1. IntroductionSecondary electron (SE) has been investigated widely and used in many electronic fields since it was reported by Austin and Strake in 1902.[1] With detecting SEs due to the irradiation of focused electron beams, one can obtain surface images on a micro scale.[2] The SE multiplication between plates can conduce to magnifying the number of electrons.[3,4] Nevertheless, the generation of SEs may deteriorate the performance of device,[5] such as electron cloud due to high energy particle collision in accelerators, and multiplication effect of microwave device in the space environment.[6] For using or restraining SE emission better, it is important to calculate the yield of SE from material.[7] Furthermore, considering the fact that material used in device is contaminated more or less,[8] the investigation of secondary electron yield (SEY) in complicated surface states is still required.
The SEs are the electrons emitted from material surface, including true secondary electrons (TSEs) and backscattering secondary electrons (BSEs).[9] Those TSEs are generated by inelastic scattering events between incident electrons and material extra-nuclear electrons, while elastic scattering events between incident electrons and material nuclei may result in BSEs.[10] Yield of SE due to electron irradiation can be calculated with both Monte Carlo (MC) numerical simulations and formulas directly.[11] Since SEY is calculated by tracing each scattering events of electron inside the material in MC simulations, the computation of those simulations is always huge, and the accuracies of their results are limited. On the other hand, the yield of SEs can be quickly obtained with some classical formulas,[12] such as Joy, Vaughan and Furman’s formulas.[13–16] Due to the simplification of direction and energy distributions of internal SEs and the ignorance of the contribution of backscattering SEs, Joy and Vaughan’s formulas have an obvious deviation from experiment data in the case of high energy electron irradiation. And the experimental data fitted Furman formula requires assigning as many as twenty parameters for each material.[17] In addition, SEY of surface contaminated materials is still unclear in physics and needs further investigations.
In this paper, we propose a synthetic semi-empirical physical model of SEY of some ordinary metals, Ag, Al, and Au. This model performs processes of internal SEs excitation, outgoing toward surface, incident angle, and backscattering revises. Penetration coefficient of incident electron is corrected for the materials. Internal SEs are treated by using a more precise model. In addition, according to experimental data, we make a further investigation on one of important surface contaminations, i.e., surface gas adsorption. This research can help understand the influences of relevant parameters and surface states on SE emission, and has significance for application and restraint of SE in relevant fields.
2. Simulation method and resultsEmission processes of SEs from a metal material are illustrated in Fig. 1, including internal SE excitation, outgoing of SEs from the inside, and SE emission across surface potential. In addition, SE emission is still related to incidence angle, backscattering, and surface contamination.
2.1. Excitation of internal secondary electronInternal SE can be generated inside the material due to energetic electron beam irradiation. Primary electron with a larger kinetic energy can reach a deeper position, and the lateral range of the internal SE is also larger. For most amorphous materials, the range of internal SE R(Epe) and the incidence electron energy Epe are related by
where
Lk is the Lane–Zaffarano constant and set to be 76 nm,
[18] ρr is the relative material density,
Ek is the standard energy (1 keV), and
α is the index parameter and presents the capability of incident electron penetration. Lane and Zaffarano thought that
α was a constant which was independent of materials, but the calculated results using the model under this presupposition cannot accord well with experimental results in the case of low-energy-electron incidence. Then, through analyzing NIST data and simulating electron scattering process with Monte Carlo method, we find the following rules.
1) A material with a higher atomic number has a larger deflection angle during the elastic scattering between incident electron and atom, and it will weaken the capability of incident electron penetration consequently.
2) A material with a higher atomic number has more orbital electrons and loses energy with a higher probability during the inelastic scattering between incident electrons and orbital electrons. It may also weaken the capability of incident electron penetration.
According to the quantitative analysis, we propose a relationship between parameter α and material atomic number (mainly for the metals: Al, Au, Ag, Cu, etc.) by
As shown in Fig. 2(a), the parameter α decreases with atom number z increasing, which means that the material with a less atom number allows primary electrons to penetrate deeper.
