Investigation of Charging Effect on an Isolated Conductor Based on a Monte Carlo Simulation

Abstract

We report calculations of charging effect on an isolated conductor, gold nanosphere, under electron beam bombardment at primary electron energies of 0.1–10 keV based on an up-to-date Monte Carlo simulation method. The calculations consider electron flow in sample, in which the electron yield is almost equivalent to the case when the electron flow is not considered. The electron yields and charging spatial distribution are obtained. For comparison, the calculation for bulk conductor is also performed, for which the time average of electric potential is found to reproduce the law of electrostatics.

1. Introduction

Specimen charging in electron beam-related techniques, for example, scanning electron microscopy (SEM), significantly influences the imaging quality and analytical capabilities of the instrument [1,2,3]. The induced electric field on specimen surface not only distorts the trajectories of incident electrons but also affects the emitted signal electrons, thereby manifesting the charging effect [4,5]. Semiconductors and insulators are susceptible to the charging effect because the deposited charges cannot flow freely in these materials. Since conductors have high conductivity, analysis specimens are generally grounded so that the metallic specimen maintains electrical neutrality in practical cases. However, if the conductor is not grounded, such as in the case where there is an insulating substrate beneath the conductive film, the deposited charges can accumulate on the metallic surface, resulting in a different charging effect from that of an insulator. The present paper studies theoretically this type of charging effect for conductors by a Monte Carlo simulation method.

The charging effect is first evident from the change in electron yields: the secondary electron yield (SEY), backscattering coefficient (BSC), and total electron yield (TEY), under the electron beam irradiation. The SEY and BSC are defined as the ratio of the number of emitted electrons at the energy lower or higher than 50 eV from a specimen to the number of incident electrons, respectively [6]. The TEY, the sum of SEY and BSC, is an essential quantity to judge whether the nature of charging is positive or negative [7,8]. When the TEY is equal to 1, the number of emitted electrons and the number of incident electrons are precisely balanced, and the specimen does not exhibit the expected charging phenomenon. In the dependence of TEY as a function of primary energy, there can be found two such primary energies where TEY equals unity.

Hypothetically, the Monte Carlo method is a most suitable tool for the investigation of the electron–solid interaction and has been extensively used in the fields of electron spectroscopy and electron microscopy [9,10,11,12]. Over the past few decades, a number of studies have developed various Monte Carlo models based on different electron scattering theories [13,14,15,16,17,18,19,20,21]. In this regard, an up-to-date Monte Carlo simulation model [22] has been developed by combing the Mott’s cross-section [23] for electron elastic scattering with a dielectric functional formalism [24] for the calculation of electron inelastic scattering cross-section. This classical trajectory Monte Carlo simulation methods have proven to be successful in applications to, such as SEM [25,26,27], Auger electron spectroscopy and reflection electron energy loss spectroscopy (REELS) [28]. The corresponding simulation codes have been developed for various applications to electron beam techniques.

With respect to the modeling of charging effects of semiconductors and insulators, several Monte Carlo calculations have already been performed and enabled a significant understanding of the dynamic process in the charging phenomena [29,30,31,32,33,34,35,36]. In contrast, while the charging behavior of floating conductors has been qualitatively discussed in a few theoretical and experimental studies [37,38], there has been, to our knowledge, no prior Monte Carlo simulation study specifically focused on conductors—especially isolated, ungrounded metallic structures. This represents a notable gap in the literature, as the unique properties of conductors—such as their rapid charge redistribution, absence of trap-assisted charge retention, and natural tendency toward equipotential states—make the charging behavior of conductors fundamentally different from that of semiconductors and insulators.

This gap is particularly critical in nanoscale applications, where even a comparably small amount of deposited charge may create quite a strong electric field on the surface of a nanoparticle due to its quite small size. To address this, we present here a Monte Carlo simulation approach to model the charging dynamics of isolated conductors. Specifically, we develop two charging models—the charge drifting model (Model I) and the transient model (Model II)—to describe conductor behavior under electron beam irradiation. We then apply our modified Classical Trajectory Monte Carlo for SEM (CTMC-SEM) code [22] to simulate charge deposition in isolated gold nanospheres across a range of primary electron energies from 0.1 keV to 10 keV. In addition, we calculate the corresponding electron emission yields to better understand the relationship between charging and emission behavior.

