoms.The thermal velocities of the determined surface atoms in the same radial ring are averaged to obtain the local surface temperature.As shown by the temperature bar in figure 3, the distribution of this average temperature exhibits the axis symmetry and an obvious radial gradient as expected.At each timestep, the surface electron emission is calculated once the surface temperature tracking is finished.

Surface atom emission and back ions
When the z-velocity of a surface atom is high enough, it can break the surface bond and escape from the cathode surface.This is the microscale theory of surface atom emission.Differentiated by the escaping mechanism, there are two types of surface atom emission.First is evaporation, in which the high z-velocities of the surface atoms are obtained from the thermal vibration under high surface temperature.Second, is the atom sputtering, which is commonly observed in the case of ion bombardment, where the surface atoms gain a high z-velocity from the direct momentum exchange between themselves and the incident ions.In [17] it was found that the relatively low ion energy in the case of cathode spot (i.e.62 eV) is insufficient to produce surface atom sputtering on a cold cathode in 300 K by single ion bombardment.However, during the continuous ion bombardment, the surface temperature will increase, leading to atom sputtering even under low ion energy.The influence of local surface temperature on the atom sputtering will be analysed in section 3.
As shown in figure 1(b), some emitted atoms are observed isolated from the surface.The isolated atoms are judged by the method of common neighbour analysis.The real-time position, velocity, and forces of these emitted atoms can be recorded in MD simulation.In comparison, in the other HD simulation works [9,10], these emitted atoms cannot be observed directly, and are calculated based on the assumption of the saturated evaporation theory.
According to the surface layer theory, the emitted atoms will be all or partially ionised by the surface emitted electrons within a relaxation zone extended to about 10 nm above the cathode surface [20,21].Zhang et al [12] assumed that the coefficients of atom ionisation α and ions returning to the cathode β were both 1.A more advanced consideration was proposed in [13], where only the atoms being ionised within a certain distance from the cathode could immediately return to the cathode surface.The coefficient of ions returning β was proposed determined by the sheath voltage and the surface temperature.When the surface temperature was higher than 4500 K, β approaches 1.
Limited by the MD method, in this simulation, the ionisation in the surface layer could not be coupled directly.Therefore, an assumption is made that all the emitted atoms are ionised when they reach +z = 10 nm, which means that the coefficients α and β are both regarded as 1.The emitted atoms above this plane will be converted into back ions with the same axial velocity as the inserted leftover plasma ions.However, the x-y positions of the back ions are uncertain as the x-and y-velocities of emitted atoms exhibit randomness, which more approaches to reality.In addition, in the headroom, the elastic collisions among the high-density plasma ions, emitted atoms, and back ions are considered.Therefore, although assigned the same initial velocity along the −z-direction, the directions and values of the ion velocities vary when they bombard the cathode surface.In the simulation, fast ions with velocity three times higher than the initial velocity are found.Also, some ions failed to bombard the surface due to the reversed velocity after elastic collisions are also recorded.However, the following results and analysis will focus on the overall phenomena of the cathode spot formation and surface atom emission.The detailed statistical analysis of the ion velocity under the effect of elastic collisions will not be presented.