The internal SEs are excited inside the material due to inelastic scattering process after primary electron irradiation. Although distributions of internal electrons at different positions are different from each other, the internal SE distributions at SE emission related surface positions are almost the same. Hence, we can treat the distribution of internal SE in deep n(z)such that the following equality holds:
Here,
R(
Epe) is the range of incident electron and
N(
Epe) is the total number of internal SEs.
2.2. Outgoing from the insideIn the processes of elastic and inelastic scatterings, part of internal SEs can reach the surface of material and exit after crossing the surface potential barrier, while others deposit inside the material and trapped by defective atoms. Bruining[19] and Wittry[20] simplified the directions of internal SEs into upward and downward. Actually, directions of internal SEs can be arbitrary. Hence, we propose absorption probability pabs(ds) of internal SE at a distance ds in each direction can be expressed as
Here,
γ is the absorption coefficient and obeys the exponential distribution. An internal SE in the direction of
φ at depth
z passes through a distance of
z/cos
φ to reach material surface as shown in the following figure. Then, the probability with which an internal SE in the direction of
φ at depth
z can reach the material surface is
where
λ is the mean free path of electron in material (for the ordinary metal Au, Ag, Cu, and Al, their values of
λ are 1.6, 1.4, 1.2, and 3.8 nm, respectively).
Meanwhile, when an internal SE moves inside the material, it may crash with atom inelastically and generate another internal SE as well, which is cascade scattering. For amorphous material, this collision is isotropic, and directions of internal SEs after this process can be thought to be homogeneous. In general, the density of internal SEs at depth z is n(z), we can obtain the number of internal SEs in the direction φ ∼ φ + dφ at the depth z ∼ z + dz which can reach the following material surface:
Combining the density of internal SEs with SE distribution at a depth, the total number of surface SEs Ns from the inside of material can be obtained from the following equation:
Here,
which is acomplicated function and can be approximated as
Figure
2(b) shows the difference between the approximated functions (
8) and (
9). In Eq. (
8), Γ is the gamma function
. Then, the total number of surface SEs
Ns is expressed as
Besides, we still have to figure out the energy distribution of internal SEs, which is related to the emission of internal SEs at the surface. Lin and Joy[13] thought that the internal SE energy equals the average ionization energy, while Streitwolf[21] set the energy distribution of internal SEs S(Es) as S(Es) ∝ (Es − EF)−2. However, there is an essential problem under this assumption, the integration of all the internal SEs whose energy larger than EF is not convergent. That means that the internal SEs could be innumerable, which conflict with realistic scenario obviously. In addition, this assumption only considers the internal SEs generated by incident electrons, and ignores the cascade scattering process.
Therefore, in this paper, we propose an energy distribution of internal SEs with taking into consideration both the integration convergence and the contribution from the cascade scattering process
Here,
Es is the energy of internal SE,
Ev is the expected value of
Es, and
EF is the Fermi level of material. According to the law of energy conservation, the sum energy of all internal SEs should equal primary electron energy. Considering the fact that the energy of internal SEs is on the basis of Fermi level
EF, the sum energy of all internal SEs can expressed as
where
SR(
Es)d
Es can represent the probability of internal SEs in an energy range of
Es + d
Es. The maximum of internal SEs is primary energy
Epe.
