Electron scattering in PMMA, including the elastic and inelastic scattering, are simulated with the Monte Carlo method. In this simulation, the PE energy is set to be less than 10 keV. The Mott cross-section^[29] is adopted to describe the elastic scattering as

where σ is the Mott scattering cross-section, Ω and θ are solid angle and scattering angle, f (θ ) and g(θ ) denote the incident and the scattering wavelength division functions, respectively. Considering the complexity of calculating the Mott differential scattering cross section, we use the Mott scattering cross-section database^[30] for different sample atoms, PE energy, and scattering angles. The total elastic scattering cross section can be obtained by

The inelastic scattering cross section σ _in is calculated with the fast secondary electron model^[31] when the PE energy E_PE is larger than 3  keV

in which E is the incident electron energy, e denotes the absolute value of the electron charge, and Ω _c is the minimum of normalized energy loss coefficient and set to be 0.001. The energy loss of inelastic scattering events is calculated with the modified Bethe energy loss formula by Joy and Luo^[31]

where the coefficient k is interpolated with the average atmospheric atom number Z̄ , Ā and J̄ are the average atomic weight and the average ionization energy, respectively, and ρ is the density of material.

When E_PE is less than 3  keV, the inelastic scattering of electrons is more suitable to be calculated with the Penn model.^[32] The corresponding inelastic differential cross section is represented by

where λ _in is the electron mean free path of inelastic scattering, Θ is the staircase function considering conservation of energy and momentum, Δ E is the electron transfer energy, E_p is plasma energy satisfying E_p = ℏ ω _p, with ω _p being plasma frequency, a₀ is the Bohr radius, is the electron free wave vector, q̄ is the root of the following dispersion equation:

where v_F is the Fermi velocity. When the energy loss during each inelastic scattering event is larger than the band gap of material, an SE will be generated, leaving a hole in the sample. The SE will travel in the sample like a PE until it exhausts its energy or escapes from the sample surface.

Due to the elastic and inelastic scattering, some electrons deposit in the sample, while others can escape as the SEs.^[33] To verify the electron scattering model for PMMA, the SE yield δ defined as the ratio of the number of SEs to the number of PEs, δ = N_SE/N_PE, is calculated and compared with experimental results. As shown in Fig.  2, the simulated δ depicted by a solid line for a PMMA sample with the flat surface is presented as a function of primary energy E_PE, compared with the experiment result from Boudaya and Blaise.^[27] Under the same neutrality surface potential, the agreement between simulation and experiment is obtained.

Fig.  2.

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	Fig.  2. Secondary electron yield δ as a function of primary energy EPE. Experiment results are plotted for comparison.

The charge transport, including both the charge drift and diffusion caused by the build-in electric field and charge density gradient, are both taken into consideration in the charge and discharge processes. During the transport, some of the deposited charges may be trapped by trapping centers which widely exist in polymer materials.^[34] The transport of holes are neglected for their low volume and velocity on charging.^[19]

During the charge process, the variation of free electron density n_F (t) in the sample with time t satisfies

where terms with subscripts scat, drift, diff, and trap represent internal electrons distributions, drift, diffusion, and trap, respectively. The term (∂ n_F/∂ t)_scat in the charge process is calculated with the Monte Carlo method mentioned above according to different energies of irradiation electron. The landing energy of PEs is modified by the sample surface potential E_land = E_PE− | V_S| . Other terms in Eq.  (7) satisfy

where μ _e is the electron mobility of the sample material, E_f is the build-in electric field intensity, D_e is the electron diffusion coefficient which is a function of μ _e in the Nernst-Einstein relation as D_e = μ _ek_BT/e (k_B is Boltzmann constant and temperature T = 300  K), n_t is the density of trapped electrons in trapping center, τ _F is the trap time constant of 1  s, and N_T = 1.0× 10¹⁵  cm^{− 3} is the density of trapping center for PMMA, respectively.^[18] In addition, the space potential V(z, t) inside the sample is calculated with the Poisson equation due to the sample net charge distribution as

where the net charge distribution n_NET (z, t) is obtained by considering hole distribution h(z, t), free electron, and trapped electron distributions as n_NET (z, t) = h(z, t) − n_F (z, t) − n_trap (z, t). The ɛ is the dielectric constant. The charge quantity per unit area Q(t_i), accumulated in the sample at 0 < t < t_i where t_i is any moment in the charge or discharge process, and can be calculated by