Nottingham heat and Joule heating
Surface electron emission has a cooling effect on a high temperature surface, known as Nottingham heat.The energy flux density of Nottingham heat q em is calculated as: The variables in (2) are consistent as introduced above.Based on the temperature of surface atoms at each timestep, j em and q em are both available.The chunk size in x-y plane is used to calculate the power of this cooling effect.The heat reduction on the surface atoms by Nottingham heat effect at one timestep is calculated based on the surface temperature obtained in the previous timestep.
Similar operations are conducted for coupling the Joule heating.Under the same meshing as described in section 2.2, the current density of the emitted electrons, leftover plasma ions, and the back ions in each x-y chunk is obtained.These currents are all assumed in the z-direction.The total current density J total is calculated by: J total = J em + J leftover ion + J back ion . ( Unlike the Nottingham heat, which is only calculated and reduced on the surface atoms, the Joule heating is a bulk effect on all the substrate atoms.To allow the calculation of the electrical parameters of the substrate atoms, another meshing operation is conducted across the whole substrate, with no more than one atom assigned to each chunk.This meshing approach is realised by the commands of LAMMPS, and the chunks obtained are named as x-y-z chunks.
In each x-y chunk, the calculated j total is assigned to all the x-y-z chunks inside it.The local electrical resistivity is calculated on each x-y-z chunk based on the size and the temperature of the atom in it.The local electrical resistivity is capped at the highest available value of 3500 K, same as [22].With the local resistance R atom calculated by the laws of resistance, the power of joule heating on each x-y-z chunk is available, and then the corresponding thermal energy is then assigned to the atoms in the x-y-z chunks.This calculation and meshing processes are shown in figure 4, in which the x-y meshing is plotted on the x-y plane, with several chunks presented as examples.The x-y-z chunks are not shown.The parameters calculated in x-y-z chunks are shown bold.From J total to q joule , the transition of parameters between the x-y chunks and the x-y-z chunks is illustrated.

Results
Under the assumption that all the emitted atoms are ionised and return to the cathode surface, the formation of an individual cathode spot is observed by this model.The applied space function f (r) is: where r is the radial position, and r 0 is a constant as one input parameter.Different sets of input parameters are investigated, as shown in table 1.For MD model, the two key parameters are plasma ions density and ion energy.According to the measured parameters in experiments, several examples are adopted as shown in set #1∼3, with corresponding ion energies as 29 eV, 45 eV, and 62 eV.The value of r 0 decides the spatial size of the cathode spot and is set as 5 nm in set #1∼3 to achieve an equivalent cathode spot and a cathode crater in the size of several nanometres.In section 3 the simulation results of set #2 will be presented as an example, and the influence of ion energy on the cathode spot formation and the surface atom emission will be presented in section 4.
In addition, in set #4, r 0 is set as 5µm to simulate the spatial size of a real cathode spot of several micrometres.However, the simulation domain is still in nano size, which means the model only simulates the central local position of the practical cathode spot.The simulation results of set #4 will be analysed Set n i0 (×10 26 m −3 ) z i Te (eV) U (V) r 0 (µm) q il (eV) in section 4 to validate the consistency of the equivalent cathode spot model with the practical cathode spot.During the formation of the cathode spot, the highest surface temperature is in the range from 4500 K to 5500 K.The high-temperature region first appears at the central position where the ion bombardment is the most frequent, and then expands outwards with time.Here the high-temperature region is defined by the area with a stable surface temperature above 4000 K.In figures 5(a) and (b), the radial distributions of the surface temperature and the current densities of leftover plasma ions, emitted electrons, and back ions at 30 ps and 250 ps are shown as examples.At 30 ps, a distinct temperature gradient is established across the entire surface, with the central temperature reaching 5000 K and the temperature of the edge still below the melting point of copper.As the ion bombardment continues, the high-temperature region gradually expands.At 250 ps, the surface temperature is overall high in the simulation domain.As a consequence of hightemperature region expansion, the molten pool and the surface crater also expand radially, which is consistent with the previous MD simulation of cathode spot formation [17].From 30 ps to 250 ps, the profile of the temperature gradient becomes aligned with the current density of leftover plasma ions, as the ion bombardment of the leftover plasma ions is the main heating source of cathode spot formation.
As shown in figure 5(a), the surface emitted electron current density surpasses the leftover ion current density and is higher than 1 × 10 11 A m 2 in the central high-temperature region with a radius of 6 nm.Based on the criterion of surface electron emission, it is perceived that in this model a cathode spot is formed.In the meantime, in the periphery region with lower temperature, even though the surface is melted and deformed, the surface emission is still insufficient to maintain the current density.From 30 ps to 250 ps, the size of the cathode spot grows from 6 nm to over 10 nm.
The back ion current density is shown as isolated points in the model.The appearance of back ion current density represents that there are back ions existing at this radial position at this time point.The distribution of back ions exhibits randomness rather than a clear radial gradient.The randomness of the back ion positions results from the randomness of the velocities of the surface emitted atoms.
Since all the back ions are assigned with the same velocity, the number of back ions at a transient timestep decides the value of back ion current density.From the consideration of the whole cathode spot, as shown in figure 5(a), at 30 ps the back ion current density is lower than the leftover ion current density, with only two points of back ions appearance.However, as shown in figure 5(b), at 250 ps the back ion current density is comparable to the leftover ion current density, with back ions appearing almost over the whole cathode spot.This comparison reveals that the surface atom emission is slight at the beginning but becomes intense enough to be comparable to the leftover plasma input.This result is different from the analysis of other cathode simulations [12,13], in which the contribution of back ions is always slight.This difference will be further analysed in section 3.2.
In figures 6(a) and (b) the radial distribution of the energy flux densities of Nottingham heat, Joule heat, and leftover plasma ions at 30 ps and 250 ps are presented.The relationship between the heating effect of the plasma ions and the cooling effect of the surface emission is similar with the previous analysis.Only within the cathode spot with surface emitted electron current density higher than 1 × 10 11 A m −2 could the cooling effect of surface emission be comparable to the heating effect of the plasma ions.
The Joule heating is a bulk effect.In figure 6 the plotted Joule heat values are taken from the surface, which is representative since the surface temperature is always the highest throughout the whole substrate.In both pictures the Joule heat is much lower than the heating effect of leftover plasma ions.Therefore, it is acceptable to ignore the contribution of Joule heating in the process of cathode spot formation.