2.3. Emission from the surfaceAfter reaching the surface, the internal SEs still need to cross the surface potential barrier before being emitted from the material surface, and its probability pe(Es) mainly depends on barrier height U and internal SE energy Es and is expressed as
and then based on the number of internal SEs, the SEY can be expressed as
When the primary energy
Epe is much larger than Fermi level
EF,
N(
Epe) can be approached as
N(
Epe) =
Epe/
Ev. Hence, we can obtain SEY
The SEY is a function of primary energy
δ(
Epe). Primary electron with peak energy
Em generates a maximum SEY
δm. Assuming its range
Rm =
R′
λ, the equation above can be expressed with
R′
As
α is given by Eq. (
2), the
R′ values of metals Al, Cu, Ag, and Au can be calculated to be 1.19, 1.39, 1.53, and 1.61, respectively. The peak energy
Em can be calculated from
and the SEY can be expressed as
Here, maximum SEY
δm satisfies
2.4. Modification considering the incident angleAs many researches think that the SEY increases with incidence angle increasing, hence, in this paper, we should treat SEY semi-empirical physical model also with an incidence angle revision. When the incidence angle is θ, the depth range of primary electron can reach Rcosθ, where R is the range in the case of vertical incidence. The distribution of internal SEs with incidence angle θin in should be revised as n(z, θin) = n(z)/cosθ. Then, the number of internal SEs that can reach the surface in a depth z ∼ z + dz is
The upper limitation of integral of the total number of surface electrons
Ns(
θ) should be
Rcos
θ. The peak energy of primary electron with incident angle
θin, and its corresponding maximum SEY
δm(
θin) can be expressed respectively as
under the assumption of
λθ =
λ/cos
θin. The SEY of metal with primary electron irradiation at incident angle
θ is
In the processes we performed above, trajectories of PEs are thought to be straight lines, which is reasonable when the incident angle is less than 60°. Hence, this incident angle revision of SEY is available within an angle range of 0–60°.
2.5. Modification with considering the electron backscatteringApart from internal electrons which can escape from material surface, electrons due to backscattering of primary electrons can also emit from the surface, and constitute the total SEs. Those backscattering SEs include two parts, one is caused by the reflection surface potential barrier, while the elastic scattering between surface material atoms and primary electrons produces the other part. The reflection yield is
Here,
η1(0) is the backscattering coefficient of PEs with low energy. For an absolutely clean surface, it should be 0, while it is set to be 0.6 for a gas-adsorbed surface. The
is set to be 12.5 eV. Combining the elastic backscattering yield in the case of vertical incidence and the contribution from the effect of incident angle, we can obtain the following elastic scattering yield with the incident angle
η2(
Epe,
θ):
Here,
η2(∞) elastic backscattering yield in high energy which is related with materials, and
EBS relative backscattering energy which is related with materials as well,
e1 and
e2 is backscattering revision parameters, which are set to be 0.8 and 0.4 for metals, respectively. Therefore, the total backscattering yield can be expressed as the sum of
η1 and
η2,
ηBS =
η1 +
η2. Then, the total SEY can be revised as
2.6. Experiment comparisonFor verifying the theory we performed above, we compare the theoretical results with experimental data of kinds of metals under electron beam irradiation. Since the ordinary metal surface is always contaminated due to oxidization and adsorption, we clean metal surface with Ar ion irradiation before testing their SE emission characteristics.
Figure 3(a) shows the measurement system of SEY. The vacuum environment of the main cavity can reach as high as 10−9 Pa. The electron gun works in a range of 20 eV–5000 eV, and its current is in a range between 0.05 nA and 2 nA. Samples are fixed on the platform which is rotatable and moveable in any direction. In addition, temperature of sample can be controlled by the heater under the stage. The default temperature of sample is ordinary room temperature 300 K.
Figure 3(b) shows the comparison of SEY curve of Ag between calculated results and experimental results. It is easy to find out that the simulation results calculated with this model accord with experimental results better than those with other models, such as Joy’s and Vaughan’s,[13,22] which can also verify the validity of our model.
Table 1 is SEY related parameters setting of metals (Ag, Al, and Au), which are corresponding with Fig. 4. Figure 4 includes the total SEY δtotal, errors of SEY test under different primary electron energy irradiations. Results of calculations for metals approximately accord with experimental data.
Table 1.
Table 1.
 Table 1.