2. Monte Carlo Model

An up-to-date Monte Carlo simulation model with the latest electron elastic and inelastic scattering cross-sections are used in the present calculations. Given the details of this procedure are described elsewhere [22], here we only outline the calculation procedure.

2.1. Electron Elastic Scattering

The Mott’s differential cross-section [23] is used to model the elastic scattering of electrons:

$${\displaystyle \frac{d{\sigma }_e}{d\Omega }}={\left|f\left(\theta \right)\right|}^2+{\left|g\left(\theta \right)\right|}^2,$$

(1)

where Ω and θ denote the scattering solid and polar angles, respectively, and $f\left(\theta \right)$ and $g\left(\theta \right)$ are the scattering amplitudes and can be calculated using the partial wave expansion method:

$$f\left(\theta \right)={\displaystyle \frac{1}{2ik}}{\displaystyle \sum }_{l=0}^{\infty }\left[\left(l+1\right)\left(e^{2i{\delta }_l^{+}}-1\right)+l\left(e^{2i{\delta }_l^{-}}-1\right)\right]P_l\left(\cos \theta \right),$$

(2)

$$g\left(\theta \right)={\displaystyle \frac{1}{2ik}}{\displaystyle \sum }_{l=1}^{\infty }\left(-e^{2i{\delta }_l^{+}}+e^{2i{\delta }_l^{-}}\right)P_l^1\left(\cos \theta \right),$$

(3)

where k represents the wavenumber, ${\delta }_l^{+}$ and ${\delta }_l^{-}$ are the spin-up and spin-down phase shifts in the lth partial wave, respectively; $P_l\left(\cos \theta \right)$ and $P_l^1\left(\cos \theta \right)$ are the Legendre and the first order associated Legendre functions, respectively.

The scattering potential is composed of three parts, i.e., the electrostatic potential, the exchange potential, and the correlation–polarization potential. The Fermi distribution and the Dirac–Fock electron density [39] are used to determine the nuclear and electronic charge-density, respectively. In addition, the Furness–McCarthy exchange potential and the correlation-polarization potential based on the local-density-approximation [40] are also considered. In this study, we use the latest ELSEPA program [41] to calculate Mott’s cross-section (1).

2.2. Electron Inelastic Scattering

The dielectric functional formalism is used to determine the inelastic scattering cross-section of electrons. In this model, the differential inverse inelastic mean free path (DIIMFP) for moving electrons in a material reads [42]

$${\displaystyle \frac{d^2{\lambda }_{\mathrm{in}}^{-1}}{d\left(ћ\omega \right)dq}}={\displaystyle \frac{2{\gamma }^2}{1+\gamma }}{\displaystyle \frac{1}{\pi a_{0}E}}\mathrm{Im}\left[{\displaystyle \frac{-1}{\epsilon \left(q,\omega \right)}}\right]{\displaystyle \frac{1}{q}},$$

(4)

where ћ is the reduced Planck constant, ћω represents the energy loss, $\gamma =1+E/\left(m{c}^2\right)$ is the relativistic correction factor with E and m the kinetic energy and mass of electron, respectively, and c the speed of light, $a_0$ is the Bohr radius, and ${\lambda }_{\mathrm{in}}$ is the electron inelastic mean free path (IMFP). $\epsilon \left(q,\omega \right)$ is the momentum q and frequency ω dependent complex dielectric function of the medium. The probability of the inelastic scattering events is determined by the energy loss function (ELF), $\mathrm{Im}\left[-1/\epsilon \left(q,\omega \right)\right]$. David Penn proposed an algorithm, known nowadays as the full Penn algorithm (FPA) [24], for the extension of the optical ELF, $\mathrm{Im}\left[-1/\epsilon \left(0,\omega \right)\right]$, from the optical limit $q \to 0$ into the $\left(q,\omega \right)$-plane. Using the Lindhard dielectric function ${\epsilon }_L\left(q,\omega ;{\omega }_p\right)$, the ELF is expressed as an integral expansion

$$\mathrm{Im}\left[{\displaystyle \frac{-1}{\epsilon \left(q,\omega \right)}}\right]={\displaystyle {\int }_0^{\infty }g\left({\omega }_p\right)\mathrm{Im}\left[\frac{-1}{{\epsilon }_L\left(q,\omega ;{\omega }_p\right)}\right]d{\omega }_p},$$