In the discharge process, since the inflow PEs stop, the term (∂ n_F/∂ t)_scat is set to be zero. Hence, according to Eq.  (7), the variation of free electron density n_F (t) with time t during the discharge process satisfies

The surface potential V(z, t) and the charge quantity Q(t_i) during the discharge process can also be calculated with Eqs.  (11) and (12), respectively. During the whole charge and discharge process, the leakage current J_L(t) satisfy the continuity equations J_L(t) = − eμ _en_F (H, t)∇ V(H, t) + eD_e ∇ n_F (H, t), and the displacement current J_D(t) can be calculated with J_D(t) = ɛ ∂ (∇ V(H, t))/∂ t.

The whole simulation procedure including both charge and discharge processes is shown in Fig.  3. With the interruption of PE irradiation, the discharging process begins from the former charging states. The discharge process finishes with the disappearance of leakage current. Combining the scattering and the transport models, we simulate the evolution of internal currents J_L(t), J_D(t), surface potential V_S(t), and total charge quantity Q(t) of the sample during both the charge and discharge processes. Simulations were carried out with default values of sample parameters as follows: the electron mobility of PMMA μ _e = 1.0 × 10^{− 10}  cm²· V^{− 1}· s^{− 1}, the sample thickness H = 10  μ m, the value of PE energy and current are set to be E_PE = 10  keV and J_PE = 50  nA· cm^{− 2}, respectively.

Fig.  3.

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	Fig.  3. Solution procedure of charge and discharge processes.

The transient properties of charge and discharge processes of PMMA are presented in Fig.  4. As shown in Fig.  4(a), due to the accumulation of the net inflow charges and the transport of internal charges, the leakage current J_L emerges at 22  s. J_L gradually reaches its maximum and then keeps dynamic balance at 40.5  s. Based on the continuity of electric current, the displacement current J_D appears immediately at the beginning of irradiation. With the emergence and increase of leakage current J_L, J_D gradually reduces to zero when the internal charges accumulated become saturation within the charge duration I as illustrated in Fig.  4(a). In the discharge duration II, when PE irradiation stops, the leakage current J_L gradually decreases to zero with the elapse of internal free charges. The displacement current J_D drops to its negative maximum immediately at the very beginning of discharge duration and then gradually approaches zero, for the continuity of electric current as J_L + J_D = 0. The discharge time T_C is defined as the duration from the beginning of discharge t₀ to the moment that leakage current J_L reaches 1% of its maximum t₁ in the discharge process, T_C = t₁ − t₀. Here, T_C is ∼ 73.1  s according to this calculation.

Fig.  4.

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	Fig.  4. Transient characteristics of charge and discharge processes of the PMMA sample. (a) Variations of leakage current JL and displacement current JD with time t. TC is the discharge time. (b) Variations of internal charge quantity Qt and surface potential VS with time t.

Due to variations of inflow and outflow currents, the charge quantity and surface potential of the sample will also change. According to Eq.  (12), the charge quantity Q_t can be integrated by displacement current J_D. As shown in Fig.  4(b), the quantity of Q_t increases with time t of irradiation until reaches its charging saturation Q_charge at the moment when J_L reaches saturation in the charge process. In the discharge process, Q_t decreases gradually to the final value Q_F. Since the de-trap process of trapped charges is negligible for its low volume and speed, the final internal charges Q_F in the discharge process is mainly composed of trap electrons. Accordingly, the discharge quantity in the form of leakage current J_L can be calculated as Q_disc = Q_charge − Q_F. The ratio of discharge quantity r_dis defined as r_dis = Q_disc/Q_charge, is 0.63 according to the results in Fig.  4(b). Moreover, in the charge process, with the increase of internal charges Q_t, the value of surface potential V_S decreases and keeps a minimum V_SC when Q_t reaches saturation. When the electron irradiation stops, V_S increases to its final value V_SF which is determined by Q_F.