Parameters setting of SEY related metals. .
| Metals |
U/eV |
α |
R′ |
Em/eV |
δm |
η2(∞) |
EBS/eV |
| Ag |
4.53 |
1.3 |
1.5 |
391 |
2.06 |
0.55 |
29.1 |
| Al |
2.56 |
1.57 |
1.11 |
281 |
5.28 |
0.46 |
46.1 |
| Au |
5.1 |
1.27 |
1.61 |
731 |
2.36 |
0.56 |
23.9 |
| Table 1.
Parameters setting of SEY related metals. . |
In addition, the comparisons shown in Fig. 4 still include SEY and their errors at different incident angles for three metals, Ag, Al, and Au. The agreement between experimental results and simulation results can also indicate the applicability of our model in incident angle revision. On the whole, comparison differences in Ag between experimental results and simulation results metals are not so notable as those in Au and Al.Error comparison results also show that this model produces more consistent results with experimental data when the incident energy is larger.
2.7. Surface contaminationFor metals in the ordinary environment in air, their characteristics of SE emission is also related to surface contamination state. Generally, surface contaminations contain adsorbed gases and oxide layers for most parts. On the one hand, those contaminations may reduce the escape depth of internal SEs, and thus change the peak energy Em. On the other hand, contaminations may also change the surface potential barrier U and the mean energy of internal SEs Ev, then affect the process of SE emission.
Table 2 shows the values of mean free path λ of four metals (Au, Ag, Cu, and Al) exposed to the air environment for a long time. We can find that λ values of those metals in air environment are lower than those with a clean surface state. Then, based on Eq. (17), the peak energy Em can be obtained.
Table 2.
Table 2.
Table 2.
Mean free paths of metals exposed to air environment. .
| Materials |
Atom number Z |
Mean free path λ/nm |
| Au |
79 |
1.4 |
| Ag |
47 |
0.7 |
| Cu |
29 |
0.9 |
| Al |
13 |
3.7 |
| Table 2.
Mean free paths of metals exposed to air environment. . |
Considering the fact that the quantity of surface adsorbed gas is hard to modify directly in experiment, we describe the quantity of surface adsorbed gas with metals heated to some certain temperatures in this research. Taking Ag for example, we try to figure out the relationship between the ultimate temperature and the desorption gas quantity. By heating five silver samples with the same initial states to different ultimate temperature TS, we obtain changes of surface adsorbed gas quantity QS of each silver sample. Those heating and testing processes are finished in vacuum environment with the SE measurement system as we mentioned above.
Figure 5 shows the desorption gas quantity QS as a function of ultimate sample temperature TS. The relationship between QS and TS can be expressed as
After adsorbed gas is released from the surface, the work function of the surface may change, and SEY varies consequently. Since the work function of metal after being heated in vacuum is hard to measure in experiment, we can adapt the work function for the measured SEY, and obtain the maximum SEY
δm and function of work function
U as a function of ultimate sample temperature
TS and desorption gas quantity
QS as shown in Fig.
6.
As shown in Fig. 6(a), we can find that the maximum SEY δm decreases linearly with ultimate sample temperature TS, while the work function U increases exponentially in the range shown in the figure. And a similar relationship of the desorption gas quantity QS is shown in Fig. 6(b), which can be fit with an exponential function
Considering the relationship between ultimate temperature
TS and desorption gas quantities
QS, the maximum SEY
δm can be expressed by the ultimate temperature
TS as
Then, with the revision of surface contamination on SEY, we compare SEY calculation results of unclean silver at three different temperatures
TS = 108 °C, 233 °C, 320 °C with experiment results as shown in Fig.
7.
Since the silver easily adsorbs gas in air environment, gases will escape from the surface after heat processing, and then the work function increases and the SEY reduces as a consequence. As shown in the figures above, calculated results accord well with experimental data after revising the work function and mean free path in the case of surface contamination.