(5)

where $g\left(\omega \right)$ is the expansion coefficient which is related to the optical ELF by

$$g\left(\omega \right)={\displaystyle \frac{2}{\pi \omega }}\mathrm{Im}\left[{\displaystyle \frac{-1}{\epsilon \left(0,\omega \right)}}\right].$$

(6)

To check the accuracy of the optical ELFs data used for the input, one may apply the oscillator strength sum rule (f-sum rule) and the perfect screening sum rule (ps-sum rule) [43]. The f-sum rule $Z_{\mathrm{eff}}$ is given by

$$Z_{\mathrm{eff}}={\displaystyle \frac{2}{\pi {\Omega }_P^2}}{\displaystyle \underset{0}{\overset{{\omega }_{\max }}{\int }}\omega \mathrm{Im}}\left[{\displaystyle \frac{-1}{\epsilon \left(\omega \right)}}\right]d\omega ,$$

(7)

where $ћ{\Omega }_P=\sqrt{4\pi n_a{e}^2/m}$, with $n_a$ the atomic density. The expectation value of $Z_{\mathrm{eff}}$ must be the atomic number $Z$ of the constitute atoms for an elemental solid, or the total number of electrons per atom or molecule for a compound, when ${\omega }_{\max }\to \infty$. The ps-sum rule $P_{\mathrm{eff}}$ can be obtained from the Kramers–Kronig relation as follows:

$$P_{\mathrm{eff}}={\displaystyle \frac{2}{\pi }}{\displaystyle \underset{0}{\overset{{\omega }_{\max }}{\int }}\frac{1}{\omega }\mathrm{Im}}\hspace{0.33em}\left[{\displaystyle \frac{-1}{\epsilon \left(\omega \right)}}\right]d\omega +\mathrm{Re}\left[{\displaystyle \frac{1}{\epsilon \left(0\right)}}\right],$$

(8)

where $\mathrm{Re}\left[1/\epsilon \left(0\right)\right]=0$ for conductors. The expectation value of $P_{\mathrm{eff}}$ is, thus, unity when ${\omega }_{\max }\to \infty$.

For the floating conductor, Au, investigated in this paper, we use the optical ELF data (Figure 1) for the calculation of IMFPs in Ref. [44]. The f-sum rule of this dataset is 75.99 with the relative error of −3.8%, and the ps-sum rule is 1.089 with the relative error of 8.9%.

3. Monte Carlo Simulation

3.1. Simulation Procedure

When an electron is directed into a sample, it then suffers elastic and inelastic collisions, so the moving direction and the kinetic energy will be frequently changed during the movement. The scattering angle and the energy loss can be sampled by the respective differential cross-section with random numbers in a Monte Carlo simulation [10,16]. In an inelastic scattering event, a secondary electron can be excited whose energy is given according to the energy loss $ћ\omega$ [21]. If $ћ\omega >E_B$, where $E_B$ is the smallest binding energy of the observable ionization edge presented in the optical ELF, the secondary electron is excited from the inner-shell and has kinetic energy of ($ћ\omega -E_B$). Otherwise, a secondary electron is assumed to be excited from the Fermi sea by transferring $ћ\omega$ energy from a primary electron to a valence electron of energy with the excitation probability being proportional to a joint density of states of free electrons. In addition, the relaxation of the excited atoms may proceed via the emission of an Auger electron or a photon. However, the contribution of Auger electrons is negligible to the emitted electron yield due to the low probability of the inner-shell ionization. An excited kinetic secondary electron can undergo the similar inelastic scattering processes as a primary electron so as to generate lower energy secondary electrons. This cascade secondary electron excitation proceeds until all the excited secondary electrons come to rest by lowering their energies to the Fermi level.