Surface atom emission
As described in section 2, the atoms being isolated from the surface are defined as emitted atoms.With the coefficients of both atom ionisation and ions backflowing are set as 1, the surface atom emission of set #2 is shown in figure 7.
Figure 7(a) counts the number of the input ions and emitted atoms during the simulation duration of 300 ps.The number of total input ions increases almost linearly with time, which is consistent with the input scheme of ion insertion as described in section 2. The number of total emitted atoms grows slowly at the beginning.However, the slope of the growth increases sharply.As a result, at 300 ps, the number of the total emitted atoms is already comparable to the number of the total input ions and is predicted to surpass the latter afterwards.
In HD simulations [12,13], limited by the method the surface atom emission cannot be involved or observed.Instead, they adopted the assumption of saturated evaporation by the Langmuir equation.The pressure effect of leftover plasma ions on the cathode was commonly simulated by a surface pressure applied on the surface of the molten pool.As a result, the surface pressure led to continuous surface deformation, and the detachment of the droplet from the rim formed at the periphery of the crater was observed as the main contribution to the mass loss.In comparison, the contribution of the back ions based on the surface atom emission, which-was regarded as solely evaporation, was negligible.However, in the MD model, the surface atom emission can be observed directly during the ion bombardment.As ion bombardment continues, the number of surface atom emission turns to be comparable to or higher than the incident leftover plasma ions, as shown in figure 7(a).As a result, the back ions current density can be comparable to the leftover plasma ions current density, as shown in figure 5(b).This surface atom emission observed is much more intense compared to the assumption of saturated evaporation adopted in HD models.
To further analyse this phenomenon, figure 7(a) also presents the evolution of the number of emitted atoms every 10 ps, compared with the theoretical value of evaporated atoms every 10 ps.The theoretical evaporation value is obtained based on the real-time surface temperature distribution of this model.From this comparison, two periods of atom emission are identified.First, before 200 ps, there is no noteworthy difference between the numbers of emitted atoms and theoretical evaporated atoms.The number of surface atom emission is always slightly larger than the expected number of theoretical saturated evaporation.On the contrary, after 200 ps, the number of emitted atoms every 10 ps grows prominently faster and becomes much larger than the theoretical evaporation number.This reveals that evaporation is not the only mechanism of surface atom emission.
From the knowledge of atom sputtering under ion bombardment, the sputtering yield is determined by the ion energy and substrate temperature.According to [14], it was concluded that for the single ion bombardment with the low ion energy of 62 eV, which is a typical value in the case of cathode spot formation, no atoms could be sputtered out from a cold substrate at 300 K.However, during the formation of cathode spot, the ion bombardment is continuous.As shown in figure 5(b), the central region is possible to reach a temperature above 4000 K.When the ion with low energy bombards the high-temperature substrate, the atom sputtering becomes possible.
In figure 7(b), the evolution of surface temperature distribution is shown.Initially, the surface is relaxed at 300 K.When the ion bombardment begins, the surface temperature of the central region is increased above 4000 K very quickly.At 100 ps, the temperature of the central region in the radius of 3 nm surpasses 4000 K, and can be regarded as the hightemperature region.Within this high-temperature region, the atom sputtering is possible with low ion energy of 62 eV.However, as the area of this high-temperature region is very small with limited number of substrate atoms, the contribution of atom sputtering to the surface atom emission is not prominent.This explains why before 200 ps the number of emitted atoms is just slightly higher than the theoretical number of evaporated atoms.Then, as the ion bombardment continues, the high-temperature region expands, hence the effect of atom sputtering becomes more obvious.At 300 ps, the temperature surpasses 4000 K nearly across the entire surface.As a result, the atom sputtering dominates the surface atom emission.
In addition, as the local substrate becomes loosened, it is possible that the atoms leaving the surface in the form of an atom group, which can be regarded as splashing.For example, figure 8 presents the cross-sectional view of the substrate (the thermostat layer and fixed layer are not shown) at 300 ps.Among all the atoms leaving the substrate, there are some atom clusters marked by the dotted black circles.The mechanism of this splashing is similar with the droplet detachment observed in HD models.However, this simulation shows that except from the big droplet detachment at the periphery of the cathode crater, the splashing of atom clusters within the spatial range of the molten pool also exists during the process of cathode spot formation.
In conclusion, there are two types of surface atom emission: evaporation and atom sputtering or splashing.When the z-velocity of an atom is sufficiently high, it breaks the surface bond and leaves the surface.When the local temperature is high, it is possible for a surface atom to gain a high zvelocity by thermal vibration and become evaporated from the surface.During the transient collision process of ion bombardment, especially on a high-temperature surface where the surface bonding is weakened, the substrate atoms can also gain a high z-velocity by this kinetic effect and leave the surface in the form of atom sputtering.The splashing happens at the location where the local atomic structure is greatly loosened under high-temperature and a group of atoms possess a high z-velocity.