Figure  5 shows the distribution of sample internal charges in the z direction at different moments during the charge and discharge processes. Due to the drift, diffusion and trap, internal charges transport towards the sample bottom. The position of the maximum charge density related with electron scattering changes slightly in the charge process, as represented by dashed lines, in 11  s, 22  s, and 34  s. However, the maximum charges density decreases significantly and moves towards the sample bottom in the discharge process. With leakage of free charges in the discharge process, from the moment of 45  s to 88  s, the internal charge density decreases and finally approach an uniform distribution with the value of trapping center N_T = 1.0× 10¹⁵  cm^{− 3}.

Fig.  5.

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	Fig.  5. Internal charge distributions in the z direction at different times.

We also simulate the charge and discharge processes for the PMMA sample of different thicknesses. The sample thickness H is 4– 20  μ m, widely used as a cover film. Figure  6(a) presents the negative charge quantity Q and surface potential V_S, as functions of H. The charge quantity in saturation Q_charge and discharge quantity Q_disc increases with the increase of H. Q_charge is composed of trapped electrons and free electrons in the sample. Q_disc is almost from free electrons. As reported in our previous work, ^[25] the quantity of trapped electrons increase almost linearly with the increase of H, while the increase of free electrons with H is more slowly. Therefore, the increase of the sample thickness H results in more increase in Q_charge than in Q_disc as shown in Fig.  6(a). The discharge quantity ratio r_dis then decreases with sample thickness H, as shown in Fig.  6(b).

Fig.  6.

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	Fig.  6. (a) Surface potential VS and charge quantity Q. (b) Discharge ratio rdis and (c) discharge time constant TC, as functions of the sample thickness H during charge and discharge processes.

Moreover, as illustrated in Fig.  6(a), the surface potential in saturation V_SC and surface potential in final state of discharge V_SF decrease with the increase in H. The surface potential of the sample is stronger for thicker samples. Meanwhile, the difference between the surface potential in charging saturation V_SC and in final state of discharge V_SF, which is determined by discharge quantity Q_disc, increases with the increase in H.

As shown in Fig.  6(c), in thicker sample, free electrons need longer T_C to leak to the grounded basement due to longer paths during the discharge process. The simulation results also show that the time constant T_C increase with the increase of sample thickness H logarithmically as T_C ∼ log (H).

The charge and discharge properties are also affected by electron mobility μ _e of the sample. For PMMA, μ _e can vary from 10^{− 9} to 10^{− 11}  cm²· V^{− 1}· s^{− 1}.^{[18, 19]} Figure  7(a) shows that the charge quantity in saturation Q_charge and the discharge quantity Q_disc decrease with the increase of μ _e almost in the same amount. Larger electron mobility μ _e makes free electrons leakage faster in the sample, which results in a less free charge quantity in charging saturation. Hence, both Q_charge and Q_disc decrease with μ _e, and the surface potential in saturation V_SC determined by Q_charge increase with μ _e. Since the difference between Q_charge and Q_disc is mainly composed of trapped electrons, which is independent of μ _e, the difference almost remains constant. Furthermore, the surface potential in the final state of discharge V_SF determined by trapped electrons keeps invariable with μ _e.

Fig.  7.

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	Fig.  7. (a) Surface potential VS and total charge quantity Q; (b) discharge ratio rdis and discharge time constant TC, as functions of the sample electron mobility μ e during charge and discharge processes.

According to the variations of Q_charge and Q_disc shown in Fig.  7(a), the discharge quantity ratio r_dis decreases with the increase of μ _e, as shown in Fig.  7(b). Since larger electron mobility promotes the transport of free electrons, less time is needed for discharge process to reach its final states. Then, T_C decreases with the increase of μ _e, as illustrated in Fig.  7(b).

It should be noted that the discharge process in this simulation begins immediately after the charging reaches saturation. If the irradiation duration is extended, the charging saturation of sample will remain, and the discharging process after irradiation will be the same. If the interval of irradiation is shorter than the discharge time, the charge quantity and the surface potential of the sample will be stronger. In some cases, in order to weaken the charging effect, e-beam irradiation with an optimum interval estimated by this method is helpful. Moreover, some other parameters, such as trapping center density, temperature and permittivity, will also affect polymer charging.^{[35– 37]} Further efforts could adopt this simulation method by taking more factors into account.