After undergoing multiple elastic and inelastic scatterings inside the sample, an electron may reach the surface. This electron may then escape from the sample surface with a certain probability, i.e., the transmission function whose quantum mechanical expression is [45]

$$T\left(E,\beta \right)=\left\{\begin{array}{ll}{\displaystyle \frac{4\sqrt{1-U_0/E{\cos }^2\beta }}{{(1+\sqrt{1-U_0/E{\cos }^2\beta })}^2}}\hfill & \mathrm{if} \hspace{0.33em}E{\cos }^2\beta >U_0;\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(9)

where $\beta$ is the angle between the electron moving direction and the surface normal, and $U_0$ is the surface barrier, which is the sum of work function and Fermi energy for conductor. According to the kinetic energy, an escaped electron is counted either as a true secondary electron (below 50 eV) or a backscattered electron (above 50 eV).

3.2. Drift Model

Charges are deposited in the sample when a primary electron and all its generated secondary electrons stop moving by having exhausted their kinetic energies. The negative deposited charges are, thus, electrons, while the positive deposited charges or the holes are the vacancies where secondary electrons are exited. These deposited charges generate an electric field, which in turn influences the motion of the deposited electrons and holes. This electric field may also modulate the trajectories of the transporting electrons, which have not exhausted their kinetic energies.

The deposited charges in a conductor in the drift model are assumed to move with a finite speed, i.e., their moving speed is proportional to the electric field strength (Model I). In the model, the drift velocity of electrons and holes are described as follows:

$$\left\{\begin{array}{l}{v}_{\mathrm{e}}=-{\mu }_{\mathrm{e}}E,\\ v_{\mathrm{h}}={\mu }_{\mathrm{h}}E,\end{array}\right.$$

(10)

where $E$ is the electric field, $v_{\mathrm{e}}$ and $v_{\mathrm{h}}$ are the velocities, and ${\mu }_{\mathrm{e}}$ and ${\mu }_{\mathrm{h}}$ are the mobilities of the deposited negative charges (electrons) and positive charges (holes), respectively. The electron mobility is set to 48.4 cm² V⁻¹ s⁻¹ for Au, and we assume the mobility of holes is the same as that of electrons because the movement of a hole can be regarded as the opposite movement of an electron.

The charge drift under an establishing electric field is modeled numerically in Model I. Considering the case that the primary beam intensity is 10 nA, the average time interval between two primary electrons arrival at sample is 1.6 × 10⁻¹¹ s. To improve the accuracy of the calculation, the drift process of charges is divided into time steps, and the established electric field is updated at each time step. The total number of steps is set to be twice the sum of the number of electrons and holes, balancing computational efficiency with adequate accuracy in tracking charge dynamics. The motion of each particle during a time step is computed using the forward Euler method, where the step size is fixed such that the particle moves approximately 1 nm per step, depending on its instantaneous velocity. Additionally, the neutralization of electrons and holes is considered at each step: if the distance between an electron and a hole is smaller than a specific threshold, which is set to 1.5 times of the lattice constant here, the electron and hole will be neutralized and eliminated. The lattice constant for gold is taken of 0.4065 nm [46].

The electric potential and electric field are computed based on a discretized spatial grid. In Model I, the grid is made of cubic cells with a side length of 2 nm. Within each cell, the electric potential and electric field are assumed to be constant. The spatial distribution of deposited charges is also discretized for calculating potential. If there are $n_{\mathrm{e}}$ electrons and $n_{\mathrm{h}}$ holes in cell A, then the electric potential, $V_{\mathrm{AB}}$, and the electric field, $E_{\mathrm{AB}}$, they generate at a different cell B are given by

$$V_{\mathrm{AB}}=k_{\mathrm{e}}{\displaystyle \frac{e\left(n_{\mathrm{h}}-n_{\mathrm{e}}\right)}{\left|r_{\mathrm{AB}}\right|}}$$

(11)

and

$$E_{\mathrm{AB}}=k_{\mathrm{e}}{\displaystyle \frac{e\left(n_{\mathrm{h}}-n_{\mathrm{e}}\right)}{{\left|r_{\mathrm{AB}}\right|}^3}}{r}_{\mathrm{AB}},$$

(12)

respectively, where $r_{\mathrm{AB}}$ is the distance vector from the center of cell A to the center of cell B, $e$ is the elementary charge, is the Coulomb constant. If cells A and B coincide (i.e., they represent the same cell), the electric field $E_{\mathrm{AA}}=0$ by assuming a single electron or a hole cannot be propelled by its own electric field, and the potential at cell A is calculated as

$$V_{\mathrm{AA}}={\displaystyle \frac{4}{\sqrt{3}}}{k}_{\mathrm{e}}{\displaystyle \frac{e\left(n_{\mathrm{h}}-n_{\mathrm{e}}\right)}{d}},$$

(13)

where $d$ is the side length of the grid cell, which is set to 2 nm. This formula is derived from dividing the cell into smaller pieces, in which the process is similar to the first subdivision of an octree. By dividing cell A into eight congruent smaller cubes, the potentials of these cubes related to the original cube (cell A) can be obtained from Formula Equation (11). The potential and electric field of cell A are the sum of the potentials and electric fields generated by all single cells at the place of cell A, respectively.