Discussion
In section 3, the simulation results of set #2 are presented.In this section, the influences of the input parameters on the simulation results are discussed.Then, the mechanism of the mass loss on the practical cathode spot is analysed.

The influence of ion energy on surface atom emission
From experiments, the measured values of the several parameters describing the leftover plasma including electron temperature T e , mean charge number of leftover plasma ions z i , and sheath fall U in a certain range.In this MD model, the influences of these three parameters on the cathode spot formation are reflected by the ion energy.As shown in table 1, z i values of 1, 1.5, and 2 are selected in simulation set #1, 2, and 3.The corresponding ion energies are 29 eV, 45 eV, and 62 eV.The influences of ion energy on the surface temperature distribution and surface atom emission are presented by figure 9.
Figure 9(a) shows the radial distributions of the surface temperature in the three sets.The higher ion energy leads to more intense surface heating.At 200 ps, the result of 45 eV ion energy exhibits an overall higher surface temperature than that of 29 eV ion energy.In comparison, just at 80 ps, the case of 62 eV ion energy not only shows high central temperature, but also exhibits more expanded high-temperature region than that of 45 eV at 200 ps.According to [17], the evolution of surface temperature can reflect the expansion of molten pool, the increase of the cathode spot radius, and the intensity of surface deformation.A clear trend is shown that with higher ion energy the formation of the cathode spot is faster.
For the surface atom emission, evaporation depends on the surface temperature, while sputtering is determined by both the surface temperature and the ion energy.The numbers of the emitted atoms every 10 ps and those of the theoretical evaporated atoms every 10 ps of the three sets are compared in figure 9(b).The first two sets are shown within the time duration of 300 ps, while the last set is only shown within 80 ps.Compared to the set of 45 eV, in the set of 62 eV the emitted atom number is significantly higher than the evaporated atom number at a much earlier timestep around 20 ps.In contrast, in the set of 29 eV the two numbers are comparable throughout the entire 300 ps.
One reason of this difference is the different surface temperature as shown in figure 9(a).As aforementioned, only within the high-temperature region can the atom sputtering takes place.In the set of 29 eV, the evolution of surface temperature is slow, which results in only a small area of hightemperature region.Moreover, for lower ion energy, the surface temperature needs to be higher to produce an obvious sputtering effect.Therefore, in the set #1 of 29 eV the sputtering yield is very low, and the slight surface atom emission is dominated by the evaporation.In comparison, in set #3 of 62 eV, the higher ion energy and bigger high-temperature region leads to much more intense atom sputtering.The surface atom emission is already dominated by the atom sputtering at a very early stage.
Considering that not all the emitted atoms return to the cathode surface in the real case, the intense surface atom emission could be the origin of mass loss of the cathode spot.Therefore, another conclusion can be drawn that during the formation of the cathode spot, the ions with higher charged-numbers are the main contributor to the surface atom emission and the mass loss.

Surface atom emission of a practical cathode spot at the central position
In simulation sets #1∼3, r 0 is set as 5 nm to develop the cathode spot within several or tens of nanometres.These models build an equivalent cathode spot in a small size and reflect the formation process of the cathode spot qualitatively.Another simulation set #4 is applied, with r 0 set as 5µm to simulate a practical cathode spot size.In this case, the simulation domain of 22.4 × 22.4 × 11.2 nm only depicts the central position of a practical cathode spot.