The drift model (Model I) is considered a more physically accurate one because it assumes that the deposited electrons and holes move at finite speeds under the influence of the local electric field. However, this approach requires repeated recalculations of particle positions and the resulting electrostatic fields, resembling a complex many-body problem. As a result, the drift model demands significant computational resources and is relatively slow in terms of simulation performance.

3.3. Transient Model

To address the computational inefficiency of the drift model, we also consider a simplified model in which the deposited charges could be regarded as redistribute instantaneously across the conductor. This assumption is valid when the velocity of the deposited charge is sufficiently high such that the charge can traverse the entire conductor granule within the time interval between successive primary electron incidence. For example, considering a deposited charge with kinetic energy as low as 1 eV above the Fermi level, the time needed to traverse a granule in size of 1 μm is about 1 ps. While in scanning electron microscopic imaging the current of electron beam is typically below 10 nA, the time interval between two successive primary electron incidence is about 10 ps. In such cases, the redistribution time is certainly negligible compared to the time interval of incidence, and the conductor can be treated as reaching electrostatic equilibrium instantaneously.

Therefore, in the transient model (Model II), the deposited charges in the conductor to be assumed to move at an infinite speed. In this model, only the total number of net charges at each step is obtained as the sum of total number of deposited holes minus the total number of deposited electrons. The assumption of infinite speed for charge movement ensures that electrostatic equilibrium is reached instantaneously, and according to Gaussian law the net charges will distribute themselves on the surface of the conductor in such a way as to maintain the conductor’s equipotential nature. For a conductor with an arbitrary shape, the charge distribution under electrostatic equilibrium must first be determined based on the equipotential property, and after the net charge is allocated according to this distribution. However, in this study, we just consider the ideal spherical conductor, for which the net charges under electrostatic equilibrium are uniformly distributed. After a primary electron and all its generated secondary electrons stopped moving, the net deposited charge is uniformly redistributed over the surface of the spherical conductor. This surface charge then alters the electric field to be experienced by the next incoming primary electron.

To account for the influence of this established electric field, the trajectory of each primary electron is computed using the same fixed-distance Euler method as in the drift model. Here too, the step size is set such that the electron moves approximately 1 nm per computational step, based on its instantaneous velocity.

If the conductor is grounded, this model assumes that there are no charges in the conductor due to charges flow with an infinite speed. That is, the common assumption for the SEM image simulation is recovered.

4. Results and Discussion

Monte Carlo simulations were performed for primary electrons incident on samples of a bulk gold and an isolated gold nanosphere. The incident electrons were initiated in vacuum at a far distance from the sample surface, while their trajectories are then influenced by the electric field once the electrons are within 100 nm from the surface.

4.1. Bulk Material

For the bulk conductor case, the semi-infinite nature of the sample makes it equivalent to a grounded conductor since the surface charge density is totally negligible due to the finite amount of incident charges. For this study, we simulated 10⁴ incident electrons at primary energies ranging from 0.1 to 1 keV in order to learn the dynamic behavior of charging establishing process, though negligible, by Model I. The spatial volume in the calculation of charging effect extends over 4 × 10⁶ nm³, i.e., 200 nm in the lateral direction and 100 nm in depth from the surface within the material. When a primary electron and all its generated cascade secondary electrons come to rest by losing their kinetic energies down to Fermi energy, the information of deposited charges is recorded. Then the charge drifting calculation is followed according to the procedure described in the drift model (Model I).

Figure 2 shows that after completing the calculation of the nth injected primary electron and the drifting of the deposited charges, the number of electrons, holes and the number of net charges remained inside the sample. Since the bulk material has no boundaries other than the surface, charges are free to move outside the region of calculation volume and are no longer considered contributing to the fluctuation of the total charges other than zero. As one sees, this behavior holds for all primary energies investigated from 0.1 to 1 keV, except the amount of the negative and positive charges changes somewhat with the primary energy. Therefore, this model illustrates that, in time, the averaged excited electrons and holes are annihilated in a finite spatial region and only very few net charges, in order of number of incident electrons, gets a chance to move far away from incident position to reach the conductor surface.

The establishing electric potentials were then recorded as soon as the tracing of trajectories of primary electrons and the cascade secondary electrons was completed. The time-averaged potential is defined as the average of all recorded potentials over a certain time period. Some examples of time-averaged potentials are shown in Figure 3. Although the shorter period is seen to produce relatively larger averaging potential values, these values still remain at the millivolt order. Additionally, the potential distributions shown in Figure 3 are nearly random, such that the macroscopic long-time averaging finally eliminates the potential fluctuation. This is a vivid demonstration of how the dynamic establishing potential leads to the equal potential of a conductor as by the law in electrostatics. This result also implies that the charging effect in this case is negligible and has minimal effect on the electron yields. As illustrated in Figure 4, neither SEY or TEY nor the BSC to be affected by the charging.

4.2. Drift Model for Isolated Granule

We then simulated the case of isolated gold granule for 10⁴ electrons incidence at primary energies ranging from 0.1 to 1 keV by Model I. The granule was modeled as a nanosphere with a diameter of 100 nm; the primary electron beam was directed toward the center of the sphere. The calculation region encompasses the entire sphere, as the deposited charges cannot move outside the granule.

Figure 5a shows the number of deposited charges at a primary energy of 0.1 keV, illustrating the deposition of negative charges inside the gold granule after a long time at this low primary energy. However, Figure 5c and Figure 5d show that at higher primary energies of 0.5 keV and 1 keV, respectively, the deposited total charges become positive quite quickly after about 10² electrons happened to complete their interactions. With the change in primary energy in the range of 0.1–1 keV, the interaction volume of the incident electrons is found to vary from smaller to larger size compared to the size of the granule. Therefore, at a high primary energy, the most incident electrons may penetrate the granule, and these electrons contribute no charge deposition. The net charging intensity depends on how many excited cascade secondary electrons can leave the granule so that the net positive charges are left and cannot be annihilated.

Figure 6 and Figure 7 show the spatial potential distributions. While the potential at a given moment reflects there are some local negative-charge clusters, the time-averaged potentials suggest that the granule behaves as an equipotential body negatively charged at 0.1 keV (Figure 6b) and positively charged at 0.5 keV (Figure 7). Thus, when a large enough number of electrons bombard an isolated conductor, the averaged effect of the electric field generated by the deposited charges is equivalent to that when the conductor is regarded as an equipotential body. Moreover, the two images in Figure 7 exhibit nearly the identical potential values, indicating that the number of charges on the granule is stabilized.

4.3. Transient Model for Isolated Granule

As discussed in Section 4.2, deposited charges can be considered to flow instantaneously to the surface of a conductor (Model II). Based on this assumption, Monte Carlo simulations were performed for 10⁶ primary electrons at primary energies ranging from 0.1 to 10 keV, bombarding a gold granule. The granule was also modeled as a sphere, but with the two diameter cases considered: 100 nm and 400 nm. Figure 8 illustrates the number of deposited charges in the gold granules over time (or the number of incident electrons), showing completely different results at different primary energies. This variation can be explained by the TEY, as shown in Figure 9.

Figure 9 shows the primary energy dependence of SEY, BSC, and TEY for the gold granule. In the granule case, the primary electrons may penetrate through the granule and become transmitted electrons. However, it is quite complicated to have a definition for so-called backscattered electrons as in the backscattered direction. All escaped electrons are categorized based on their energy as either secondary or backscattered electrons, and all those electrons with energies above 50 eV are counted as backscattered electrons even though some of them are actually transmitted ones. On the curve of Figure 9b, without the charging effect one finds two energies, $E_1$ and $E_2$, such that the TEY is 1. At low primary energies below $E_1$, both the TEYs with and without electric field included are less than 1. Hence, the conductor granule accumulates negative charges in this case and an electric field is generated to repel further incoming primary electrons. As more and more negative charges are accumulated on the granule surface, the landing energy of the primary electrons is observed to decrease progressively. As soon as the accumulated negative charges reach a certain amount, the electric potential to be that of accelerating voltage of the primary electron beam, and the primary electrons may no longer enter the material and to be deflected by the electric field. Consequently, TEY and BSC tend to 1 and SEY tends to 0. This phenomenon is found to occur at primary energies up to about of 0.2 keV for Au.