The simulation results of set #4 during the first 40 ps are compared with set #3, as shown in figure 10. Figure 10(a) presents the evolution of the surface temperature at 0 ps, 20 ps, and 40 ps in both sets.From the same initial temperature, the evolution of central temperature is similar because the ion bombardment frequencies are similar.However, as set #4 represents only the central part with a radius of 10 nm of a practical cathode spot, the ion bombardment in this simulation domain is uniformly high.Therefore, set #4 exhibits a behaviour that the whole surface is heated almost with the same rate, as the temperature distribution is flat.This phenomenon is consistent with the HD simulation results of cathode spot formation with a size of several micrometres.In comparison, in set #3 the surface temperature drops rapidly towards the outer region.
Figure 10(b) presents the evolution of the total numbers of input ions and emitted atoms.The two sets both show a trend that the number of emitted atoms is initially increased steadily and then escalated drastically.Compared to set #3 where r 0 is 5 nm, the surface atom emission in set #4 where r 0 is 5 µm is much more intense.As the ion energy is the same, this difference can be explained by the much larger high-temperature region in set #4.In addition, the larger high-temperature region also induces a stronger splashing of atom cluster at the early stage, as shown in figure 11.For a better observation of the  The intense surface atom emission in set #4 is helpful to understand the contact erosion and the mass loss in a real cathode spot.In this MD model, the mass input from ions is considered, and it is found that evaporation is not the only pattern of surface atom emission of cathode spot.The effect of sputtering or splashing is also significant, especially in the high-temperature region.From the simulation results, for ion energy of 45 eV the surface temperature is required to be above around 4000 K to generate atom sputtering.For lower ion energy such as 29 eV the temperature required is higher.With the bigger high-temperature region, the contribution of atom sputtering is more obvious to be observed.
From the understanding by the MD simulation, during the input of leftover ions, the central region is undergoing the most frequent ion bombardment and hence heated the fastest.Then, the pressure effect of ion bombardment causes the surface deformation, where the central surface is sunken.In the meantime, under the effect of heat conduction and less frequent ion bombardment, the outer area is heated relatively slowly.Eventually, the high-temperature region expands, and a temperature gradient is established across the surface.This is consistent with the HD simulation results [13], where a similar temperature gradient is obtained, with an area of 3 ∼ 4 µm under the high temperature over 4000 K.However, on the contrary to the conclusions of [12] and [13] as introduced in sections 1 and 3, in the MD simulation the pressure effect of ion bombardment not only leads to the sunken surface in the high-temperature region, but also the intense atom sputtering or splashing in this region.As a result, a dense atom source is generated to form the self-generated plasma and leads to the mass loss in this individual cathode spot.Based on the simulation findings of both HD model and MD model, the possible patterns of mass loss of an individual cathode spot are presented in figure 12.
Over the whole cathode spot including the periphery rim, the surface atom emission occurs in the central region and is negligible in the outer region.From figure 10(b), compared to the leftover ions input, the surface atom emission in the central high-temperature region of a cathode spot is so intense that it acts as the main atom source and forms the self-generated plasma, as well as causes the mass loss.