As the primary energy increases up to $E_2$, the TEY without considering electric field is obtained to exceed 1. In this case, the conductor granule accumulates positive charges, although this accumulation cannot be persistent because low energy electrons escaped from the surface can be attracted back to the conductor to annihilate positive charges. Let us define $f\left(E_s|E_p\right)$, the energy spectrum of escaped electrons of energy, $E_s$, for a primary energy, $E_p$, without considering electric field, such that the integral of $f\left(E_s|E_p\right)$ with respect to $E_s$ from 0 to $E_p$ gives the TEY. Then the number of accumulated charges is stabilized when there exists a potential, $V_0$, such that the integral

$$I\left(eV|E_p\right)={\displaystyle \underset{eV}{\overset{E_p+eV}{\int }}f\left(E_s|E_p+eV\right)\mathrm{d}{E}_s},$$

(14)

where e denotes the charge of the electron and $V$ is the potential of the conductor, is stable at 1. That is, the potential, $V_0$, satisfies $I\left(e{V}_0|E_p\right)=1$, and there exists a neighborhood of $V$ at $V=V_0$ such that

$$\left(V-V_0\right)\cdot \left[I\left(eV|E_p\right)-1\right]\le 0.$$

(15)

As a result, the SEY converges to 1 due to the number of both the injected and escaped electrons being nearly equal. The potential $V_0$ on the conductor will attract secondary electrons with energies below $e{V}_0$, significantly reducing the SEY.

At high primary energies above $E_2$, the condition (15) may not be satisfied. This is because primary electrons penetrating the granule, along with excited electrons escaping the granule’s electric field, can cause TEY to be somewhat above 1, as shown in the 10 keV case in Figure 9a. If the primary electrons do not penetrate the granule, the TEY can drop below 1 again. In this situation, the granule accumulates negative charges until the landing energy of the injected electrons is reduced to $E_2$, as shown in Figure 9b. The SEY is nearly zero because emitted electrons gain energy from the charged granule often exceeding 50 eV, which prevents their reclassification as secondary electrons.

The energies $E_1$ and $E_2$ have been extensively studied in semi-conductors and insulators, where charging effects are more pronounced than in conductors due to the lack of free charge carriers [1,2,3,4]. In conductive materials, $E_1$ and $E_2$ were found to be less pronounced than that of insulators, but their importance was observed to be still worth noting when some granules fail to ground. Within the simulations performed for this study, $E_1$ represents an unstable point, as one finds at 0.2 keV primary energy in Figure 8a. At this energy, the number of deposited charges on the granule fluctuates somewhat higher than vanishing, but the granule starts accumulating quite a large amount of negative charges above 10⁵ primary electrons due to instability. $E_2$ seems to be a stable point Indeed, as soon as the primary energy exceeds, but not significantly, $E_2$, the energy of injected electrons converges to $E_2$, although the speed of convergence is relatively slow. This behavior is well seen at 5 keV primary energy in Figure 8b.

5. Conclusions

In this paper, we have investigated the charging effects of a conductor by using a Monte Carlo simulation method, focusing on the behavior of deposited charges, the influence of electric fields, and their impact on electron yields. Simulations were performed for both bulk materials and isolated gold granules, incorporating the electric field’s effects on all charged particles. The simulations suggest that deposited charges redistribute almost instantaneously, allowing the conductor to be regarded as achieving electric equilibrium at each step of the simulation.

The charging effects are found to be more significant in isolated gold granules, where system stability depends on the total electron yield. At low primary energies, the granules are found to accumulate negative charge, creating electric fields to be strong enough to repel incoming primary electrons from entering the granules. At high primary energies, all primary electrons may enter the granules; however, the secondary electron yield decreases significantly. This reduction is attributed here to the granule’s potential, or cutoff energy, which to recapture quite a number of secondary electrons to be otherwise detected.