Comparison with experiments and other simulations
Limited by the technology of experimental observations, the detailed dynamics of an individual cathode spot are hard to be detected by experiments.However, from the experiments, the crater profiles were obtained to assess the cathode spot.In the spot chains observed on the cathode [3,5], the craters have a radius distributed in the range from hundreds of nanometres to tens of micrometres, but with similar profiles.This variety could be explained by the fact that the radius of the cathode crater is determined by the spatial size of the plasma ions working as the input.
In this work, a much more developed MD model compared to [17] is built to consider more effects involved in the cathode spot processes, with the radial distribution of temperature, current density, and energy flux density of an individual cathode spot shown as the results.The peak temperature observed locates at the central position of the cathode spot and is in the range of 4000 − 4500 K, which is similar to the simulation results of other cathode spot simulations [12,13].The crater profile is relatively flat, with the temperature overall high in most of the cathode spot area.This also agrees well with the temperature distribution calculated in [15].As a function of the surface temperature, the electron current density is as expected high to maintain the current continuity of the cathode spot.
As for the mass loss, in HD models, the mass input from leftover plasma ions was massive but was not considered, while the mass output of surface atom emission could not be observed and was assumed as saturated evaporation.As a result, contact erosion during the development of the cathode spot cannot be discussed.On the contrary, sputtering has been observed as the result of the kinetic effect of ion bombardment from other MD simulations [16].In this work, intense sputtering was found in the high-temperature region of the cathode spot.This result answers the mechanism of contact erosion from a cathode spot from a quantitative view.Moreover, the most intense surface atom emission in the central position agrees well with the experimental measurements and simulation results of the density distribution of the spotgenerated plasma [23,24].
This model simulated the generation of a cathode spot based on the MD method, with the species behaviours in the surface layer incorporated by simplifications.The interaction between the MD simulation and a direct surface layer simulation would be required in the future to advance the model and to present a quantitative analysis of the surface atom emission and the mass loss of the cathode spots.

Summary
This paper applies molecular dynamics simulation to investigate the formation of an individual cathode spot and the mechanism of surface atom emission associated.One innovative uniqueness of this work lies in its modelling approach.Firstly, the boundary and bulk effects on the cathode spot formation are all simultaneously considered.The boundary effects include the contributions of external plasma, surface electron emission, surface atom emission, back ions, and evaporation, and the bulk effects include Joule heating and heat conduction.The leftover plasma ions are assigned with certain velocities or energies and are inserted to bombard the surface.Secondly, the thermal effects of heat conduction, evaporation, Joule heating, and Nottingham heat are coupled.Finally, the contribution of back ions derived from the ionisation of emitted atoms is also achieved by particle conversion and ion bombardment.As a result, the surface deformation and surface temperature are derived, based on which the surface electron emission is calculated.This ensures a comprehensive cathode spot model is achievable.The simulation results of surface electron emission demonstrate the process of cathode spot formation.
The evolutions of the radial distribution of surface temperature and the number of surface atom emission are the main simulation results from this MD model.Two patterns of surface atom emission are observed, including evaporation and atom sputtering or splashing.The evaporation exists across the whole surface and is relatively moderate.The atom sputtering appears in the high-temperature region and can be much more intense than theoretical saturated evaporation.The highest surface temperature is in the range of 4500 ∼ 5500 K, while the temperature required for intense atom sputtering depends on the ion energy.Ions with larger charge number possess higher energy and lead to severer atom sputtering, while the ions with z i = 1 and energy of 29 eV are insufficient to produce obvious atom sputtering even under the highest surface temperature.
The sputtered atoms in the central high-temperature region are locally denser than the leftover plasma ions required to generate a new cathode spot.Therefore, this pattern of surface atom emission is proposed to be one of the main atom sources of self-generated plasma.Also, it is perceived as a main cause of mass loss of an individual cathode spot.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Figure 1 .
Figure 1.Cross-section of the MD simulation box: (a) the copper substrate, the thermostat layer, the fixed layer, and the headroom.(b) The leftover plasma ions, the evaporated atoms, and the back ions.The arrows in (b) represent the directions of the velocity.

Figure 2 .
Figure 2. The procedures of calculating the various particle effects and thermal effects within the individual cathode spot.

Figure 3 .
Figure 3.An example of the detected surface atoms of the deformed substrate.(a) The upper view with the distribution of the average temperature of the surface rings.(b) The front view.

Figure 4 .
Figure 4.The method of considering the Joule heating and Nottingham heat by the two meshing operations.Parameters being bolded represent that they are calculated based on the atom-sized chunks, while the rest are calculated based on the x-y chunks.Dotted arrows represent the transition of information between chunks and atoms.

3. 1 .
The evolution of surface temperature, current densities, and energy flux densities during the cathode spot formationThe parameters of interest to describe the cathode spot are the surface temperature, the composition of the current densities, and the composition of the energy flux densities.Since the ion energy in set #2 is moderate among the three sets, the simulation results of set #2 are chosen as the example.Figures5 and 6present the simulation results of set #2 to illustrate the contributions of the various effects on this individual cathode spot.To analyse the radial distribution of parameters in this 3D model, the values of these parameters within each radial ring, as shown in figure3(a), are averaged.In figures 5 and 6 the error bars are shown for the temperature distribution, based on ten times of simulation.The surface emitted electron current density is directly determined by the surface temperature thus the corresponding averaged value is shown.

Figure 5 .
Figure 5.The simulation results of set #2 at (a) 30 ps, and (b) 250 ps, including the surface temperature, leftover plasma ion current density, emitted electron current density, and back ion current density.

Figure 6 .
Figure 6.The simulation results of set #2 at (a) 30 ps, and (b) 250 ps, including the energy flux density of Nottingham heat, Joule heat, and leftover plasma ions.

Figure 7 .
Figure 7. simulated results of set #2 within 300 ps.(a) The number of the total input ions, the total emitted atoms, the number of emitted atoms every 10 ps, and the number of the theoretical evaporated atoms every 10 ps.(b) Radial distribution of surface temperature at 0 ps, 100 ps, 200 ps and 300 ps.

Figure 8 .
Figure 8.The substrate domain of set #2 at 300 ps.

Figure 9 .
Figure 9. comparison among set #1-3.(a) The radial distribution of the surface temperature.(b) The number of emitted atoms and the theoretical number of evaporated atoms.

Figure 10 .
Figure 10.The comparison of set #3 and #4.(a) The radial distribution of the surface temperature.(b) The number of emitted atoms and the theoretical number of evaporated atoms.

Figure 11 .
Figure 11.The substrate domain of set #4 at 20 ps, with most of the single emitted atoms deleted.

Figure 12 .
Figure 12. three possible patterns of mass loss in the practical cathode spot.

Table 1 .
The various sets of input parameters.